Lectures on Torus Embeddings and Applications (Based on Joint Work with Katsuya Miyake)

Lectures on Torus Embeddings and Applications (Based on joint work with Katsuya Miyake)

By Tadao Oda

Tata Institute of Fundamental Research Bombay 1978 Lectures on Torus Embeddings and Applications (Based on joint work with Katsuya Miyake)

By Tadao Oda

Published for the Tata Institute of Fundamental Research, Bombay Springer–Verlag Berlin Heidelberg New York 1978 c Tata Institute of Fundamental Research, 1978

ISBN 3–540–08852–0 Springer–Verlag Berlin. Heidelberg. New York ISBN 0–387–08852–0 Springer–Verlag New York. Heidelberg. Berlin

No part of this book may be reproduced in any form by print, microﬁlm or any other means without written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005

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Introduction

In recent years, the theory of torus embeddings has been finding many applications. The point of the theory lies in its ability of translating meaningful algebra-geometric and analytic phenomena into very simple statements about the combinatorics of cones in affine space over the reals. In different terminology, it was first introduced by Demazure [8] and then by Mumford et al. [63], Satake [57] and Miyake-Oda [40]. There is already a good and concise account on it in [63]. Nevertheless, we produce here another. For one thing, we wanted to supply the details of the partial classification of complete non-singular 3-dimensional torus embeddings announced in [40]. Besides, we wanted to make the theory, in its most general form, ac- cessible to non-algebraic geometers in view of its possible applications in other branches of mathematics. We can state at least the main results without using algebraic geometry, although for the proof we cannot avoid using it. This was made possible by the following down-to-earth description due to Ramanan of normal affine torus embeddings over a field k of finite type as the set of unitary semigroup homomorphisms

U(σ) = Hom .(σ ˇ M, k), unit.semigr ∩ r r where (i)N Z and σ is a convex rational polyhedral cone in NR R with σ ( σ) = 0 , (ii) for the dual Z-module M of N, ∩ − { } σˇ M = m M : m, y 0 for all y σ ∩ { ∈ h i≥ ∈ } is a ﬁnitely generated additive semigroup and (iii) k is considered to be a semigroup under the multiplication.

v vi 0. Introduction

When we have a suitable collection of such cones, an r.p.p. de- △ composition, then U(σ)′ s can be canonically patched together to form a normal and separated algebraic variety over k locally of ﬁnite type

T emb( ) N △ which has an eﬀective action of the algebraic torus

T = N Z k∗ = Hom (M, k∗) = k∗ ... k∗ N ⊗ gr × × with a dense orbit. Such a variety is called a torus embedding, since it is a partial compactification of TN. Conversely, we get all normal torus embeddings in this way (Theorem 4.1). Important Algebre-geometric phenomena can most often be described purely in terms of (Theorems △ 4.2, 4.3, 4.4 and Corollary 4.5). We may say that contains all the relevant information, in a unified △ and globalized way, about the “exponents of monomials” necessary to describe such varieties. When an algebraic variety or a morphism can be described solely in terms of monomials, then there is a good chance that it can most effectively be described in terms of torus embeddings. For instance, a normal irreducible affine algebraic variety V A defined ⊂ r by equation of the form

a1 a2 ar b1 b2 br X1 X2 ... Xr = X1 X2 ... Xr can be expressed as U(σ) for some σ. (cf. (7.9)) We try to avoid overlaps with Demazure [8] and Mumford et al. [63] as much as possible. For later convenience for reference, we collect together in . 4 the first main theorem in their most general form and leave § their proof to . 5. Various standard examples are collected together in § . 7. § In . 6, we deal with torus embedding which can be embedded into § projectives spaces. The results are slightly more general than those in [8] but less so than those in [63]. . 8 and . 9 are devoted to the classification of complete non- § § singular torus embeddings. We are reduced to the classification of certain weighted circular graphs and that of weighted triangulations of the vii

2-sphere. As by-products, we obtain many interesting complete non- singular rational three folds. (cf. Prop. 9.4) Besides, we see that torus embeddings provides us with a good testing ground for important con- jectures on birational geometry in higher dimension. There are many basic results we left out : For the cohomology of equi variant coherent sheaves on torus embeddings as well as the description of the automorphism groups of torus embeddings, we refer the reader to Demazure [8]. Mumford et al. [63] generalizes the notion of torus embedding to that of toroidal embeddings and proves very important semi-stable reduction theorem. Torus embeddings have also been used effectively in the compactification problem of the moduli spaces by Satake [57], Hirzebruch [19], Mumford et al.[61], Namikawa [46], Nakamura [44], Rapoport [54], Oda-Seshadri [49] and Ishida [26]. Here we deal with more elementary but illustrative applications in Chapter 2. When the ground field k is the field C of complex numbers or one with a non-archimedean rank one valuation, then U(σ) = Homunit.semigr. (σ ˇ M, k), hence T emb( ), has the topology induced from that of k ∩ N △ by the valuation. Let CTN be the maximal compact torus of TN in this topology. Then we can usually draw the picture of the quotient of a torus embedding T emb( ) by CT and get better geometric insight. N △ N The quotient was introduced by Mumford et al. [61]. We call it the manifold with corners after Borel-Serre [4] and denote it by Mc(N, ) = T emb( )/CT . △ N △ N Using this we will be able to visualize the construction and degenerations of complex tori, Hopf surfaces and other class VII0 surfaces introduced by Inoue. (cf. . 11, . 13, . 14 and . 15). Using Suwa’s § § § § classification of hyper elliptic surfaces om [60], Tsuchihashi [62] were able to describe their degenerations and the compactification of the moduli space in terms of torus embeddings. There are many recent results on the actions of algebraic and analytic groups other than algebraic tori on algebraic varieties and complex manifolds. See, for instance, Akao [1], Popov [52], [53], Orlik- Wagreich [50] and Ishida [25]. viii 0. Introduction

Complete non-singular 2-dimensional torus embeddings over C give 2 2 rise to rational compactifications of C and (C∗) . Compactifications of C2 were shown to be always rational by Kodaira [31] and were classified by Morrow [37]. There are, however, many non-rational, even non- 2 algebraic, compactifications of (C∗) . They were recently classified by simha [58] and Ueda [64]. These notes are based on a joint work with K. Miyake of Nagoya University and grew out of the lectures which the author gave at Tata In- stitute of Fundamental Research, University of Paris-Sud, Orsay, Nagoya University, Tohoku University, Instituto Jorge Juan, Harvard University and various other places. He would like to thank the mathe- maticians at these institutions for the hospitality and the patience shown to him. The notes taken by K.Makio at Tohoku University was very helpful.1 Recall that for a connected locally noetherian scheme X, its dualizing complex RX is determined uniquely up to quasi-isomorphism, dimension shift and the tensor product of invertible 0X-modules. (cf. Hartshorne, Residue and duality, Lecture Notes in Math. 20, Springer- Verlag, 1966). Let T = TN be an algebraic torus and consider the normal torus embedding Temb( ) corresponding to an r.p.p.decomposition (N, ). Let △ △ us fix an orientation for each cone σ and define, in case dim τ ∈ △ − dim σ = 1, the incidence number [σ : τ] = 0, 1 or 1 in an appropriate − manner so that we have a complex

d C( , Z) = (... 0 C0 C1 ... Crank N 0 ...) △ → → → → → → → where C j is the free Z-module generated by the set of j-dimensional cones in and where △ d(σ) = [σ : τ](τ). dim τXdim σ=1 − 1After these notes were written, M.Ishida of Tohoku University obtained the following seemingly deﬁnitive result on the Cohen-Macaulay and Gorenstein properties in relation to torus embeddings. ix

Let X be a T-invariant closed connected reduced subscheme of Temb( ) and consider the complex R of 0 -modules defined by △ X X R j = 0 X ⊕ Y where the direct sum is taken over all the T-invariant closed irreducible subvarieties of X which is of codimension j in Temb( ) and where δ : j j+1 △ RX RX is defined as follows : X = orb(σ) for a star closed → σ S∈ subset = σ ; orb(σ) X . When Y =Porb(σ), then for fσ 0 , { ∈△ ⊂ } ∈ Y P δ( fσ) = [σ : τ](the restriction of fσ to orb(τ)) X with the sum taken over τ with dim τ dim σ = 1. ∈△ − Theorem 0.1 ((Ishida)). If X is a T-invariant closed connected reduced subscheme of a normal torus embedding Temb( ), then R defined above △ X is the dualizing complex for X. For ρ E, let = σ ; σ < ρ and consider the complex of ∈ ρ { ∈ } k-modules C( ρ, k)P with the coboundaryP map induced by d. Corollary 1 ((Ishida))P . Let X be as above. Then X is Cohen-Macaulay if and only if there exists ℓ such that for any ρ , we have ∈ H j( , k) = 0 for j , Pℓ Xρ We immediately see that Tamb(δ) itself is Chohen-Maculay (cf. Re- mark after our Prop. 6.6 and Hochster [20]). On the other hand, if Temb( ) = A is the affine space, we get the results in Reisner (Cohen- △ r Macaulay quotients of polynomial rings, Advances in Math. 21(1976), 30-43) and Hochster (Cohen-Macaulay rings, combinatorics and simplicial complexes, in Ring Theory II, Lecture Notes in Pure and App. Math. 26(1977). Dekker, pp. 171-223). In a special case, the complex RX. already appears in [44]. Tadao Oda Mathematical Institute Tohoku University Sendai, 980 Japan

Contents

Introduction v

1 Torus embeddings 1 1 Algebraictori ...... 1 2 Torus embeddings and Summaries theorem ...... 3 3 Rational partial polyhedral decompositions ...... 4 4 Firstmaintheorems...... 8 5 The proof of theorems in .4 ...... 10 § 6 Projective torus embeddings ...... 19 7 Example of torus embeddings and morphisms ...... 27 8 Torus embeddings of dimension 2 ...... 38 ≤ 9 Complete non-singular torus embeddings in dimension 3 45

2 Applications 91 10 Manifolds with corners associated to torus embeddings . 91 11 Complextori ...... 96 12 Compact complex surfaces of class VII ...... 111 13 Hopf surfaces and their degeneration ...... 113

14 Inoue’s examples of class VII0 surfaces...... 119

15 Hilbert modular surfaces and class VII0 surfaces . . . . 127

Chapter 1

Torus embeddings

For simplicity, we work over an algebraically closed field k of arbi- 1 trary characteristic. All k-algebras are commutative with unity and all k-algebra homomorphisms preserve unity. Although rather unconven- tional, we mean by an algebraic variety a reduced, irreducible and separated scheme over k which is only locally of finite type, i.e. possibly an infinite union of open subsets which are usual affine varieties of finite type.

1 Algebraic tori

In this section, we recall basic facts about algebraic tori which we use later. We denote by k∗ the multiplicative group of non-zero elements of k considered as an algebraic group over k. It is usually denoted by Gm 1 and is the aﬃne algebraic group Spec(k[t, t− ]) endowed with the co- multiplication t t t on the coordinate ring. It is more convenient 7−→ ⊗ to consider it as a group functor which assigns to a k-algebra B its multiplicative group B∗ of units. An algebraic torus over k is an algebraic group T isomorphic to a ﬁnite direct product k k . ∗ ×···× ∗ Mutually dual free Z-modules M and N with the pairing , : M h i × 1 2 1. Torus embeddings

N Z give rise to an algebraic torus →

T = Hom (M, k∗) = N Z k∗. gr ⊗ 2 Conversely, an algebraic torus T gives rise to the character group

M = Homalg.gr(T, k∗)

and the group of 1-parameter subgroups

N = Homalg.gr(k∗, T),

both written additively, together with the duality pairing

, : M N Hom (k∗, k∗) = Z. h i × → alg.gr For m in M, we denote by e(m) : T k the corresponding charac- → ∗ ter. The coordinate ring p(0T ) is the group algebra of M over k with e(m); m M forming a k-basis and with e(m + m ) = e(m)e(m ). For { ∈ } ′ ′ n in N, the corresponding 1-parameter subgroup sends t in k∗ to the ?,n m,n element th i of T = Homgr(M, k∗) which maps m in M to th i in k∗. A homomorphism f : T T of algebraic tori correspond in one- → ′ to-one fashion with a homomorphism f : N N′ and a homomor- ∗ → phism f : M M, where N and M are, respectively, the group of ∗ ′ → ′ ′ 1-parameter subgroups and character group of T ′. The following fact is quite relevant to us later: f is surjective if and only if f has ﬁnite ∗ cokernel (resp. f ∗ is injective). The main reason why things work out so well later is that algebraic tori are the only algebraic groups which, besides being connected and commutative, satisfy the following basic complete reducibility property:

3 Theorem. Every algebraic representation of an algebraic torus is completely reducible, and is in fact a direct sum of one dimensional representations, i.e. characters.

For the proof of this theorem as well as other basic facts about algebraic groups, we refer the reader to Borel [3]. 2. Torus embeddings and Summaries theorem 3

2 Torus embeddings and Summaries theorem

An algebraic action of an algebraic torus T on an algebraic variety X is a morphism T X X satisfying the usual axioms (tt )x = t(t x) × → ′ ′ and ex = x for t, t T, x X and e = (the identity of T). When ′ ∈ ∈ algebraic tori T and T ′ act on algebraic varieties X and X′, respectively, an equivariant morphism consists of a homomorphism f : T T and → ′ a morphism f¯ : X X such that f¯(tx) = f (t) f¯(x) for t T and x X. → ′ ∈ ∈ Here is one of the basic results about torus actions: Theorem (Sumihiro). If an algebraic torus T acts algebraically on a normal algebraic variety over k locally of finite type, then X is covered by T-stable affine open subsets of finite type. We do not repeat the proof here and refer the reader to [59] or [63, I.2, Thm.5]. Note that since Pic(G) is finite by [55, Cor. VII 1.6], the argument in [63] can be modified to cover the general case in [59]. Ishida pointed out that if an algebraic group G acts on X locally of finite type, then X is covered by G-stable open sets of finite type to which [59] ap- 4 plies. Remark. As was pointed out by H.Matsumura, the assertion of the theorem is not true unless X is normal. Indeed, T = k∗ acts on the rational curve X with a node obtained by identifying the zero and the point at infinity of the projective line P1. The node does not have any T-stable affine open neighborhood. When k = C, C∗ acts analytically on an elliptic curve E, which is the quotient of C∗ by an infinite cyclic subgroup. Obviously, there is no C -stable open neighborhood. (cf. . 11). ∗ § When an algebraic torus T acts on X, not much is lost, even if we assume the action to be (even scheme-theoretically) effective. Indeed, since T is commutative, we can replace T by its quotient torus by the kernel of the canonical homomorphism from T to the automorphism group functor Aut(X). In these notes, we are interested in almost homogeneous (or sometimes called pre-homogeneous) algebraic torus actions on irreducible algebraic varieties, i.e. those which have a dense orbit. The dense orbit is necessarily Zariski open, and is isomorphic to T when the action is effective. Thus we are led to the following: 4 1. Torus embeddings

Deﬁnition. A torus embedding T X consists of an algebraic variety X ⊂ containing T as a Zariski open dense subset and an algebraic action of T on X which extends the group law of T, i.e. we have a commutative diagram T X / X ×

S S T T / T. × 5 An equivariant dominant morphism from a torus embedding T X ⊂ to another T X is a dominant morphism (i.e. with dense image) ′ ⊂ ′ f : X X whose restriction to T induces a surjective homomorphism → ′ f T : T T and which is equivariant with respect to the actions, i.e. | → ′ the following diagram is commutative:

T X / X × ( f T) f f | × / T X / X ′ × ′ ′ Thus we have the category of torus embeddings. Although dominant morphisms look too restrictive, those are the only equivariant morphisms which can be described in terms of r.p.p. decompositions. We will give typical examples of torus embeddings in . 7. § 3 Rational partial polyhedral decompositions

We denote by Z, Q and R the set of integers, rational numbers and real numbers, respectively. Z0, Q0 and R0 are the sets of non-negative elements in them, respectively. r For a free Z-module N Z of rank r, let M = HomZ(N, Z) be its dual Z-module with the canonical pairing , : M N Z. We denote h i × → their scalar extensions to R by NR = R ZN, MR and , : MR NR R. ⊗ h i × → Deﬁnition. A subset σ NR is called a strongly convex rational poly- ⊂ hedral cone with apex 0 (or simply a cone later) if σ ( σ) = 0 and ∩ − { } there exist n1,..., ns in N such that 3. Rational partial polyhedral decompositions 5

σ = R n1 + + R n1 = a1n1 + + asns; a1,..., as R . ◦ ··· ◦ { ··· ∈ ◦} n1,..., ns, under the condition that they are irrredundant with each n1 a 6 primitive element of N), are uniquely determined by σ and are called the fundamental generators of σ. dim σ is the dimension of the R-vector subspace σ + ( σ) in NR generated by σ. − When rank N = 3, the following are example of cones.

0 0 0 0

Deﬁnition. Let σ be a cone in NR. A subset σ′ of σ is a face of σ, and denoted σ <σ, if there exists m in M such that m, y 0 for all y σ ′ h i≥ ∈ and σ′ = y σ; m, y = 0 = σ m⊥. { ∈ h i } ∩ Here are examples when rank N = 2.

Deﬁnition. A rational partial polyhedral decomposition (r.p.p. decom- 7 position, for short) is a pair (N, ∆) consisting of a free Zmodule N of ﬁ- nite rank and a collection ∆ of cones in NR such that (i) if ∆ σ, σ > τ, ∋ 6 1. Torus embeddings

then τ ∆ and (ii) if σ and τ belong to ∆, then the intersection σ τ ∈ ∩ is a face of σ as well as of τ.(N, ∆) is called a ﬁnite rational partial polyhedral decomposition(f. r. p. p. decomposition for short ) if ∆ is ﬁnite.

The relative interiors of σ ∆ are disjoint and ﬁll a part of NR. ∈ When rank N = 2, ∆ looks like this:

Deﬁnition. A map h : (N, ∆) (N , ∆ ) between r.p.p. decompositions → ′ ′ is a Z- homomorphism h : N N with ﬁnite cokernel such that for → ′ each σ ∆, there exists σ ∆ with the scalar extension h : NR NR ∈ ′ ∈ ′ → ′ satisfying h(σ) σ . Thus we have the category of r.p.p. decomposi- ⊂ ′ tions.

Deﬁnition. A cone σ in NR is called simplicial if its fundamental generators n1,..., ns are R- linearly independent. σ is called non- singular if its fundamental generators form a part of a Z-basis if N.

8 Given a strongly convex rational polyhedral cone σ in NR, we denote byσ ˇ its dual in MR

σˇ = x MR; x, y for ally σ , n ∈ h i≥ ∈ o where M = N∗ is the dual of N.σ ˇ can be written as the set of R -linear combinations of a finite number of elements of M and is a convex◦ rational polyhedral cone, although it no longer satisfies the strong convexity σˇ + ( σˇ ) = 0 . Instead, it satisfiesσ ˇ + ( σˇ )MR. For the general theory − n o − 3. Rational partial polyhedral decompositions 7 convex polyhedral cones, we refer the reader for instance to Grünbaum [13] and Rockafellar [56]. For a subset τ of σ, we denote

τ⊥ = x MR; x, y = 0 for all y τ . n ∈ h i ∈ o Then y σ is in relative interior of σ if and only ifσ ˇ y = σ , i.e for ∈ ∩ ⊥ ⊥ all x σˇ not in σ , we have x, y > 0 ∈ ⊥ h i The following propositions will be useful later.

Proposition 3.1. Let σ be a cone in NR and σˇ be its dual in MR. Then the map τ σˇ τ⊥ 7−→ ∩ is na order reversing bijection

faces of σ ∼ faces of σˇ . n o −→ n o Proof. By deﬁnition, a face ofσ ˇ is of the formσ ˇ y for y σ. y ∩ ⊥ ∈ belongs to the relative interior of a face τ of σ if and only ifτ ˇ y = τ ∩ ⊥ ⊥ , i.e.σ ˇ τ . ∩ ⊥ Proposition 3.2. Let σ be a come in NR and τ a face of σ. Then there 9 exists x σˇ τ such that ∈ ∩ ⊥

τˇ = σˇ + τ⊥ = τˇ + R ( x). ◦ − Proof. Since τ is a face of σ, there exists x σˇ such that τ = σ x . ∈ ∩ ⊥ We have inclusionsτ ˇ σˇ +τ⊥ σˇ +R ( x) of convex polyhedral cones ⊃ ⊃ ◦ − in MR. The dual τ of the ﬁrst and the dual σ R ( x)∨ = σ x⊥ of the ∩ ◦ − ∩ third coincide, and we are done.

Proposition 3.3. The correspondence

σ σˇ M 7−→ ∩ establishes a bijection between the set of strongly convex rational polyhedral cones in NR and the set of subsemigroups S of M which satisfy the following properties: 8 1. Torus embeddings

(1) S 0 and is ﬁnitely generated as a semigroup. ∈

(2) S generates M as a group.

(3) S is saturated, i.e. S contains m M if there exists a positive ∈ integer a such that am S. ∈

Proof. σˇ M obviously contains 0 and is a saturated subsemigroup. ∩ Since σ ( σ) = 0 by definition, we haveσ ˇ + ( ˇσ) = MR, hence ∩ − { } − (ii). The finite generation ofσ ˇ M as a semigroup is what is known as ∩ Gordan’s lemma and can be proved as follows : We may assume thatσ ˇ is of the form R m1 + + R ms for R-linearly independent elements ◦ ··· ◦ m ,..., m M, since generalσ ˇ is a finite union of convex cones of 1 S ∈ this form by Carathoeodry’s theorem (see for instance Grünbaum [13]). 10 Let M = M (Qm + + Qm ). Thenσ ˇ M = σˇ M and M = ′ ∩ 1 ··· S ∩ ∩ ′ ′′ Zm + + Zm , is a submodule of finite index in M . Sinceσ ˇ M = 1 ··· s ′ ∩ ′ Z m1 + + Z ms, we are done. Conversely, let S satisfy (i), (ii) and ◦ ··· ◦ (iii). By (i), there exist elements m1,..., ms M such that Z m1 + + ∈ ◦ ··· Z mS . Thenσ ˇ = R m1 + + R ms is a convex rational polyhedral ◦ ◦ ··· ◦ cone in MR satisfyingσ ˇ + ( σˇ ) = MR. Henceσ ˇ = σˇ is a strongly − convex rational polyhedral cone. It remains to show that S = σˇ M. ∩ Again by Caratheodory’s theorem, and element inσ ˇ M is a Q linear ∩ ◦ combinations of m1,..., ms. Thus a positive integral multiple of it is contained is S . Hence we are done by (iii).

4 First main theorems

For later convenience, we state the ﬁrst main theorems relating torus embeddings and r.p.p. decompositions in this section, and leave their proofs of . 5. § The following theorem is slightly more general than those in De- mazure [8, . 4], Mumford et al.[63, I.2, Thm.6] and Miyake-Oda [40]. § 4. First main theorems 9

Theorem 4.1. Given k, there exists an equivalence of categories

normal and separated torus r.p.p.decompositions embeddings over k  −→   locally of ﬁnite type      (N, ∆) TN TN emb(∆)  7−→ ⊂ where T = N Z k = Hom (N , k ) and T emb(∆) is obtained as the 11 N ⊗ ∗ gr ∗ ∗ N canonical patching of its aﬃnce open subsets

Hom (σ ˇ N∗, k). unit.semigr ∩ Here k is considered as a unitary semigroup under the multiplication and T acts on it by (tx)(m) = t(m)x(m) for t T , m N and x N ∈ N ∈ ∗ ∈ Hom (σ ˇ N , k). unit.semigr ∩ ∗ TN emb(∆) is of finite type over k if and only if ∆ is finite. Remark . This down-to-earth description of the functor as well as the simplified proof in . 5 was pointed out by Ramanan. § Theorem 4.2. Let (N, ∆) be an r.p.p decomposition. (i) The map σ orb(σ) = Hom (σ⊥ N∗, k∗) 7−→ gr ∩ is a bijection

orb : ∆ ∼ TN orbits in TN emb(∆) , −→ n − o such that orb( 0 ) = T and dim σ + dim orb(σ) = dim T . More- { } N N over, τ<σ if and only if the closure of orb(τ) contains orb(σ).

(ii) The map

σ U(σ) = orb(τ) = Homunit.semigr(σ ˇ N∗, k) 7−→ aτ<σ ∩

establishes an order and intersection preserving bijection ∆ ∼ 12 −→ T -stable affine open sets in T emb(∆) . In particular, T emb(∆) { N N } n is affine if and only if ∆ consists of the faces of a fixed cone in NR. 10 1. Torus embeddings

(iii) For σ ∆, let N¯ be the quotient of N by the subgroup generated ∈ by σ N. Then T ¯ = Hom (σ N , k ). The closure orb(σ) of ∩ N gr ⊥ ∩ ∗ ∗ orb(σ) in TN emb(∆) is normal and orb(σ) = orb(τ). σ<τa∆ ∈ ¯ It coincides with TN emb(∆), where ∆ is the r.p.p.decomposition of NR consisting of the images τ of σ < τ ∆ under NR NR. ∈ → Theorem 4.3. Let (N, ∆) be an r.p.p.decomposition. Then the corresponding T emb(∆) is non-singular if and only if each σ ∆ is non- N ∈ singular, i.e. its fundamental generators of each form a part of a Z-basis of N. In this case, the closure of each TN-orbit orb (σ) is again non- singular.

Remark . When TN emb(∆) is non-singular, the collection of the sets fundamental generators of σ with running through ∆ is a “fan” of De- mazure [8, . 4, n .2]. § ◦ Theorem 4.4. Let h : (N, ∆) (N , ∆ ) be a map of r.p.p. decom- −→ ′ ′ positions, and let f : T emb(∆) T emb(∆ ) be the corresponding N → N′ ′ equivariant dominant morphism. Then f is proper if and only if for each σ ∆ the set σ ∆; h(σ) σ is ﬁnite and h 1(σ ) is the of its ′ ∈ ′ ∈ ⊂ ′ − ′ members. n o

13 Corollary 4.5. Let (N, ∆) be an r.p.p.decomposition. Then TN emb(∆) is complete (i.e. proper over Spec k), if and only if ∆ is ﬁnite and

NR = σ. σ[∆ ∈ 5 The proof of theorems in .4 § In this section, we prove theorems stated in . 4 in a way slightly diﬀer- § ent from Mumford et al.[63]. We informally deal with k-valued points only, although, to be rigorous, we should deal with points with values in arbitrary k-algebras. Here is the key observations relating cones and normal aﬃne torus embeddings. We were inspired by Hochster [20]. 5. The proof of theorems in .4 11 §

5.1 Let N and M be mutually dual free Z-modules of ﬁnite rank with the pairing <,>. Let T = T = N Z k . Then the correspondence N ⊗ ∗

σ U(σ) = Hom . (σ ˇ M, k∗) 7−→ unit semigr ∩ is a bijection from the set of (strongly convex rational polyhedral) cones σ NR to the set of normal aﬃne T -embeddings (of ﬁnite type) over ⊂ N k.

Proof. By Proposition 3.3,σ ˇ M is a finitely generated saturated sub- ∩ semigroup of M which generates M as a group. Thus its semi group algebra A = A(σ) = ke(m) is a subalgebra of finite type of the 14 m Lσˇ M coordinate ring k[T] =∈ ∩ ke(m) of T. It is T-stable under algebraic mLM representation of T on k[T∈ ] defined by f (t) f (t t) for f (t) k[T] 7−→ ′ ∈ and t T. Sinceσ ˇ M generates M as a group, the quotient field of ′ ∈ ∩ A coincides with the quotient field k(T) of k[T]. Sinceσ ˇ M is satu- ∩ rated, A is integrally closed in k(T). Indeed, the integral closure A′ is of finite type over k, hence the representation of T on A′ as above is also algebraic. Thus by the complete reducibility theorem, A′ has a k-basis consisting of elements of the form e(m ) with m M. e(m ) satisfies an ′ ′ ∈ ′ equation c c 1 e(m′) + a e(m′) − + + a = 0 1 ··· c with ai in A, which we may assume to be a k-multiple of an element e(m ), m σˇ M. Obviously there exists a non-vanishing a . Then we i i ∈ ∩ i have cm = m + (c i)m , i.e. im = m σˇ M, hence m σˇ M.A ′ i − ′ ′ i ∈ ∩ ′ ∈ ∩ (k-valued) point of U(σ) = Spec A is a k=algebraic homomorphism u : A K, which is determined uniquely by u(e(m)) k for m M, such → ∈ ∈ that u(e(0)) = 1 and u(e(m + m )) = u(e(m))u(e(m )), i.e., u e : M K ′ ′ ◦ → is a homomorphism of unitary semigroups. T obviously acts on U(σ) as in the statement of Theorem 4.1. Let m ,... m be generators ofσ ˇ M 1 s ∩ as a semigroup. Then M is generates by m ,..., m and (m + +m ). 1 s − 1 ··· s Thus k[T] is the localization of A by e(m + +m ). Hence U(σ) contains 1 · s T as an open set.

Conversely, let T Spec A be a normal aﬃne T-embedding of ﬁnite 15 ⊂ 12 1. Torus embeddings

type. Hence A is a k-subalgebra of finite type of k[T], is normal with the quotient field k(T) and is T-stable under the algebraic representations of T on k[T] and is T-stable under the algebraic representations of T. Thus by the complete reducibility, it has a k-basis consisting of T-semi invariants, i.e. A = ke(m) for a finitely generated subsemigroup S ∋ mLS 0 of M. S generates M∈ as a group. Indeed, for m M, the denominator ′ ∈ ideal of e(m′) is a non-zero T-stable ideal of A, hence contains some e(m)with m S , and e(m)e(m ) A. S is obviously saturated, since ∈ ′ ∈ A is integrally closed in k(T). Hence by proposition 3.3, there exists a unique cone σ NR such that S = σˇ M. ⊂ ∩

Remark. Non-normal aﬃnce TN -embedding can also be written as

S pec( ke(m)) Mm S ∈ for a subsemigroup S 0 which generates M as a group. The simplest ∋ non-trivial example is the curve Spec (k[t2, t3]) with an ordinary cusp. In this case, M = Z, and S is generates by 2 and 3. In general, it is diﬃcult to describe such non-saturated S . For the special case of dimension one, we refer the reader to Delorme [7], Herzog [17] and Herzog-Kunz [21].

5.2 Let h : N N be a homomorphism with finite cokernel and let → ′ f : T T be the corresponding surjective homomorphism of alge- N → N′ braic tori. Let T U(σ) and T U(σ ) be the normal affine torus N ⊂ N′ ⊂ ′ 16 embeddings corresponding to cones σ NR and σ N as in (5.1). ⊂ ′ ⊂ R′ Then f can be extended to a unique equivariant dominant morphism f¯ : U(σ) U(σ ), if and only if of the scalar extension h : NR N → ′ → R′ satisfies h(σ) σ . ⊂ ′

Proof. Let M and M′ be the duals of N and N′, respectively. Then h induces an injection h : M M, which gives rise to f : T = ∗ ′ → N Hom (M, k ) T = Hom (M , k ). Obviously, h(σ ) σ if and gr ∗ → N′ gr ′ ∗ ≤ ⊂ ′ only if h (σ ˇ M ) σˇ M, and in this case it induces a morphism ∗ ′ ∩ ′ ⊂ ∩ f¯ : U(σ) = Hom . (σ ˇ M, k) U(σ ) = Hom . (σ ˇ unit semigr ∩ −→ ′ unit semigr ′ ∩ M′, k). 5. The proof of theorems in .4 13 §

5.3 Let σ be a cone in NR and let M be the dual of N. Then the corresponding normal aﬃne T -embedding U(σ) = Hom . (σ ˇ M, k) N unit semigr ∩ is the disjoint union of TN-orbits

orb(τ) = Hom (τ⊥ M, k∗) gr ∩ with τ running through the faces of σ. The closure of orb τ in U(σ) is

orb(τ) = Hom . (σ ˇ τ⊥ M, k) unit semigr ∩ ∩ and it is the disjoint union of orb τ′ with τ′ running through the faces of σ with τ < τ′. Proof. The following argument, which considerably simplifies our original proof, is due to Ramanan. For simplicity, we denote S = σˇ M. ∩ Let g : S k be a unitary semigroup homomorphism, i.e. g(0) = 1 and → g(m + m ) for m, m S . Then S is the disjoint union of I = g 1(0) and ′ ′ ∈ − S = g 1(k ). S is a subsemigroup containing 0 of S and S + I I. We 17 ′ − ∗ ′ ⊂ first show that a decomposition S = S ′ I is obtained exactly by taking a face F ofσ ˇ and letting S ′ = F M.` Indeed, if F is a face ofσ ˇ , there ∩ exists y σ such that F = σˇ y . Certainly, F M is a subsemigroup ∈ ⊂ ⊥ ∩ containing 0 of S , and for z and w in S with w not in F M, z + w is not ∩ in F M. ∩ Conversely, let S ′ be a subsemigroup containing 0 of S such that its complement I satisfies S + I I. Hence m S is in S of (S + ⊂ ∈ ′ m) S , φ. Replacingσ ˇ by its smallest face containing S , we may ∩ ′ ′ assume that there exists m S in the relative interior ofσ ˇ . We claim ′ ∈ ′ (S + m) Z m′ , φ for m S , hence m S ′ by assumption and ∩ ◦ ∈ ∈ Z m′ S ′. Indeed, since m′ is in the relative interior ofσ ˇ , we see ◦ ⊂ that m , n > 0 for the fundamental generators n ,..., n of σ. Thus h ′ ii 1 s S + m = m M; < m n > < m, n > for 1 i s . Choose a { ′′ ∈ ′′ i ≥ i ≤ ≤ } positive integer a such that a < m , n > < m, n >for 1 i s. Then ′ i ≥ i ≤ ≤ am′ is in S + m. By proposition 3.1, we know that F = σˇ τ for a unique face τ ∩ ⊥ of σ. Thus we see that Homunit.semigr(σ ˇ M, k) = Homunit.semigr(σ ˇ ∩ τ<σ ∩ τ M) generates τ M as a group, hence Hom` (σ ˇ τ ⊥ ∩ ⊥ ∩ unit. semigr ∩ ⊥ ∩ 14 1. Torus embeddings

M, k ) = Hom (τ M, k ) = T ε(τ), where ε(τ) is the trivial group ∗ gr ⊥ ∩ ∗ N · 18 homomorphism τ M k sending every element to 1. Let A = ⊥ ∩ → ∗ m σˇ M ke(m) and L ∈ ∩ P(τ) = ke(m). mMσˇ M m<σˇ∈ τ∩ M ∩ ⊥∩ Then P(τ) is a prime ideal of A with the subring ke(m) isomor- m σˇ⊕τ⊥ M phic to A/P(τ). Hom (σ ˇ τ M, k) is obviously∈ ∩ the disjoint unit. simigr ∩ ⊥ ∩ union of orb (τ′) with τ′ running through the faces of σ with τ < τ′ and is precisely (the set of k-valued points of) the closed set Spec (A/P(τ)). Remark. Let A and P(τ) be as above. Then the correspondence τ P(τ) 7−→ establishes an order preserving bijection

faces of σ ∼ T stable prime ideals of A . { } −→ N − P(σ) is the largest TN-stable proper ideal of A. Let n1,..., ns be the fundamental generators of σ, hence Ron1,..., Rons are the one- dimensional faces of σ. Then P(Ron1),..., P(Rons) are the TN-stable

height one prime ideals of A. The localization AP(Roni) is a discrete valuation ring, hence we have a surjective homomorphism ord : k(T) Z, i ∗ → valuation ring onto Z . Composed with e : M k(T)∗, it gives rise to a ◦ → surjective homomorphism ord e : M Z, which is exactly the primi- i ◦ → tive element n N. i ∈

19 5.4 Let σ NR be a cone and let U(σ) = Hom (σ ˇ M, k) be ⊂ unit.semigr ∩ the corresponding normal aﬃne TN -embedding. Then the map τ U(τ) = Hom (ˇτ M, k) 7−→ unit.semigr ∩ is a bijection

faces of σ ∼ T stable aﬃne open subsets of U(σ) . { } −→{ N − } Moreover, we have U(τ) = orb(τ′). τ <τ Q′ 5. The proof of theorems in .4 15 §

Proof. Let A = ke(m) so that U(σ) = Spec(A). If τ is a face of σ, m σˇ M ∈L∩ there exists, by proposition 3.2, an element m0 σˇ τ⊥ M such that ∈ ∩ ∩ 1 τˇ M = (σ ˇ M) + Z ( m0). Hence ke(m) = A[e(m0) ], whose ∩ ∩ ◦ − m τˇ M ∈L∩ spectrum U(τ) is obviously a TN -stable aﬃne open set.

Conversely, let spec B be a TN-stable aﬃne open set of U(σ). Then by (5.1), There exists a cone τ σ in NR such that e(m); m τˇ M ⊂ { ∈ ∩ } form a k-basis of B. It remains to show that τ is a face of σ. The following argument is again due to Ramanan. Replacing σ by its smallest face containing τ, we may assume that there exists n τ N in the ∈ ∩ relative interior of σ. Thenτ ˇ n is face ofσ ˇ and its intersection with ∩ ⊥ σˇ is exactlyσ ˇ n = σ . Thus the ideal P(σ)B generated by the prime ∩ ⊥ ⊥ ideal P(σ) of A is a proper ideal of B. Thus orb(σ) = spec(A/P(σ)) is contained in the TN-stable aﬃne open set spec B. Since the closure of any TN-orbit in U(σ) contains orb(σ) by (5.3), any TN-orbit in U(σ) is contained in spec B and we are done. U(τ) is the disjoint union of orb(τ′), τ′ < τ, by (5.3). Combining (5.3) and (5.4), we have the following: 20

5.5 Let σ NR be a cone and let U(σ) = Hom (σ ˇ M, k) be ⊂ unit.semigr ∩ the corresponding normal aﬃne TN-embedding. Then

τ orb(τ) = Hom (τ⊥ M, k∗) 7−→ gr ∩ is a bijection

orb : faces of σ ∼ T orbits in U(σ) . { } −→{ N − } Moreover, τ U(τ) = Hom (ˇτ M, k) 7−→ unit.semigr ∩ is a bijection

faces of σ ∼ T stable aﬃne open subsets of U(σ) . { } −→{ N − } For each τ<σ, they satisfy the following properties : 16 1. Torus embeddings

(i) dim τ + dim orb(τ) = dim T , orb( 0 ) = T and orb(σ) is the N { } N unique closed orbit of U(σ).

(ii) U(τ) = orb(τ) τ <τ `′ (iii) The closure of orb(τ) in U(σ) is

orb(τ) = orb(τ′) = Hom (σ ˇ τ⊥ M, k). unit.semigr ∩ ∩ τ<τa′<σ It is the normal aﬃne embedding of orb(τ) = Hom (t M, k ) gr ⊥ ∩ ∗ corresponding to the image of σ under the map from NR to its quotient by the subspace σ + ( σ) generated by σ. −

(iv) The morphism ρτ : U(σ) orb(τ) induced by the inclusionσ ˇ → ∩ .((τ M ֒ σˇ M is a retraction such that U(τ) = ρ 1(orb(τ ⊥ ∩ → ∩ τ− Remark. The map orb can also be described as in Mumford et al. [63] as follows : For t k and n N, consider the element τ ?,n of T = ∈ ∗ ∈ h i N 21 Homgs(M, k∗). Let U(τ) = Homunit. semigr(σ ˇ M, k) be the normal aﬃne ?,n ∩ TN -embedding. The limit of th i as t tends to zero exists in U(σ) if and only if m, n 0 for all m σˇ M, i.e. n σ N. In that case, the h i ≥ ∈ ∩ ∈ ∩ limit is the semigroup homomorphism formσ ˇ M to k sending those ∩ m with m, n = 0 to 1 k and those with m, n > 0 to 0 k, i.e. the h i ∈ h i ∈ identity element ε(τ) of orb(τ) = Hom (τ M, k ), where τ is the face gr ⊥ ∩ ∗ of σ containing n in its relative interior, by Proposition 3.1.

5.6 Let σ NR correspond to the normal aﬃne T -embedding ⊂ N U(σ) = spec A. Let P(σ) be the largest TN-stable proper ideal of A as in the remark after (5.3). Then the following are equivalent:

(i) U(σ) is non-singular.

(ii) The local ring AP(σ) is regular.

(iii) σ is non-singular, i.e. the fundamental generators of σ form a part of a Z-basis of N. 5. The proof of theorems in .4 17 §

Proof. (i) obviously implies (ii). Let us assume (ii) and show (iii). Ob- viously, there exist m ,..., m , s = height (P(σ)), such that e(m ),..., 1 s { 1 e(m ) is a minimal set of generators of the maximal ideal of the lo- s } cal ring. It is easy to see that e(m), m M is contained in the local ∈ ring if and only if m σˇ M. But such m can be written uniquely as ∈ ∩ m = m + a m + + a m with a Zo and m σ M. Hence 22 0 1 1 ··· s s i ∈ 0 ∈ ⊥ ∩ a Z-basis of σ M together with m ,..., m form a Z-basis of M. ⊥ ∩ 1 s Among the dual basis of N, we can choose n ,..., n σ such that 1 s ∈ m , n = δ . Then σ = R n + + R n . It remains to show that h i ji i j o 1 ··· o 1 (iii) implies (i). Let the fundamental generators n1,..., ns of σ be extended to a Z-basis n1,..., nr of N. Let m1,..., mr be the dual basis of M. Thenσ ˇ M = (Zom1 + + Zoms) + (Zms+1 + + Zmr), ∩ 1 1··· ··· hence A = k[u1,..., ur, u1− ,... u−s ] with ui = e(mi), and spec A is non- singular.

5.7 Proof of Theorem 4.1 Let (N, ∆) be an r.p.p.decomposition. Let M = N be the dual of N. For σ ∆, let U(σ) = Hom (σ ˇ ∗ ∈ unit.semigr ∩ M, k) be the corresponding normal affine open TN embedding of finite type. For σ, τ ∆,σ τ is a face of σ and τ. Thus by (5.4), U(σ ∈ ∩ ∩ τ) is canonically a TN-stable affine open set of U(σ) and U(τ). Thus we can paste U(σ)’s together along U(σ τ) to obtain an irreducible ∩ normal scheme TN emb(∆) locally of finite type. Obviously, TN acts algebraically on it with the dense orbit U( 0 ) = T . It is separated, { } N since U(σ) U(τ) = U(σ τ) is an affine open set and the coordinate ∩ ∩ ring of U(σ τ) is generated by those of U(σ) and U(τ). Indeed, they ∩ k-bases consisting of elements of the form e(m) and (σ ˇ M)+(ˇτ M) = ∩ ∩ (σ τ)v M. ∩ ∩ A map h : (N, ∆) (N , ∆ ) of r.p.p. decompositions obviously → ′ ′ gives rise to an equivariant dormant morphism f : T emb(∆) T N → N′ emb(∆ ). On the other hand, suppose f is given. It induces h : N N . 23 ′ → ′ Then for σ ∆, the unique closed T -orbit orb (σ) of U(σ) is mapped ∈ N by f to a T -orbit f (orb(σ)) = orb(σ ) with some σ ∆ . Hence by N′ ′ ′ ∈ ′ (5.5) we have f (U(σ)) U(σ ), i.e. h(σ) σ by (5.2). ⊂ ′ ⊂ ′ Let T X be a normal and separated torus embedding over k. Thus ⊂ there exists N such that T = TN . Consider the collection of T-stable 18 1. Torus embeddings

affine open subsets of X. Since each of them is a normal affine T- embedding of finite type, there exists, by (5.1), a collection ∆ of cones in NR such that U(σ); σ ∆ is the set of T-stable affine open subsets of { ∈ } X. By Sumihiro’s theorem in . 2, U(σ) s convex X. We now show that § ′ (N, ∆) is an r.p.p.decomposition. If σ is in ∆ and τ is a face of σ , then τ is in ∆ by (5.4). For σ and τ in ∆, U(σ) U(τ) is a T-stable affine open ∩ set, hence equals U(ρ) for a ρ ∆, since X is separated. The coordinate ∈ ring of U(ρ) is generated by those of U(σ) and U(τ). Hence looking at T-semiinvariants in them, we see that (σ ˇ M) + (ˇτ M) = ρˇ M. Thus ∩ ∩ ∩ σ τ = ρ. Since U(ρ) is an affine open subset of U(σ) and of U(τ), we ∩ have ρ<σ and ρ < τ, again by (5.4). X is of finite type over k if and only if ∆ is finite, since each U(σ) has only a finite number of T-stable affine open subsets by (5.4)

24 5.8 Proof of Theorem 4.2: We first show (ii). By the construction of T emb (∆), it is covered by T -stable affine open sets U(σ); σ ∆ , N N { ∈ } and U(σ) U(τ) = U(σ τ). Hence the map σ U(σ) is injective. ∩ ∩ 7−→ Let U be a TN -stable affine open set of TN emb (∆). Then U is a normal affine T -embedding, hence by (5.1) there exists a unique cone σ NR N ⊂ such that U = U(ρ). Let σ be in ∆. Then since U(σ) U(ρ) is affine, it is equal as in (5.7) ∩ to U(σ ρ) which is a T -stable affine open set of U(σ) and U(ρ). Thus ∩ N by (5.5), we see that U(ρ) is covered by U(τ) with τ running through the elements of ∆ with τ < ρ. Thus the unique closed orbit orb (ρ) of U(ρ) belongs to some U(τ) with ∆ τ < ρ. On the other hand, ρ is then a face of τ by (5.5), and we are ∋ done. We next show (i). For σ ∆, orb(σ) is the unique closed orbit of ∈ U(σ) by (5.5) (i). On the other hand, each TN-orbit of TN emb (∆) is contained in a TN -stable affine open set, hence by (5.5) is the unique closed T -orbit of a unique σ ∆. N ∈ It remains to show (iii). The closure orb(σ) of orb(σ)in TNemb (∆) is the union of its closures in U(τ) with τ running through the elements of ∆ with σ < τ. In particular, orb(σ) is the disjoint union of orb(τ) with σ < τ ∆ by (5.5) (iii). orb(σ) is the union of normal affine ∈ 6. Projective torus embeddings 19 embeddings of T = Hom (σ M, k ) corresponding to the imageτ ¯ N gr ⊥ ∩ ∗ of τ under NR N¯R, again by (5.5) (iii). The collection ∆¯ of thoseτ ¯ s → ′ is obviously an r.p.p.decomposition of N¯ .

5.9 Proof of Theorem 4.3: This follows easily from (5.6) and theorem 25 4.2 (iii). Note that if τ is non-singular and σ < τ, then its imageτ ¯ under the map from NR to its quotient N¯R by the subspace generated by σ is again non-singular.

5.10 Proof of Theorem 4.4: For simplicity, let X = TN emb(∆) and X′ = TNemb(∆′). By the valuate criterion of properness (see for instance [10] and Mumford [39]), f : X X is proper if and only if it is of → ′ finite type and, moreover, each discrete valuation ring R k which is ⊃ contained in the function field k(X) of X and which dominates a local ring of X′ necessarily dominates a local ring of X. Let ord :k(X) Z be the valuation corresponding to R. Let us ∗ → denote by n = ord e its composite with e : M k(X) . ◦ → ∗ Hence n is an element of N. Each non-zero element n N is obtained ∈ in this way. R dominates a local ring of X if and only if it contains the coordinate ring of one of one of the TN stable affine open sets, i.e. there exists σ ∆ such that m, n 0 for all m σˇ M, i.e. n σ. Similarly, ∈ h i≥ ∈ ∩ ∈ since M ֒ M, R, dominates a local ring of X if and only if there exists ′ → ′ σ ∆ such that n σ . ′ ∈ ′ ∈ ′ f is of finite type if and only if for each σ ∆ , there exist only ′ ∈ ′ a finite number of U(σ)’s with f (U(σ)) U(σ ), i.e. σ ∆; h(σ) ⊂ ′ { ∈ ⊂ (σ) is finite, by (5.2). The rest of the proof is obvious. ′}

6 Projective torus embeddings

For simplicity, we restrict ourselves to complete normal torus embed- 26 dings and try to generalize Demazure’s results in [8] on the ampleness of invertible sheaves to this case. Mumford et al. [63] deal more generally with T-stable fractional ideals on not necessarily complete T- embeddings. 20 1. Torus embeddings

In this section, we ﬁx an f.r.p.p.decompositioon (N, ∆) with NR = σ. Note that ∆ consists of the faces of the maximal dimensional σ ∆ conesS∈ σ ∆, i.e. dim σ = rank N. For simplicity, we denote T = T and ∈ N X = TN emb(∆), which is complete and normal. As before, we denote by M the Z-module dual to N with the canonical pairing , : M N Z. h i × → As usual, a Weil divisor D on X is a ﬁnite Z-linear combination of reduced and irreducible closed subvarieties of codimension one. D gives rise to a fractional ideal OX(D). A Cartier divisor D is a locally principal Weil divisor, which gives rise to an invertible fractional ideal OX(D). We denote by Pic (X) the group of isomorphism classes of invertible sheaves on X, i.e. the group of linear equivalence classes of Cartier divisors on X. Let the 1-skeleton S k1(∆) = σ ,...,σ be the set of 1-dimensional { 1 d} 27 cones in ∆. Let ni be the fundamental generator of σi. By Theorem 4.2, D ,..., D , where D = orb(σ ), is the set of T-stable irreducible Weil { 1 d} i i divisors on X and forms a Z-basis of the group

ZDi 1Mi d ≤ ≤ of T-stable Weil divisors on X.

Proposition 6.1 ((Demazure)). For a complete normal T-embedding X, we have exact sequences

/ div / ZD / linear equivalence / 0 / M / i / classes of Weil / 0 1 i d L≤ ≤ divisors on X

S S / div / T-stable Cartier / / 0 M divisors on X Pic(X) 0 n o where the ﬁrst arrows send m M to ∈ div(m) = m, n D h ii i 1Xi d ≤ ≤ which is the divisor of the character e(m) as a rational function on X 6. Projective torus embeddings 21

Proof. For m M, the divisor of the rational function e(m) on X is equal ∈ to m, ni Di by the remark after (5.3). This vanishes if and only if 1 i dh i m =≤P≤0, since X is complete. For a non-zero rational function f on X, its divisor on X is T-stable if and only if f is a T-semiinvariant. Since X Di = T is factorial, any Weil divisor on X is linearly equivalent − 1 i d ≤S≤ to a linear combination of Di′ s

Lemma 6.2. Let D = aiDi be a T-stable Weil divisor. Then the 28 1 i d following are equivalent.≤P≤ (i) D is a Cartier divisor. (ii) D is principal on each T-stable aﬃne open set U(σ) of X with σ ∆. ∈ (iii) For all σ ∆, there exists m(σ) M such that ∈ ∈ m(σ), n = a h ii − i for all ni contained in σ.

Proof. D deﬁnes a T-stable fractional ideal 0X(D) on X. The space of its sections over the aﬃne open set U(σ) has a k-basis consisting of e(m) with m in

u(σ, D) = m M; m, n a for all n σ { ∈ h ii≥− i i ∈ } by the complete reducibility. Thus (ii) and (iii) are obviously equivalent and imply (i). It remains to show (i) = (ii). Let A(σ) be the coordinate ⇒ ring of U(σ) and let L(σ) be the A(σ)- module of sections of OX(D) over U(σ). Then, as is well-known (see Cartier [5], OX(D) is a Cartier divisor on U(σ) if and only if L(σ).(A(σ) : L(σ)) = A(σ). In terms of T-semi invariants in them, it amounts to

u(σ, D) + m′ M; m′ + µ(σ, D) σˇ m = σˇ M. { ∈ ∈ ∩ } ∩ Since the right hand side contains 0, there exists m(σ) µ(σ, D)such ∈ that m(σ) + u(σ, D) σˇ M. We thus conclude that µ(σ, D) = m(σ) + − ⊂ ∩ σˇ M. Hence L(σ) = A(σ) e(m(σ)). ∩ · 22 1. Torus embeddings

Lemma 6.3. Let D = aiDi be a T-stable Weil divisor. The frac- 1 i d ≤P≤ 29 tional ideal OX(D) is generated by its global sections if and only if for all σ ∆, there exists m(σ) M such that ∈ ∈ m(σ), n a for 1 i d h ii≥− i ≤ ≤ with the equality holding if n σ. i ∈ Proof. The space of global sections of OX(D) has a k-basis consisting of e(m) with m in

λ(D) = m M; m, n a for 1 i d . { ∈ h ii≥− i ≤ ≤ }

The suﬃciency is obvious, since m(σ) is in λ(D) and µ(σ, D) = m(σ) + σˇ M. Let us assume that O (D) is generated by its global ∩ X sections. Hence, ﬁrst of all, D is a Cartier divisor. Thus by lemma 6.2, there exists m (σ) µ(σ, D) with m (σ), n = a for n σ such ′ ∈ h ′ ii − i i ∈ that µ(σ, D) = m (σ) + σˇ M for each σ ∆. On the other hand ′ ∩ ∈ µ(σ, D) = λ(D)+σˇ M by assumption. Hence there exists m(σ) λ(D) ∩ ∈ such that m (σ, D) = m(σ) + σˇ M and we are done. ′ ∈ ∩ Theorem 6.4. Let X= Temb (∆) be a complete normal T-embedding, and let D = aiDi be a T-stable Weil divisor. Then OX(D) is an 1 i d ample invertible≤P≤ sheaf if and only if there exists a positive integer b and, for all maximal dimensional σ ∆, (a unique m(σ) M, such that ∈ ∈ m(σ), n ba for 1 i d h ii≥− i ≤ ≤

with the equality holding if and only if ni is in σ.

Proof. Let OX(D) be ample. Then there exists a positive integer b such that OX(bD) is very ample, hence there exists a projective embedding

f : X P( ke(m)) → m⊕λ ∈ 30 where λ(bD) is as in the proof of Lemma 6.3 and is ﬁnite, since X is 6. Projective torus embeddings 23

complete. OX(bD) is generated by its global sections, hence for each σ ∆ there exists m(σ) M such that m(σ), n ba for 1 i d ∈ ∈ h ii≥− i ≤ ≤ with the equality holding if ni is in σ. If dim σ is maximal, i.e. σ generates N , then such m(σ) is unique. We now show that m(σ), n R h ii− bai if ni is not in a maximal dimensional σ. The restriction of f to the. T-stable aﬃne open set U(σ) induces an open immersion of U(σ) into the spectrum of the k-subalgebra k[e(m m(σ)); m λ(bD)] k[T], − ∈ ⊂ hence into its normalization, which corresponds by (5.1) to the cone σ = y NR; m m(σ), y 0 for all m λ(bD) . Thus by (5.4), ′ { ∈ h − i ≥ ∈ } σ is a face of σ′. Since σ is maximal dimensional, we conclude that σ = σ . Hence if n is not in σ, there exists m λ(bD) such that ′ i ∈ 0 > m m(σ), n and we are done. h − ii Let us now prove the suﬃciency. Suppose there exists a positive integer b and, for each maximal dimensional σ ∆, m(σ) M such that ∈ ∈ m(σ), n ba for 1 i d with the equality holding if and only if h ii≥− i ≤ ≤ ni is in σ. In particular, m(σ) is in λ(bD). Thus by Lemma 6.3, OX(bD) is generated by its global sections, and there exists a morphism

f : X P( ke(m)). −→ mMλ(bD) ∈ For each maximal dimensional σ ∆, let V(σ) be the aﬃne subspace of 31 ∈ the projective space whose homogeneous coordinate corresponding to m(σ) does not vanish. To show that f is a closed immersion by possibly replacing b by its multiple, it is enough to show that

(i) f 1(V(σ)) = U(σ) for all maximal dimensional σ , and − ∈△ (ii) for a multiple of b, the restriction f U(σ) : U(σ) V(σ) is a | → closed immersion.

For (i), it is enough to show that

1 f − (V(σ)) U(σ′) = U(σ) U(σ′) ∩ ∩ for all maximal dimensional σ , since X is covered by those U(σ )’s. ′ ∈ ∧ ′ The right hand side is contained in the left hand side and equals U(σ ∩ 24 1. Torus embeddings

σ′) by Theorem 4.2 (ii). The left hand side is the T-sable aﬃne open set of U(σ ) deﬁned by the non-vanishing of e(m(σ) m(σ )), thus it ′ − ′ corresponds to the face σ = σ (m(σ) m(σ )) of σ . Hence we ′′ ′ ∩ − ′ ⊥ ′ are reduced to showing σ = σ σ . Both of those are faces of σ , and ′′ ∩ ′ ′ σ σ σ . Let n be in σ but not in σ. Then m(σ) m(σ ), n > 0 ∩ ′ ⊂ ′′ i ′ h − ′ ii by assumption, thus ni is not in σ′′. It remains to show (ii). It is enough to show that as a semigroup σˇ M is generated by m m(σ); m λ(bD) by possibly taking a ∩ { − ∈ } multiple of b. Let m ,..., m be generators ofσ ˇ M as a semigroup. 1 s ∩ Let b be a positive integer. Thus for n σ and 1 j s, we have ′ i ∈ ≤ ≤ 32 m , n 0, hence m + b m(σ), n b ba . On the other hand, for h j ii ≥ h j ′ ii≥− ′ i n < σ and 1 j s, we have m + b m(σ), n b ba if b is large i ≤ ≤ h j ′ ii≥− ′ i ′ enough. Thus m λ(b bD) for 1 j s and we are done. j ∈ ′ ≤ ≤ Remark. The inequalities in Theorem 6.4 can be interpreted as follows: The convex hull in MR of m(σ); σ maximal dimensional has { ∈ △ } exactly d facets (i.e. codimension one faces) F1,..., Fd perpendicular to n1,..., nd, respectively. Moreover, m(σ)’s are exactly the vertices,

and the intersection of Fis is the vertex m(σ) if and only if ni1 ,..., nis are the fundamental generators of σ.

Remark . We refer the reader to Mumford et al. [63] for the interpre- tation of these inequalities in terms of the “concavity” and the “strict concavity” of certain functions. In our language, they show in Theorem 13, p.48 that even if X is not complete, D = aiDi is ample if and 1 i d only if there exists a positive integer b and, for≤P each≤ σ , m(σ) M ∈ △ ∈ such that m(σ), n ba with the equality holding if and only if h ii ≥− i n σ. i ∈ Corollary 6.5 (Demazure). Let X = temb(∆) be a complete nonsingular T-embedding, and let D = aiDi. Then the following are equivalent 1 i d : ≤P≤

(i) D is very ample

(ii) D is ample. 6. Projective torus embeddings 25

33 (iii) For each maximal dimensional σ ∆, the unique element m(σ) ∈ ∈ M defined by m(σ), n = a for n σ h ii − i i ∈ satisfies m(σ), n > a for n < σ h ii − i i Proof. (ii) (ii) is obvious. ⇒ (ii) (iii). By Theorem 6.4, there exists a positive integer b and, for ⇒ each maximal dimensional σ , m (σ) M such that m (σ), n ∈ △ ′ ∈ h ′ ii ≥ ba for 1 i d with the equality holding if and only if n σ. Since − i ≤ ≤ i ∈ X is non-singular, the fundamental generators of σ form a Z-basis of N. Hence m′(σ) = bm(σ), where m(σ) is as in (iii). It remains to show (iii) (i). From what we saw in the proof of ⇒ Theorem 6.4, it is enough to show that for each maximal dimensional σ ∆, σˇ M is generated as a semigroup by m m(σ); m λ(D) . ∈ ∩ { − ∈ } We may assume that n1,..., nr are the fundamental generators of σ, hence form a Z-basis of N. Since X is complete and non-singular, there exists, for each 1 i r, a maximal dimensional cone σ ∆ such ≤ ≤ i ∈ that the fundamental generators of σ σ are n ,..., i,..., n . Let the i ∩ { 1 r} remaining fundamental generator of σi be ni, with r < i′ < d. Then m(σ ), n a with the equality holding if and only if j = i or h i ji≥− j ′ 1 j r. Let m ,..., m be the basis of M dual to n ,..., n . They ≤ ≤ { 1 r} { 1 r} generateσ ˇ M. We see that m(σ) = a jm j and m(σi) m(σ) = 34 ∩ 1 i d − am , where a = m(σ ) m(σ), n is positive≤P≤ by assumption. Since i h i − ii m(σ), n a for 1 j d, we see easily that m + m(σ), n a h ji≥− j ≤ ≤ h i ji≥− j for 1 j d, i.e. m + m(σ) λ(D). ≤ ≤ i ∈ Remark . We show in . 8 that any 2-dimensional normal torus em- § bedding of finite type is quasi-projective, using Theorem 6.4. On the other hand, there are many are non-projective 3-dimensional complete non-singular torus embeddings, as we see in . 9. § Proposition 6.6 (Demazure). Let X = temb(∆) be a non-singular T- 1 embedding. Then the canonical invertible sheaf det(ΩX) equals ω = 0 ( D ), X X − i 1Xi d ≤ ≤ 26 1. Torus embeddings

where D = orb(σ ) and S k1( ) = σ ,...,σ . i i △ { 1 d} Proof. For each σ ∆, there exists a Z-basis n ,..., n of N such that ∈ { 1 r} n ,..., n are the fundamental generators of σ. Let m ,..., m be the 1 s { 1 r} dual basis of M, and let ui = e(mi). Then the coordinate ring of U(σ) equals k[u1,..., ur, 1/us+1,..., 1/ur]. Consider the rational section

ξ = (du /u ) ... (du /u ) 1 1 ∧ ∧ r r 1 of det(ΩX). Its divisor on U(σ) equals orb(Roni). We are done, − 1 i s since for a different Z basis of N, the rational≤P≤ section of det(Ω1 ) de- − X fined in this fashion for another affine open set differs from ξ only by sign.

35 Remark. Mumford et al. [63, Thm.9, p.29 and Thm.14, p.52] show that any normal T-embedding X = temb(∆) is Cohen-Macaulay. Moreover, 1 its dualizing sheaf ωX coincides with the double dual of det(ΩX) and equals the fractional ideal associated with the Weil divisor Di, − 1 i d ≤P≤ where D = orb(σ ) and S k1(∆) = σ,...,σ . (See the end our Intro- i i { d} duction for a recent generalization of this result by Ishida.)

Proposition 6.7. Let temb( ) be a complete non-singular torus embed- △ ding. For a 1-codimensional cone τ , there exist exactly two max- ∈ △ imal dimensional cones σ, σ such that τ<σ and τ<σ . Let ′ ∈ △ ′ n1,..., nr 1 be the fundamental generators of τ. Let the additional fun- − damental generator of σ(resp. .σ′) be n(resp. .n′). Then there exist a Z, 1 i r 1 such that i ∈ ≤ ≤ −

n + n′ + aini = 0. 1 Xi r 1 ≤ ≤ − Moreover, we have

2 ai = D1 Di Dr 1 1 i r 1 ··· ··· − ≤ ≤ −

where Di = orb(Roni). 7. Example of torus embeddings and morphisms 27

Proof. Since X is complete, NR is the union of cones in ∆. The ﬁrst assertion, for which the non-singularity is not necessary, is a well-known fact ibn convex geometry. Since. X is on-singular, n, n1,..., nr 1 and { − } n′, n1,..., nr 1 are Z-bases of N. Thus n′ is a Z-linear combination of { − } n, n1,..., nr 1 with the coeﬃcient of n equals 1, since σ and σ′ are 36 − − on the opposite side of τ. As for the assertion, it is enough to restrict ourselves to i = 1. The divisors D = orb(R n ) 1 i r 1 intersect i o i ≤ ≤ − transversally with the intersection orb(τ) P1. Let m, m1,..., mr 1 { − } be the basis of M dual to n, n1,..., nr 1 . Then div(m1) is the sum of { − } D + m , n D and a divisor disjoint from orb(τ), where D = orb(R n ). 1 h 1 ′i ′ ′ o ′ Since m , n = a and D intersects transversally with orb(τ), we are h 1 ′i − i ′ done.

7 Example of torus embeddings and morphisms

In this section, we give typical examples of torus embeddings and equivariant dominant morphisms. We need some of them later.

r 7.1 Aﬃne spaces The r-dimensional aﬃne space Ar = k is obviously r r a (k∗) -embedding. It corresponds to (N, ), where N Z with a △ Z basis n ,..., n and − { 1 r} ∆= the faces of R n + + R n . { o 1 ··· o r}

s r s 7.2 More generally for 0 s r, k (k∗) corresponds to (N, ∆) ≤ ≤ × − with

∆= the faces of R n1 + + R ns . { ◦ ··· ◦ }

r r 7.3 k 0 is again a (k∗) - embedding. It corresponds to (N, ∆), where −{ } ∆ consists of the proper (i.e. not equal to itself) faces of R n1+ +R nr. ◦ ··· ◦ When r = 2, it looks like this: 28 1. Torus embeddings

0 37

7.4 Projective spaces: The r-dimensional projective space Pr is again r obviously a (k∗) -embedding. The corresponding (N, ∆) is deﬁned as follows : N Zr with a Z- basis n ,..., n . Let n = (n + + n ). { 1 r} 0 − 1 ··· r Then consists of the faces of σ ,...,σ , where △ 0 r i σi = R n0 + + + + R nr. ◦ ··· ∨ ··· ◦

0 0

The canonical morphism kr+1 0 P is equivariant and corre- −{ } → r sponds to the homomorphism N˜ Z˜ r+1 N sending elements of the → basis n˜ ,..., n˜ of N˜ to n ,..., n . { o r} { 0 r} In this connection, we have the following characterization of the projective space due to Mabuchi [32].

Theorem 7.1 (Mabuchi). Let X be an r-dimensional complete non- singular T-embedding. Then the following are equivalent. 7. Example of torus embeddings and morphisms 29

38 (1) X Pr equivariantly.

(2) The tangent bundle θX of X is ample. (3) The normal bundle of each T-stable irreducible divisor (which is non-singular by Theorem 4.3) is an ample invertible sheaf. (4) The number of T-ﬁxed points of X (which equals the Euler number of X by Iversen [27]) is exactly r + 1. (5) The number of T-stable irreducible divisors is exactly r + 1. (6) Pic (X) = Z. Let X = T emb( ). Then (5) means that S k1(∆) has r + 1 cones. N △ (4) means that ∆ has r + 1 maximal dimensional cones. (1) (4) (5) is easy to show in view of Proposition 6.7. ⇔ ⇔ (6) (5) follows easily from the exact sequence in Proposition 6.1. ⇒ (5) (6) follows from the exact sequence and the fact that Pix(X) is ⇒ torsion free. (See Demazure [8, p.566]). (1) (2) (3) is well-known. ⇒ ⇒ (3) (4) is due to Mabuchi, who showed (1) (2) (3) as a ⇒ ⇔ ⇔ special case of Hartshorne’s conjecture [16] (H r) to the eﬀect that − an r-dimensional non-singular complete variety with the ample tangent bundle is necessarily Pr. (H 2) is known to be true. (H 3) was proved recently by Mabuchi − − [32] and Mori-Sumihiro [41]. (6) (1) can also be proved directly by means of results in Mori [35], ⇒ as Sumihiro and Ishida pointed out.

7.5 Products We leave the proof of the following to the reader. 39

Proposition 7.2. Let (N′, ∆′) and (N′′, ∆′′) be r.p.p. decompositions. Then we have

T emb(∆′) T emb(∆′′) = T emb(∆′ ∆′′) N′ × N′′ N′ xN′′ × where ∆ ∆ is the r.p.p.decomposition of N N consisting of cones ′ × ′′ ′ × ′′ σ = σ′ σ′′ × 30 1. Torus embeddings

with σ′ and σ′′ running through ∆′ and ∆′′, respectively.

7.6 Equivariant ﬁber bundles More generally, we have the following description of equivariant ﬁber bundles.

Proposition 7.3. Let h : (N, ∆) (N , ∆ ) be a map of r.p.p. decompo- → ′ ′ sitions, and let f : X = T emb(∆) X = T emb(∆ ) be the corre- N → ′ N ′ sponding equivariant dominant morphism. Consider N′′ = ker[h : N ∆ = ∆ → N′] and an r.p.p. decomposition (N′′, ′′) with X′′ TN′′ emb( ′′). Then f : X X is an equivariant fiber bundle with the typical fiber X → ′ ′′ if and only if the following conditions are satisfied:

(i) h : N N is surjective. → ′ (ii) There exists a lifting ∆˜ ∆ of ∆ , i.e. (N, ˜ ) is an r.p.p. decom- ′ ⊂ ′ △′ position and for each σ , there exists a unique σ˜ ˜ such ′ ∈ △′ ′ ∈ △′ that h induces a bijection

h : σ˜ ∼ σ′. ′ −→

40 (iii) consists of the cones △

σ = σ˜ + σ′′ = y˜ + y′′; y˜ σ˜ , y′′ σ′′ ′ { ′ ′ ∈ ′ ∈ } with σ˜ and σ running through ˜ and . ′ ′′ △′ △′′ Again we leave the proof to the reader. Note that the lifting ˜ itself △′ deﬁnes an open set of X which is an equivariant T -bundle over X X N′′ ′ · is associated to it via the TN′′ -action on X′′.

(7.6′) Equivariant Pr-bundles: As a special case of (7.6), let D0′ ,..., Dr′ = be T ′-stable Cartier divisors on X′ and let Li′ 0X′ (Di′). Then the totally decomposable Pr- bundle

f : X = P(L′ L′ ) X′ 0 ⊕···⊕ r →

has a lifting of TN′ -action determined by Di′’s Moreover, X has a ﬁber- r r wise action of (k∗) and is a T (k∗) -embedding. A change of Cartier N′ × 7. Example of torus embeddings and morphisms 31 divisors in the linear equivalence classes gives rise to isomorphic torus embeddings. In terms of r.p.p.decompositions, it can be described as follows: Let N = Zr with a Z- basis ℓ ,...,ℓ , and let N = N N . For each ′′ { 1 r} ′ × ′′ 0 i r and σ , there exists m (σ ) M = (N ) such that the ≤ ≤ ′ ∈ △′ i′ ′ ∈ ′ ′ ∗ A(σ′)-module Li′(U(σ′)) is generated by e(mi′(σ′)), by Lemma 6.2. Let ℓ = (ℓ + + ℓ ), and let σ˜ be the image of σ under the linear map 0 − 1 ··· r ′ ′ .(NR′ ֒ NR sending y′ to (y′, mi′(σ′), y′ ℓi → − 0 i rh i ≤P≤ The collection ˜ = σ˜ ; σ ∆ forms a lifting of ∆ to NR. Then 41 △′ { ′ ′ ∈ ′} ′ X = T emb( ), where consists of N △ △

σ = σ˜′ + σ′′ with σ˜′ running through ˜′ and σ′′ running through ′′ = the faces of △ i △ { σ ,...,σ , with σ = R ℓ + + + + R ℓ . 0′′ r′′} i′′ o 0 ··· ∨ ··· o r Example. Let X = P and r = 1. For a Z , consider the rational ruled ′ 1 ∈ o surface X = P(0P 0P (a)) 1 ⊕ 1 which is usually denoted by Fa or Σa and called a Hirzebruch manifold. The corresponding r.p.p.decomposition of N = Z2 with a Z- basis n , n looks like this: { 1 2}

0 0

Remark. When E′ is a T ′-linearized vector bundle on a T ′-embedding X′, the bundle P(E′) has a lifting of T ′-action. But it is not a torus embedding unless E′ is totally decomposable as above. Nevertheless, it 32 1. Torus embeddings

provides us with a typical example of varieties with torus action which need not be a torus embedding. Thus it is of some interest to classify equivariant vector bundles on torus embeddings. This was partly car- ried out by Kaneyama [28], who, in particular, showed the existence of equivariant but not homogeneous vector bundles on P2.

7.7 Quotient singularities: The quotient singularities explained by Mumford et all. [63, p.16-19] ﬁt in nicely with our formulation. Let N Zr be a free Z-module of rank r with the Z basis n ,..., n . { 1 r} Let n ,..., n N be primitive elements which are R-linearly indepen- 1′ r′ ∈ dent in NR, and let N N be the Z-submodule of ﬁnite index generated ′ ⊂ by n1′ ,..., nr′ . Consider the simplicial cone

σ = R n′ + + R n′ NR = N′ . o 1 ··· o r ⊂ R We thus have a map of r.p.p.decompositions h : (N , ∆) (N, ∆), ′ → where ∆ consists of the faces of σ. Hence we have an equivariant surjective morphism

f : T emb(∆) = kr X = T emb(∆), N′ −→ N which is the quotient map under the canonical action of the (scheme- theoretic) kernel ker[ f : T T ]. When the characteristic of the N′ → N ground ﬁeld k does not divide the order of N/N′, then the kernel is (non- canonically) isomorphic to N/N′. Indeed, let M (resp. M′) be the Z- module dual to N (resp. N′). Then M is canonically a submodule of ﬁnite index of M′, with a pairing

, : M′ N Q h i × −→ which extends the canonical pairings M N Z and M N Z, and × → ′ × → which, moreover, induces a non-degenerate pairing

, : (M′/M) (N/N′) Q/Z h i × −→ Then ker[ f : T T ] = Hom (M /M, k ), which is (non- N′ → N gr ′ ∗ 42 canonically) isomorphic to N/N′ via the above pairing. 7. Example of torus embeddings and morphisms 33

For instance, let r = 2. We may choose a Z-basis n , n of N so { 1 2} that

n1′ = n1

n2′ = an1 + bn2 where a, b Z with (a, b) = 1 and 0 a < b. Then the action of ∈ ≤ ker[ f : T T ] on k2 coincides with the action of Z/bZ on k2 with N′ → N the generator acting via

2 a 2 k (z, w) (ζ− z, ζw) k , ∋ 7−→ ∈ where ζ is a primitive b-th root of 1 and z = e(m1′ ), w = e(m2′ ) with m , m the basis of M dual to n , n . { 1′ 2′ } ′ { 1′ 2′ }

7.8 Equivariant blowing up: The maps of the form h : (N, ∆ (N, ∆) ′ → give rise to equivariant birational morphisms f of the corresponding torus embeddings. Obviously, f is an open immersion if and only if ∆ ∆. ′ ⊂ The most interesting case is when ∆′ is a subdivision of ∆. It was shown by Mumford et all. [63, Thm.10, p.31] that is this case, f is the normalization of the blowing up along a T-stable fractional ideal. 43 They show [ibid., Thm.11, p.32] that given ∆, there always exists a subdivision ∆′ which is non-singular, i.e. any torus embedding has an equivariant resolution of singularities . 34 1. Torus embeddings

The minimal resolution of singularities of a 2-dimensional normal torus embedding can be described very simply as in [ibid., p.35-40]. Note that the singularities in this case are isolated, and the blowing up an isolated singular point is automatically normal, although its description in terms of r.p.p. decompositions is rather complicated and closely related to continued fractions. In this connection, we refer the reader also to Gonzalez-Sprinberg [12], who dealt with Nash transforms of 2-dimensional normal torus embeddings We need later the following description of a blowing up in the non- singular case. Proposition 7.4. Let T X be a non-singular torus embedding cor- ⊂ responding to an r.p.p.decompositon (N, ∆). For σ ∆, the blowing ∈ up of X along the T-stable non-singular closed subvariety orb(σ) is a non-singular T-embedding corresponding to the subdivision (N, ∆∗) obtained from ∆ by “starring at its barycenter” as follows: Let n1,..., ns be the fundamental generators of σ, and let n = n + + n be the 0 1 ··· s “barycenter”. For ∆ τ>σ and 1 i s. Let τ N be the cone ∋ ≤ ≤ i ⊂ R 44 obtained from τ by replacing one of its fundamental generators ni by n0 and leaving the other generators as they are. Then

∆∗ = (∆ τ ∆; τ>σ ) ( the faces of τ ; 1 i s ). −{ ∈ } ∪ { i ≤ ≤ } σ<τ[∆ ∈ Proof. orb(σ) is non-singular by Theorem 4.3. Obviously it is enough to describe the blowing up on the aﬃne open set U(τ) with U(τ) ∩ orb(σ) , Q, i.e. σ < τ ∆ by Theorem 4.2. Let n ,..., n , with s s ∈ 1 s ≤ ′ be the fundamental generators of τ. Since τ is non-singular, they can be extended to a Z-basis n ,..., n of N. Let m ,..., m be the dual ba- { 1 r} { 1 r} sis of M = N∗, and let ui = e(mi). Then the coordinate ring A of U(τ) is

the localization A = k[u1,..., ur]u + ur and the ideal of orb(σ) U(τ) s′ 1··· ∩ is generated by u ,..., u . For 1 i s, let A = A[u /u ,..., u /u ]. 1 s ≤ ≤ i 1 i r i Then the inverse image of U(σ) in the blowing up is covered by Spec A with 1 i s. Obviously, Spec A is a normal aﬃne T-embedding i ≤ ≤ i corresponding to the cone τi = Ron0 + Ron j. 1 j s′ ≤Pj,≤i 7. Example of torus embeddings and morphisms 35

Corollary 7.5. Let T X be a 2-dimensional non-singular torus em- ⊂ bedding corresponding to an r.p.p.decompostion (N, ∆). For a 2 dimensional cone σ = Ron1 + Ron2, the blowing up of X along the T-ﬁxed point ord(σ) is a non-singular T-embedding corresponding to (N, ∆∗), where ∆∗ is obtained from ∆ by removing σ and adding the faces of σ1 = Ro(n1 + n2) + Ron2 and σ2 = Ron1 + Ro(n1 + n2).

Corollary 7.6. Let T X be a 3-dimensional non-singular torus em- ⊂ bedding corresponding to an r.p.p.decomposition (N, ∆).

(i) For a 3-dimensional cone σ = Ron1 + Ron2 + Ron3, the blowing up of X along the T-ﬁxed point orb(σ) is a nonsingular T-embedding corresponding to (N, ∆∗), where ∆∗ is obtained from ∆ by removing σ and adding the faces of σ1,σ2 and σ3 as in the picture below.

(ii) For a 2-dimensional cone σ = Ron1 + Ron2, let τ = Ron1 + Ron2 + Ron3 and τ′ = Ron1 + Ron2 + Ron4 be the 3-dimensional cones in ∆ having σ as a face (cf. Prop.6.7). Then the blowing up of X along the T-stable curve orb(σ) is a non singular T-embedding corresponding to (N, ∆∗), where ∆∗ is obtained from ∆ by removing σ, τ, τ′ and adding the faces of τ1, τ2, τ1′ , τ2′ as in the picture below. 36 1. Torus embeddings

0 46

47 7.9 Algebraic varieties of monomial type: As Mumford et al. [63] pointed out in the it introduction the theorey of torus embeddings is 7. Example of torus embeddings and morphisms 37 a unified and globalized treatment of the “exponents of monomials”. It can be said that when we have algebraic varieties or morphisms defined in terms of monomials only, then there is a possibility of formulating them more transparently in terms of torus embeddings. We encounter many examples in Chapter 2. Here is a motivating example: Let X¯ A = kr be a closed algebraic set defined polynomials ⊂ r f (t),..., fν(t) k[t ,..., t ], each of which is of the form 1 ∈ 1 r f (t) = ta1 tar tar tb1 tbr tbr 1 2 ··· r − 1 2 ··· r with non-singular integers a1,..., ar, b1,..., br. Then X¯ is invariant the coordinatewise multiplication of elements of the group

r T¯ = X¯ (k∗) , ∩ r This T¯ is an algebraic subgroup of (k∗) but may not be an algebraic torus. It may neither be connected nor reduced. But when T¯ is an algebraic torus, then X¯ is a T¯-embedding, although it may not be normal in general. Consider, for instance the rational curve with a cusp

X¯ = (t , t ) A ; t2 = t3 . { 1 2 ∈ 2 2 1}

We have analogues in Pr or its generalization in Mori [34] Here is a more general formulation in the aﬃne case Let M Zr and 48 let 0 S be a ﬁnitely generated sub semigroup of M which generates M ∈ as a group. Hence X = Homu.s.g(S, k) is a T-embedding with

T = Homgr(M, k∗).

Let m1,..., mν, m1′ ,..., mν′ S . Then consider the quotient d∈ M¯ = M/ (the subgroup gen by m m ,..., mν m ) and the image 1 − 1′ − ν′ S¯ of S under the canonical projection M M¯ . Then S¯ is the quotient → of S with respect to the equivalent relation

s s′ ∃ s′′ S, a ,..., aν, b ,..., bν Z such that ∼ ⇔ ∈ 1 1 ∈ o 38 1. Torus embeddings

s = s′′ +Σaimi +Σbimi′

s′ = s′′ +Σbimi +Σaimi′ Let

T¯ = Hom (M¯ , k∗) T algebraic subgroup gr ⊂ X¯ = Hom . . (S¯ , k) X closed algebraic set. u s g ⊂ Then X¯ contains T¯ as a dense open subset and is invariant under T¯. If m m ,..., mν m generates a pure subgroup of M, then T¯ is an 1 − 1′ − ν′ algebraic torus and X¯ is a T¯-embedding. If, moreover, S = σˇ M for ∩ a cone σ NR as in (5.1), then the normalization of X¯ corresponds to ⊂ the cone σ N¯ R, where N¯ = N m m ,..., mν m . ∩ ∩{ 1 − 1′ − ν′ }⊥ 8 Torus embeddings of dimension 2 ≤ 49 It is easy to see that 1-dimensional normal torus embeddings are k∗, A1 = k and P1, up to isomorphism. 2 From now in this section, Let N Z and T = TN . A 2-dimensional normal T-embedding of ﬁnite type looks like this:

In particular, a complete normal 2 dimensional T-embedding is determine by a ﬁnite cycle of primitive elements n1,..., nd in N going counterclockwise once around 0 in the given order such that det (ni, n + ) > 0 for 1 i d (n + = n ), i.e. n + lies strictly before n i 1 ≤ ≤ d 1 1 i 1 − i (which may not belong to the cycle). 8. Torus embeddings of dimension 2 39 ≤

Proposition 8.1. Any 2-dimensional normal torus embedding T X 50 ⊂ of ﬁnite is quasi-projective, i.e. can be equivariantly embedded into a projective space.

Proof. Let X = Temb (∆). By ﬁlling in the complement NR σ − σ ∆ by appropriate cones if necessary, we can embed X equivariantly intoS∈ a complete normal T-embedding. (For the existence of equivariant com- pletions in general, we refer the reader to Sumihiro [59].) We may thus assume that X itself is complete. Let n1,... nd be the fundamental generators of the 1-dimensional cones in ∆ arranged, as above, in such a way that they go around 0 counterclockwise once in this order. By Theorem 6.4, X is projective if and only if there exist a ,..., a Z such that for 1 d ∈ each 2-dimensional cone σ ∆, there exists a unique m(σ) M with ∈ ∈ (σ), n a 1 i d and with the equality holding if and any if h ii≥− i ≤ ≤ n σ. As we remarked after the proof of Theorem 6.4, this means that i ∈ the convex hull of

M(σ) : ∆ σ 2-dimensional { ∋ } in MR has exactly d edges F1,..., Fd going around 0 in this order with Fi perpendicular to ni.

It is thus enough to show, by induction on d, the existence of a convex polygon P in MR with the vertices in MQ such that it has exactly d edges F1,..., Fd going around 0 in this order with Fi perpendicular to ni Obviously d 3 and the case d = 3 is trivial. If d 4, there 51 ≥ ≥ certainly exists nd, say, such that the cycle n1,..., nd 1 still deﬁnes an f.r.p.p. decomposition. Let P be a polygon with− d 1 edges ′ − 40 1. Torus embeddings

F1′ ,..., Fd′ 1 satisfying the required property. At the vertex of P′ which − is the intersection of edges F1′ and Fd′ 1, we can cut the corner by a line − perpendicular to nd and obtain a required P with d edges.

A complete non-singular 2-dimensional T-embedding is determined by a cycle of elements n ,..., n N as above with 1 d ∈

det(n , n + ) = 1 1 i d. i i 1 ≤ ≤ This condition is rigid enough to allow a complete classiﬁcation.

Theorem 8.2. Up to isomorphism, a complete non-singular 2 - dimensional T-embedding is obtained from

P or F = P(0P 0P (a)) 1 , a 0 2 a 1 ⊕ 1 ≥ by a finite succession of blowing ups along T-fixed points. It can be 52 transformed to P2 by a finite succession of blowing ups and blowing downs along T-fixed points. Given non-singular T-embeddings X and X′, there exists X′′ obtained both from X and from X′ by a finite succession of blowing ups along T-fixed points.

Proof. Let X =Temb (∆) with ∆ determined by a cycle n ,..., n N 1 d ∈ going around 0 once counterclockwise. Certainly d 3. If d = 3, then ≥ n1 + n2 + n3 = 0 and X = P2 (cf. Theorem 7.1). We may thus assume d 4. ≥ Lemma 8.3. If d 4, there exist i and j with n + n = 0. ≥ i j 8. Torus embeddings of dimension 2 41 ≤

Sublemma 8.4. Let n, n and ℓ,ℓ be Z-bases of N. If ℓ lies in the { ′} { ′} interior of the sector R n + R n and if ℓ is in the sector R n + R ( n), o o ′ ′ o ′ o − then ℓ = n or ℓ = n. ′ ′ ′ −

Proof of the sublemma: By assumption, there exist positive integers a, a′ such that ℓ = an + a′n′ and non-negative integers b, b′ such that ℓ = bn + b n . We are done, since 1 = det(ℓ,ℓ ) = ab + a b by ′ − ′ ′ ′ ′ ′ assumption.

Proof of the lemma: Suppose the assertion of the lemma is false. By re-numbering the primitive elements if necessary, we may assume that starting counterclockwise from n1, we have more than half of the prim- 53 itive elements before n , which does not coincide with any of the n s − 1 i′ by assumption. Let n ( j > 2) be th last primitive element before n . j − 1 Since n + does not coincide with n and n , we see by sublemma j 1 − 1 − 2 8.4 and the convexity that n + is is strictly between n and n . Thus j 1 − 2 − j we conclude that n , n ,..., n and n + , n + ,..., n , n are all 2 3 j − j 1 − j 2 − d − 1 diﬀerent and lie between n and n . 2 − 1

0 0

This is obviously a contradiction in view of Sublemma 8.4. 42 1. Torus embeddings

Proof of Theorem 8.2 continued: By Lemma 8.3, we may assume that n + n = 0 for some i. Let n = n = n and n = n . Thus n, n is a d i ′ − d i 1 { ′} basis of N. If d = 4, we see that n = n and n = n + an for some integer a. 2 ′ 3 − ′ This gives rise to X = F . By symmetry, we may assume that a 0. If a ≥ a = 1, n′ is the sum of the adjacent primitive elements, hence by blowing down we can eliminate n′. Thus we may assume d 5 and at least one primitive element lies ≥ between n = n1 and n′ = ni. It remains to show that

0 0

54 there exists 1 < j < i such that n j = n j 1 + n j+1. For each 1 j i, − ≤ ≤ there exist non -negative integers b j, b′j with n j = b jn+b′jn. On the other hand for each j, there exists an integer a j such that n j 1 +n j+1 +a jn j = 0 − by Proposition 6.7. Obviously a j < 0. Consider the function

c( j) = b j + b′j > 0.

Then c( j 1)+c( j+1)+a c( j) = 0. Moreover, since c(1) = c(i) = 1, − j there exists j so that c( j) > c( j + 1) and c( j) c( j 1). For this j, we ≥ − have (2 + a )c( j) > 0 and conclude a = 1. This means that if d 5, j j − ≥ we can successively blow X down to some Fa. (cf. the Farey series in additive number theory.) This fact can also be proved sing Nagata’s classiﬁcation [42] of rel- atively minimal models of rational surfaces. Note that an exceptional curve of the ﬁrst kind is automatically T-stable, hence the blowing down is equivariant. 8. Torus embeddings of dimension 2 43 ≤

As for the second assertion of Theorem 8.2, we may assume X = Fa. If we insert the sum of n + an and n between them, then n + an is − ′ − ′ − ′ the sum of n + (a 1)n and n . − − ′ ′

0 0

This means that by blowing up a long a T-fixed point of Fa and then 55 blowing down a T-stable curve, we get Fa 1. This process is usually called an elementary transformation of a ruled− surface. By a finite repe- tition of this process, we transform Fa to F1, which can be blown down to P2. The last assertion of Theorem 8.2 follows easily from the decom- posability of birational morphisms of non-singular surfaces into blowing ups along closed points. But it can also be proved in terms of r.p.p.decompositions as follows: Let X = Temb (∆) and X′ = Temb (∆ ). Then the decomposition of N obtained by the intersections σ σ ′ R ∩ ′ with σ ∆ and σ ∆ need not be non singular, but it has a non sin- ∈ ′ ∈ gular subdivision ∆′′. Thus it is enough to show that a non singular subdivision ∆′′ of a non-singular ∆ is obtained by finite succession of operations in Corollary 7.5. This can be seen exactly as in the last part of the proof of the first assertion. Let us now derive certain consequences from Theorem 8.2 which 56 we need in the next section and which are also of independent interest. Let a cycle n1,..., nd of primitive elements in N determine a complete non-singular 2- dimensional torus embedding X =Temb (∆). Let 44 1. Torus embeddings

Di = orb(Roni). Then by Proposition 6.7, we have

ni 1 + ni+1 + aini = 0 1 i d − ≤ ≤ and 2 ai = Di .

Thus we have a cycle D1,..., Dd of non-singular rational curves, which can be expressed, as usual, by a circular graph with weights ai. Note that this weighted circular graph determines the 2-dimensional T-embedding X up to isomorphism.

weighted circular X graph

The cycle a1,..., ad of integers cannot be arbitrary. For instance, we have the following necessary but not suﬃcient condition as a conse- 2 57 quence of Noether’s formula K + c2 = 12 and c2 = d (cf. Iversen[I6]):

a + + a = 3(4 d). 1 ··· d − We have the following characterization as a consequence of Theorem 8.2.

Corollary 8.5. A weighted circular graph with d vertices corresponds to a 2-dimensional complete non-singular torus embedding if and only if it is obtained from those with d = 3 or d = 4 below by a successive application of the following operation: insert one vertex with weight 1 − and subtract 1 from the weights of the two adjacent vertices. For d 6, ≤ we have the following possible cases. 9. Complete non-singular torus embeddings in dimension 3 45

1 0 0 0

0 0

Remark. A 2-dimensional complete non-singular torus embedding can be considered as a compactiﬁcation of k2, the aﬃne plane, in many different ways. The graph obtained from the corresponding wighted circular graph by removing two adjacent vertices is the graph of the complement of k2. cf. Morrow [37].

9 Complete non-singular torus embeddings in dimension 3

In this section, we give a partial classiﬁcation of 3-dimensional com- 59 plete non-singular torus embeddings. As by-products we get many non- projective complete non singular threefolds and birational morphisms which cannot be written as a succession of blowing ups along non- singular centers. These are of some independent interest even if the torus action is ignored. Torus embeddings provides us with a good testing ground for the birational geometry of non-singular varieties, since many questions are reduced to the elementary geometry of r.p.p.decompositions. Consider, for instance, the following basic question. By abuse of language, a blowing up Y X along a non-singular center of X is also called a blowing → down of Y along a non-singular center. 46 1. Torus embeddings

Question: Let X and X′ be birational non-singular algebraic varieties.

(strong version) Does there exist X′′ which can be obtained both from X and from X′ by a ﬁnite succession of blowing ups along non-singular centers?

(weak version) Can X′ be obtained from X by a finite succession of blowing ups and blowing along non-singular centers? 60 The answer is affirmative dimension 2, but is not known even in dimension 3. According to Hironaka [18], we have the following weaker affirmative answer in characteristic 0: There exists X′′ obtained from X by a finite succession of blowing ups along non-singular centers such that there exists a morphism from X′′ to X′. In the case of torus embeddings, the question is completely reduced to one on non-singular r.p.p.decompositions and looks much easier. Nevertheless, we were unable to prove the following conjecture even in dimension 3. We have already shown in Theorem 8.2 that the conjecture is true in dimension 2.

Conjecture

(strong version) Let X and X′ be non-singular T-embeddings. Then there exists a T-embedding X′′ obtained both from X and from X′ by a ﬁnite succession of T-equivariant blowing ups along T-stable non- singular centers.

(weak version) Any complete non-singular r-dimensional torus embedding is obtained from Pr by a finite succession of equivariant blowing ups and blowing downs along non-singular centers. From now on, we fix N Z3 and the ground field k. Let T = TN. The classification of 3-dimensional complete non-singular T - em- 3 beddings X is reduced to that of f.r.p.p.decompositions ∆ of NR R such that NR = σ and that the fundamental generators of each 3- σ ∆ dimensional σ ∆S∈ form a Z-basis of N. ∈ 61 It is slightly more complicated to describe such ∆ in a computable way than in the 2-dimensional case. 9. Complete non-singular torus embeddings in dimension 3 47

2 Deﬁnition . Let S = S be a sphere in NR centered at 0. Consider a triangulation of S by spherical triangles.

(i) By an N weighting of the triangulation. we mean a primitive element − of N attached to each spherical vertex. (ii) By a double Z weighting of the triangulation, we mean a pair of − integers attached to each spherical edge with one integer on the side of one vertex and with the other on the side of the other vertex. Let v1,..., vs be the vertices adjacent to a vertex v of the triangulation, and let ai be the Z-weight on the edge vvi which is on the side of vi. The weighted link of v is the spherical polygon v v v v together with 1 2 ··· s 1 weights ai at vi.

weighted circular graph

Let X = Temb (∆) be a 3-dimensional complete non-singular torus 62 embedding. If we intersect ∆ with a sphere S N centered at 0, we ⊂ R get a triangulation of S

S = (σ S ). ∩ σ[∆ ∈ We have a canonical N-weighting for this triangulation. Indeed, each 48 1. Torus embeddings

spherical vertex is of the form

(R n) S ◦ ∩ for a primitive n N, which we attach to the vertex. Obviously, ∆ is ∈ determined completely by this N-weighted triangulation of S . Deﬁnition. An N-weighted triangulation of S 2 is admissible if it is obtained from a complete non-singular X = Temb (∆) in this way. The admissibility means the following : For each spherical triangle, the N-weights n, n , n at the three vertices from a Z-basis of N and { ′ ′′} the cone R n + R n′ + R n′′ cuts out the original triangle on S . On the◦ other hand,◦ an◦ admissible N-weighting for a triangulation of S gives rise to a double Z-weighting. Indeed, each spherical edge is of the form τ S ∩ for a 2-dimensional cone τ ∆. Let n , n be the fundamental gen- ∈ { 1 2} 63 erators of τ. By Proposition 6.7, τ is the common fact of exactly two 3-dimensional cones σ and σ . Let n, n , n and n , n , n be the fun- ′ { 1 2} { ′ 1 2} damental generators of σ and σ , respectively. There exist a, b Z such ′ ∈ that ( )n + n′ + an + bn = 0. ∗ 1 2 We attach a pair (a, b) to the edge τ S , with a on the side of (R n ) S ∩ o 1 ∩ and b on the side of (R n ) S . o 2 ∩

0 9. Complete non-singular torus embeddings in dimension 3 49

In this case, consider a vertex v with N-weight n and its link with the vertices v1,..., vs with N-weights n1,..., ns going around v in this order. Let D = orb(R n) and D = orb(R n ) 1 i s be the cor- o i o i ≤ ≤ responding T-stable irreducible divisors on X = Temb (∆). Again by Proposition 6.7, we have

( )ni 1 + ni+1 + aini + bin = 0 1 i s, ∗∗ − ≤ ≤ where n0 = ns, ns+1 = n1, and a = D2 D. i i · Since D is a 2-dimensional complete non-singular torus embedding (cf. 64 Theorem 4.3) with stable curves D D 1 i s, we see that the i ∩ ≤ ≤ weighted link at v is exactly the weighted circular graph of D described immediately before Coro 8.5.

Deﬁnition . A doubly Z-weighted triangulation of S 2 is called admissible if, around each vertex, the equations ( )1 i s in unknowns ∗∗ ≤ ≤ n, n1,..., ns are compatible and if, moreover, the weighted link of each vertex is a weighted circular graph obtained as in Corollary 8.5. Deﬁnition. An isomorphism from an admissible N-weighted triangulation to another is an automorphism of N which induces an isomorphism from the corresponding f.r.p.p.decomposition to another. A combinatorial isomorphism from an admissible doubly Z- weighted triangulation 50 1. Torus embeddings

of S 2 to another is an incidence and Z-weight preserving bijection from the set of spherical simplices of one triangulation to that of the other.

65 Proposition 9.1. We have canonical bijections

combinatorial isom. classes of isom. classes isom. Classes of 3 dim 1 com- of admissible    −  ∼   ∼ admissible doubly plete n.s. torus −→ N-weighted  −→       Z-wei-ted triangns embeddings/k  triangns of s        of S              Proof. From what we have seen so far, it is enough to prove the sur- jectivity of the second map. Let n, n , n be a Z -basis of N. Pick an { ′ ′′} arbitrary triangle of an admissible doubly Z-weighted triangulation of S , and give N-weights n, n′, n′′ to its three vertices in any way. Then using the equality ( ), we can successively determine the N-weights of ∗ the other vertices of the triangulation. This N-weighting is admissible. Indeed, for each vertex v with N-weight n, the vertices v1,..., vs with N-weights n ,..., n of its link satisfy the equalities ( ). Since the dou- 1 s ∗∗ ble Z-weighting is admissible, n1,..., ns go around n once in this order, by Proposition 6.7 applied to N/Zn and the images in it of n1,..., ns. Thus the spherical triangles on S cut out by

R n + R ni + R ni+1 1 i s ◦ ◦ ◦ ≤ ≤ together ﬁll up a neighborhood of (R n) S combinatorially isomor- ◦ ∩ phic to the star of v in the original triangulation which is the spherical polygon with vertices v1,..., vs. Since S is simply connected, we are done.

66 Remark. The advantage of using admissible double Z-weighting is the following: Once a combinatorial type of a triangulation of S 2 is given, possible admissible double Z-weighting on it are computable in princi- ple, although it is more convenient sometimes to use information’s on the N-weights. To state the ﬁnal result, however, it is less cumbersome to choose a convenient Z-basis of N and describe one possible admissible N-weighting corresponding to it in terms of the Z-basis. 9. Complete non-singular torus embeddings in dimension 3 51

Convention Given an admissible N-weighted triangulation of S and a vertex with the N-weight n, we call that vertex the vertex n, for simplicity, although n itself need not be on S . By Corollary 7.6, the blowing up of X = Temb(∆) along orb(σ) corresponds to the following subdivision of the N-weighted triangulation (τ S ). τ ∩ S∈△

(i) (ii)

The following result, whose proof is immediate, will turn out to be 67 useful in our classiﬁcation below.

Corollary 9.2. Given an admissible N-weighted triangulation of S = 2 S , let n be a vertex and let n1,..., n2 be the vertices of its link going around n in this order.

(i) If s 4, there exist i, j such that the vertices n , n, n are collinear ≥ i j on S , i.e are on a great circle.

(ii) If the vertex n is 3-valent, i.e. s = 3, there exists an integer b such that n1 + n2 + n3 + bn = 0. The vertex n can be eliminated by a blowing down along a non-singular center if and only if b = 1. − 52 1. Torus embeddings

The double Z-weights around the vertex n are as in the picture below.

(iii) If the vertex n is 4-valent, i.e s = 4, there exist integer a, b, c such that after re-numbering ni’s, we have n2 +n4 +bn = 0 (in particular, the vertices n2, n, n4 are collinear) and n1 + n3 + an2 + cn = 0. The vertex n can be eliminated by a blowing down along a non-singular center if and only if b = 1. The double Z-weights around the − vertex n are as in the picture below.

(iv) If the vertex n is 5-valent, i.e. s = 5, there exist integer a, b1, b2, b3, b , b with b = b + b and ( 1)b + (a 1)b = ( a)b + ( 1)b 4 5 1 3 4 − 2 − 3 − 4 − 5 such that, after re-numbering n ’s we have n +n +(a 1)n +b n = i 1 3 − 2 2 0; n + n n + b n = 0, n + n n + b n = 0, n + n an + 2 4 − 3 3 3 5 − 4 4 4 1 − 5 68 b5n = 0 and n2 + n5 + b1n = 0 (in particular, the vertices n2, n, n5 are collinear) The double Z-weights around the vertex n are as in picture below.

1 1 0 0 0 1

1 1

3-valent 4-valent 5-valent

To show that a given complete non-singular X = Temb(∆) is projective, i.e. there exist integers ai satisfying the condition (iii) of Corollary 6.5, we have to check many inequalities. The following suﬃcient condition for non projectively is very convenient in concrete applications. 9. Complete non-singular torus embeddings in dimension 3 53

Proposition 9.3. Let S = (σ S ) be an admissible N-weighted σǫ∆ ∩ triangulation of S = S 2. ThenS the corresponding complete non-singular Temb(∆) is non-projective if there exists a spherical polygon n1n2 ... n n in the triangulation satisfying the following: For 1 i g, let g 1 ≤ ≤ σ S be one of two triangles having n n + as edge (n + = n ). Let ℓ i ∩ i i 1 g 1 1 i be the remaining vertex of σ S , i.e. ℓ , n , n + are the fundamental i ∩ { i i i 1} generators of σ . There exists n S not on the polygon n n ... n n i ∈ 1 2 g 1 and real numbers α , β , γ with α 0 and γ > 0 such that 69 i i i i ≥ i

ℓ + α + n + = β n + γ 1 i g, i i 1 i 1 i i i ≤ ≤ i.e. the edge ℓini+1 intersects the great circle passing through ni and n strictly inside the arc n n( n ). i − i

Proof. If Temb(∆) were projective, then Corollary 6.5, there exists ai, b Z and m = m(σ ) M for 1 i g such that. i ∈ i i ∈ ≤ ≤

m ,ℓ = a , m , n = b , m , n + = b + h i ii − i h i ii − i h i i 1i − i 1 and mi 1,ℓi > ai, mi 1, ni+1 > bi+1, h − i − h − i − for 1 i g, where m0 = mg. We see immediately that 0 < mi 1, ≤ ≤ h − ℓi + ai = αi+1 mi 1, ni+1 + bi+1 + γi mi 1, n mi, n , hence i − {h − i } {h − i − h i}

mi 1, n > mi, n 1 i g, h − i h i ≤ ≤ a contradiction. 54 1. Torus embeddings

Let n, n , n be a Z-basis of N, and let ℓ = (n + n + n ). Let P 70 { ′ ′′} − ′ ′′ be the surface of the tetrahedron in NR with vertices n, n′, n′′, ℓ, which has 0 as the bary center. For 0 , y NR, let us call (R y) P the point y ∈ ◦ ∩ on P. For d 7, consider the following subdivision P of P, where the ≥ d bottom triangle nn′n′′ is left as it is.

n1 = n, n2 = n′, n3 = n′′

n4 = ℓ   n = n + n  5 3 4   n6 = n1 + n4   = + n j n6 ( j 6)n4 (6 j d 1) − ≤ ≤ −  nd = n2 + n4   

Proposition 9.4. For d 7, let ∆ be the f.r.p.p.decomposition of N ≥ d R obtained by joining 0 with the simplices of the above subdivision pd. Then we have the following:

(i) Xd = Temb(∆d) is a non-projective complete non-singular 3 - dimensional tours embedding.

(ii) There exists a T-equivariant surjective morphism from Xd to P3.

(iii) Xd cannot be (not necessarily equivariantly) blown down along any non-singular center. 9. Complete non-singular torus embeddings in dimension 3 55

(iv) The tours embedding obtained by blowing up Xd along the curve orb(Ron3 + Rond) is projective, and is obtained from P3 by a suc- 71 cession of blowing ups along T-stable curves.

Proof. It is immediate to see that Xd is complete and non-singular. Xd is non-projective by proposition 9.3 applied to the polygon nn′n′′n of (σ S ), with (Roℓ) S to be taken as n in Proposition 9.3 (ii) is σǫ∆ ∩ ∩ Sd obvious, since Pd is a subdivision of P. (iii) If Xd were obtained by blowing up a non-singular variety along a non-singular center, then the exceptional divisors is automatically T-stable. But it is easy to check that Xd cannot be equivariantly blown along a non-singular center. (iv) is immediate.

Remark. The case d = 7 is the simplest non-projective complete non- singular 3-dimensional torus embedding, which was already described in in Miyake-Oda [40]. See the next page for the picture of T-orbits under the process described in (iv) Demazure [8, Appendice] previously gave an example of non projective complete non-singular 3-dimensional torus embeddings which has 32 T-stable irreducible divisors instead of 7 in our case. We latter get many other non-projective examples as the by-products of our partial classiﬁcation below. 56 1. Torus embeddings

72 9. Complete non-singular torus embeddings in dimension 3 57

By analogy, consider instead the 3-dimensional cone 73

σ = Ron + Rn′ + Ron′′ and let ℓ = n + n + n . For d 7, let ∆ be the subdivision of σ which ′ ′ ′′ ≥ d′ looks like this:

Then the following example gives as answer to a question raised by Fujiki. Corollary 9.5. The equivariant proper morphism

3 f : Y = Temb(∆′ ) U(σ) = k d d → is non-projective and cannot be written as a succession of blowing ups along non-singular centers. But the blowing up Yd′ of Yd along

orb(Ron′′ + Ro(n′ + ℓ′)) is obtained ﬁrst by blowing up U(σ) along the T-ﬁxed point and then by successively blowing up along T-stable curves. Proof. All except the non-projectivity of f are immediate. Consider the tetrahedron with vertices n, n , n and ℓ = (n + n + n ), and its ′ ′′ − ′ ′′ subdivision which ∆d′ induces on the face nn′n′′. By Proposition 9.3, we have a surjective morphism from a non-projective complete non- singular torus embedding to P3. Since f is obviously its restriction to 74 U(σ) P , we are done. ⊂ 3 58 1. Torus embeddings

It is possible to describe projective equivariant birational morphism directly in terms of the corresponding subdivision. We refer the reader to Mumford et al. [63, Chap. III, p. 152-154 in particular]. See also Namikawa [47]. Remark. On the other hand, we can show in a similar fashion that the ′′ following subdivision ∆d of σ = Ron + Ron′ + Ron′′ gives rise to a projective morphism Temb(∆′′ ) U(σ) k3 which cannot be written d → as a ﬁnite succession of blowing ups along non-singular centers, but can ′′ be if Temb(∆d ) is blown up once along orb(Ron′ + Ro(3n + n′ + 2n′′)).

75 Let d be the number of spherical vertices of a triangulation of S 2. Then obviously we have 3d 6 = the number of spherical edges − 2d 4 = the number of spherical triangles. (*) − If the triangulation is of form S = (σ S ) with X = Temb(∆) com- σǫ∪∆ ∩ plete non-singular, then d = the cardinality of S k1(∆), 3d 6 = the num- − ber of 2-dimensional cones in ∆, 2d 4 = the number of 3-dimensional − cones in ∆. By Proposition 6.1, we have d 3 = the Picard number of X. − 9. Complete non-singular torus embeddings in dimension 3 59

Furthermore for v 3, let p be the number of v-valent spherical ≥ v vertices of a triangulation of S 2. Then we see that

(6 v)p = 12. (**) − v Xv 3 ≥ By Proposition 9.1, the classification of complete non-singular 3- dimensional torus embedding is reduced to (i) the classification of combinatorially different triangulations of S 2 and

(ii) the different admissible N-weightings (or admissible double Z − weightings) on them. According to Grünbaum [13, Chap.13] [14], we have the following relevant facts: By Steinitz’s theorem (which is known to be false in higher dimension) the combinatorial equivalence classes of the triangulation of S 2 are in one to one correspondence with the combinatorial types of convex simplicial polyhedral in R3. Given d, there seems, however, to be no formula for the number of the latter. It is empirically 76 known for smaller values of d as follows: d 4 5 6 7 8 9 10 11 12 1 1 2 5 14 50 233 1249 7595 In this connection, Eberhard’s theorem [13, 13.3] looks intriguingly relevant to us, in view of our description of blowing ups along non-singular centers immediately before Corollary 9.2. Anyway for d 8, we have the list in the next page of the combina- ≤ torially different triangulation of S 2, where we stereographically project them onto the plane from one of the spherical vertices, and we express circular arcs on the plane simply be liner arcs. Among 7595 different triangulation for d = 12, we have the one induced by an icosahedron. For the classification of complete non-singular 3-dimensional torus embeddings, it remains to find possible admissible N-weightings on them. Obviously, it is enough to find those without any vertices which can be eliminated by blowing downs along non-singular centers. 60 1. Torus embeddings

Combinatorial types of the triangulation of S 2 for d 8 77 ≤ 9. Complete non-singular torus embeddings in dimension 3 61

Icosahedron.

Theorem 9.6. Let X be a 3-dimensional complete non-singular torus 78 embedding which cannot be blown down along any no-singular center. Then X is isomorphic to one among the projective space P3

P bundles over P P(0P 0P (a) 0P (b)) a, b Z 2 − 1 1 ⊕ 1 ⊕ 1 ∈ P -bundles P(0 L) over complete non-singular 2-dimensional torus 1 Y ⊕ embeddings Y and L Pic(Y) and, if the Picard number d 3 5, ∈ − ≤ Temb( ) corresponding to the following 13 diﬀerent sequences with in- △ tegral parameters of admissible N-weighted triangulations of S 2, where n, n , n is a Z-basis of N = Z3. { ′ ′′} 62 1. Torus embeddings

79 9. Complete non-singular torus embeddings in dimension 3 63

Remark. This theorem was already announced in Miyake-Oda [40] ex- 80 cept that [8-12] was not counted there. This fact, as well as the con- siderable simplification of the proof below, was pointed out by Nagaya [43]. Remark . Some of the torus embeddings listed in Theorem 9.6 can be blown down along a non-singular center if the integral parameters take special values. P3, P2-bundles, P1-bundles, [7-2], [8-2] and [8-10] are all projective. On the other hand, [7-5] and [8 5 ] are X and X of − ′′ 7 8 Proposition 9.4, respectively, hence are non-projective. We can show that [8 5 ](a , 0), [8-8], [8-12], [8 14 ] and [8 14 ] are non- − ′ − ′ − ′′ projective, using Proposition 9.3. [8 13 ] and [8 13 ] are not pro- − ′ − ′′ jective, except when they can be blown down for special values of the integral parameters. For these torus embeddings we can easily check the weak version of the conjecture at the beginning of this section. For simplicity, we adopt the following definition. Definition . An admissible N-weighted triangulation of S 2 is called weakly minimal if the corresponding complete non-singular center, i.e. the N-weighted triangulation has no 3-valent or 4-valent vertex which can be eliminated by blowing down along any non-singular center. 64 1. Torus embeddings

81 Proof of Theorem 9.6. If d = 4, then X P3 as a special case of Theorem 7.1. in the remainder of the proof, we repeatedly use Corollary 9.2, without explicit reference, to determine admissible N-weightings or the corresponding double Z-weightings for a given triangulation of S 2 with d 5. The argument depends very much on the distribution of the ≥ valency of the vertices.

Lemma 9.7. For d 5, let [d 1] be the triangulation of S 2 which ≥ − looks like the picture. A weakly minimal admissible N-weighting on it is necessarily of the following form up to isomorphism and corresponds to a projective torus embedding:

(1) d = 5, and n , n , n are collinear. In this case, the corresponding { 3 4 5} torus embedding is a P2-bundle over P1.

(2) d 5, n + n = 0 and n , n , n are collinear for all 3 i d. In ≥ 1 2 { 1 i 2} ≤ ≤ this case, the corresponding torus embedding is a P1-bundle over a 2-dimensional complete non-singular torus embedding determined by the images of n , 3 i d, in N/Zn . i ≤ ≤ 1

Proof. If d = 5, look at the 4-valent vertices n3, n4, n5. Then by Corollary 9.2, either n , n , n are collinear hence on a great circle, { 3 4 5} or n , n , n are collinear for all 3 6 i 5. Thus we easily get (1) or { 1 i 2} ≤ (2) d = 5 by 7.6′.

82 If d = 6, the triangulation is symmetric with respect to the pairs 9. Complete non-singular torus embeddings in dimension 3 65

n , n , n , n , and n , n . Again by Corollary 9.2 applied to the 4- { 1 2} { 3 5} { 4 6} valent vertices, we may assume that n1 and n2 are antipodal, i.e. n1+n2 = 0. Hence we get (2) d = 6, by 7.6′ When d 7, look at the 4-valent vertices n ,..., n . If n , n , n were ≥ 3 d { 1 i 2} collinear for at most one 3 i d, then n , n ,..., n would be on a ≤ ≤ { 3 4 d} great circle by Corollary 9.2. Since the valency of n is d 2 5, the 2 − ≥ weighted link of n would have weight 1 at some n , 3 6 j 6 d, which 2 − j could be eliminated by blowing down. If n , n , n were collinear for { 1 i 2} exactly two i s, i = 3 and j say, then n , n , n , n would be on a ′ { 1 3 2 j} great circle, and n , n ,..., n and n , n + ,..., n would be collinear. { 3 4 j} { j j 1 d} Since n and n would then be antipodal, and since n , n , n would 3 j { 3 2 j} be collinear, the weighted link of n would have weight 1 at some 2 − ni, i , 3, j, which could again be eliminated. Thus we conclude that n , n , n are collinear for more than two, hence for all 3 i d, since { 1 i 2} ≤ ≤ n1 and n2 are necessarily antipodal. In view of 7.6′ again, we are done. Lemma 9.8. For d 6, let [d 2] be the triangulation of S 2 which ≥ − looks like the picture below. A weakly minimal admissible N-weighting on it satisﬁes the following conditions: n , n ,..., n are collinear, 83 { 3 4 d} n , n = n + n . There exists 4 j d 3, such that n , n , n + and 1 2 3 d ≤ ≤ − { j 1 j 2} n , n , n + are collinear. { j 2 j 2}

The corresponding torus embeddings are always projective. In particular, we have d 7, and the only possible weakly minimal ≥ admissible N-weightings for d = 7, 8 are [7-2] and [8-2] in Theorem 9.6 up to isomorphism. 66 1. Torus embeddings

Proof. If d = 6, look at the weighted link of the 5-valent vertex n2. Its weight at n and n cannot be 1, since otherwise n or n can be elimi- 3 6 − 3 6 nated. Thus n , n , n , n , n , n and n , n , n are collinear again by { 3 2 6} { 1 4 2} { 1 5 2} Corollary 9.2. This means that n1 and n2 are antipodal, a contradiction to the strong convexity of cones. Let d 7. Since n and n cannot be antipodal, there can be at ≥ 1 2 most one 4 i d 1 such that n , n , n are collinear, hence on ≤ ≤ − { 1 i 2} a great circle. If there were one such i, the weighted link of n2 would have weight 1 at some n , 3 j d, j , i, but n , n ,..., n and − j ≤ ≤ { 3 4 i} n , n + ,..., n would be collinear, a contradiction. Thus we conclude { i i 1 d} that n , n ,..., n are collinear. Now look at the weighted link of { 3 4 d} 84 n (resp.n ). Then weight 1 is possible only at n (resp.n ). Thus 1 2 − 2 1 n1 + n2 = n3 + nd, and together with n3,..., nd it lies on a great circle. Moreover, n , n , n + and n , n , n + are collinear for some { j 1 j 2} { j 2 j 2} 4 j d 1. If d = 7 or 8, there is only one possible such weighted link. ≤ ≤ − The projectivity of the corresponding torus embeddings follows easily from Corollary 9.5 in view of the projectivity of 2-dimensional torus embeddings (Proposition 8.1) and the fact that n1 + n2 = n3 + nd, n3 ..., nd are on a great circle.

Lemma 9.9 (Nagaya). For d 7, the triangulation [d 3] below has ≥ − no weakly minimal admissible N-weightings. Proof. Suppose there were a weakly minimal admissible N-weighting. Since n and n cannot be antipodal, there is at most one 5 j d 1 1 2 ≤ ≤ − such that n , n , n are collinear, hence on a great circle. { 1 j 2} 9. Complete non-singular torus embeddings in dimension 3 67

If there were one such j, look at the 4-valent vertex n . n , n , n 4 { 1 4 5} could not be collinear, since otherwise they would be on a great circle disjoint from the great circle above. Hence n , n , n would be { 2 4 3} collinear. Then look at the weighted link of n . The weight 1 could 5 − not be at n nor n . If n , n , n were collinear, then they would be on 3 4 { 1 5 4} a great circle disjoint from another great circle, a contradiction. Thus 85 n , n , n would be collinear, hence n , n are antipodal, a contradic- { 2 5 3} 2 3 tion again. We thus conclude that n , n , n are not collinear for all { 1 i 2} 5 i d 1, hence n , n ,..., n are collinear. Look at the vertex ≤ ≤ − { 5 6 d} n again. If n , n , n were collinear, the Z- weight of the edge n n at 4 { 1 4 5} 2 4 n4 would coincide with that of n3n4 at n4, which is 1. Then look at the weighted link of n . There would exist 6 j d such that n , n , n 2 ≤ ≤ { 4 2 j} are collinear. Thus it has weight 1 at some n , 6 k d, k , j, 4, 1, − k ≤ ≤ 5, a contradiction. Thus n , n , n are collinear. Look at the weighted link of n . Since { 2 4 3} 5 it cannot have weight 1 at n and n , either (i) n , n , n are collinear, − 3 4 { 1 5 4} i.e. on a great circle, hence n , n , n are collinear, or (ii) n , n , n { 4 1 5} { 2 5 3} are collinear, hence n , n are antipodal, and n , n , n are collinear. 2 3 { 2 1 3} In both cases, the weighted link of n would have weight 1 at some 1 − n , 6 k d, a contradiction, since n , n ,..., n are collinear. k ≤ ≤ { 5 6 d} Lemma 9.10 (Nagaya). For d 7, the triangulation [d 4] below has ≥ − no weakly minimal admissible N-weighting.

Proof. Suppose the contrary. Since the weighted link of n5 cannot have weight 1 at n and n , n , n , n are collinear. Look at the weighted 86 − 3 4 { 3 5 4} link of n . Since its weight at n cannot be 1, we have three possibili- 6 3 − ties: 68 1. Torus embeddings

(i) n , n , n are collinear, hence on a great circle. Since n , n , n { 1 6 5} { 1 5 6} are then collinear, the weighted link of n has weight 1 at n , a 5 − 4 contradiction. (ii) n , n , n are collinear. In this case, the weighted link of n { 3 6 7} 5 and the Z-weights around n5 can be determined completely by means of the compatibility in Corollary 9.2. We see easily that the weighted link of n1 has positive weights at n4 and n5, a contradiction. (iii) n , n , n are collinear. Again using the compatibility of the Z- { 2 6 3} weights around n , we easily see that n , n , n are collinear, 5 { 4 2 6} hence n and n are antipodal. Thus n , n , n are also collinear. 3 4 { 3 1 4} Since n , n cannot be antipodal, n , n , n can be collinear for 2 2 { 1 i 2} at most one 5 6 i 6 d 1. Suppose n , n , n were collinear, − { l j 2} hence on a great circle. If j = 5, then n3 could be eliminated. If 6 6 j, then a pair n3, n4 of antipodal points would lie strictly on one side of the great circle, a contradiction. Thus n , n , n { 1 i 2} are not collinear for any 5 i d 1, hence n , n ,..., n are ≤ ≤ − { 6 7 d} 87 collinear. Look at the weighted link of n . Since n , n , n are 2 { 4 2 6} assumed to be collinear, there exists 7 k d such that its weight ≤ ≤ at n is 1, a contradiction. k −

Lemma 9.11 (Nagaya). For d 8, the triangulation [d 6] below has ≥ − no weakly minimal admissible N-weighting. Proof. Suppose the contrary. Since n , n cannot be antipodal, n , n , 1 2 { 1 i n can be collinear for at most one 6 i d 1. 2} ≤ ≤ − 9. Complete non-singular torus embeddings in dimension 3 69

(1) Suppose for one 6 j d 1, n , n , n were collinear, hence on ≤ ≤ − { 1 j 2} a great circle. Then n , n + ,..., n are collinear. Since n , n , n { j j 1 d} { 1 2 j} are collinear, the consideration of the weighted link of n2 shows that j = d 1. Look at the weighted link of n . Since n , n , n − 4 { 1 4 5} cannot be on a great circle, we see that n , n , n are collinear. { 3 4 6} Look then at the weighted link of n5. From what we have seen so far, we can conclude that n , n , n are collinear, hence n , n { 3 5 6} 3 6 are antipodal and are strictly on one side of the great circle passing through n1, n2, nd 1, a contradiction. − (2) We thus conclude that n , n , n are not collinear for all 6 i { 1 i 2} ≤ ≤ d 1, hence n , n ,..., n are collinear. − { 6 7 d}

Look at the weighted link of n4. 88 (2-i) If n , n , n are not collinear, then n , n , n are collinear. The { 1 4 5} { 3 4 6} weighted link of n forces n , n , n to be collinear, hence n , n are 5 { 3 5 6} 3 6 antipodal and n , n , n are collinear. Look at the weighted link of { 3 1 6} n and n simultaneously. Their weights at n , 7 i d, coincide. 1 2 i ≤ ≤ (2 i a) Suppose n , n , n , 6 j < k d, were collinear. Since − − { j 2 k} ≤ ≤ n3, n4 are antipodal, n2n jn j+1 ... nkn2 cannot be a great circle, hence n j, nk are antipodal. The sequence of weights of the weighted link of n2 at n j+1,..., nk 1 is a part of that of the weighted link of n1 at n7, n ,..., n , n , n .− We easily see that this is a contradiction. (2 i b) 8 d 1 5 − − If n , n , n , 6 j d, are collinear, then n n n n is a great cir- { 1 2 j} ≤ ≤ 1 j 2 1 cle, which determines a hemisphere containing a pair n3, n6 of antipodal points, a contradiction. (2-i-c) Suppose n , n , n , 6 j d, are { 5 2 j} ≤ ≤ collinear. Then the consideration of the weights at n j+1,..., nd, n1 of the weighted link of n2 show that j = d. Look then at the weighs at n6, n7,..., nd 1 of the weighted link of n2. There would be 1 some- − − where, a contradiction. (2 ii) Suppose n , n , n are collinear, hence on a great circle and − { 1 4 5} n , n , n are collinear. Look at the weighted link of n and n . As in { 4 1 5} 1 2 (2 i), we arrive at a contradiction. − Lemma 9.12 (Nagaya). For d 8, the triangulation [d 7] has no 89 ≥ − weakly minimal admissible N-weighting. 70 1. Torus embeddings

Proof. Suppose the contrary. Again n , n , n can be collinear for at { 1 i 2} most one 6 i d 1. ≤ ≤ −

(1) Suppose n , n , n , 6 j d 1, are collinear, hence on a great { 1 j 2} ≤ ≤ − circle. Hence n , n , n , n , n ,..., n and n , n + ,..., n are { 1 2 j} { 6 7 j} { j j 1 d} collinear. Looking at the weighted link of n2, we again see that j = d 1, n6,..., nd 1 are collinear and n1n2nd 1n1 is a great − { − } − circle. Look then at the weighted link of n . Obviously, n , n , n 4 { 5 4 6} are not collinear, hence n , n , n are collinear and n , n , n are { 1 4 3} { 4 5 6} not. At 5-valent n , n , n , n are necessarily collinear, but we have 5 { 1 5 3} a contradiction, since then n1, n3 are antipodal and n1n2nd 1n1 is a great circle. − (2) We conclude that n , n , n are not collinear for all 6 i d 1, { 1 i 2} ≤ ≤ − hence n , n , , n are collinear. Look at the weighted link of n . { 6 7 ··· d} 4 (2 i) If n , n , n are collinear, hence on a great circle, the compati- − { 5 4 6} bility of the Z-weights around 5-valent n5 shows that the weighted link of n2 has a positive weight at n5. 90 (2 ii) If n , n , n are not collinear, then n , n , n are collinear. At 5- − { 5 4 6} { 1 4 3} valent n , we see that n , n , n are collinear, hence n , n are antipodal 5 { 1 5 3} 1 3 and n , n , n are collinear, Again by the compatibility of the Z weights { 1 6 3} around n5, we see that the weighted link of n2 has a positive weight at n5. In both cases (2 i) and (2 ii), look at the weighted link of n . − − 2 Since it has a positive weight ar n , n , n , n are collinear for some 5 { 5 2 k} 9. Complete non-singular torus embeddings in dimension 3 71

7 k d, hence has weight 1 at some n , 6 i d, i , k, a ≤ ≤ − 1 ≤ ≤ contradiction. Lemma 9.13. A weakly minimal admissible N-weighting for the triangulation [7-5] is necessarily of the form in Theorem 9.6 up to isomorphism. Proof. By symmetry, we may assume that n , n , n are collinear. { 3 1 6} Look at the weighted link of n2.

Since the weights at n and n cannot be 1, n , n , n are necessar- 1 7 − { 1 2 7} ily collinear. Since its weight at n is then 1, n , n , n are collinear. 3 − { 2 3 5} Thus n , n , n and n , n , n are collinear for the same reason. Look- { 3 4 7} { 1 5 4} ing then at the weighted link of n , we see that n , n , n are collinear. 6 { 5 6 7} From what we have seen so far, an admissible double Z weighting is uniquely determined as in the picture, by Cor.9.2.

0 1 0 1 1 0 1 1 0

1 0 0 0 0 1 1 0

1 72 1. Torus embeddings

91 n = n5, n′ = n1, n′′ = n3 form a Z-basis of N. The double Z weighting successively determines the other N-weights as follows: n2 = nn , n = n n , n = n n and n = n n . − ′′ 4 − − ′ 6 − ′ − ′′ 7 − − ′′ Lemma 9.14. A weakly minimal admissible N-weighting on the triangulation [8-5] is either [8 5 ] or [8 5 ] of Theorem 9.6 up to iso- − ′ − ′′ morphism.

Proof. Neither n , n , n nor n , n , n are collinear, since otherwise { 1 4 2} { 1 8 2} the Z weights around n would be 1. 3 −

In particular, n , n are not antipodal. Thus n , n , n are collinear 1 2 { 1 i 2} for at most one 5 i 7. If there were no such i, then n n n n n n ≤ ≤ 4 5 6 7 8 4 would necessarily be a great circle and the weighted link of n4 would have weight 1 at n , a contradiction. By symmetry, it thus suﬃces to − 3 consider the following two cases:

1. n , n , n are collinear, hence n , n , n , n are collinear. Look { 1 5 2} { 5 6 7 8} at the weighted link of n . We easily see that n , n , n are 4 { 3 4 5} 92 collinear. Look then at the weighted link of n . If n , n , n were 2 { 5 2 7} collinear, then n n n n n would be a great circle, thus n , n , n 5 2 7 1 5 { 2 7 1} would be collinear, a contradiction. Thus we conclude that n , n , { 6 2 n are collinear. In particular, n , n are antipodal, hence n , n , 8} 6 8 { 6 1 n are also collinear. Then the weighted link of n has necessar- 8} 1 ily weights 2, 1, 2 at n , n , n , respectively. Looking at the − − − 3 4 5 Z weights around n4 determined so far, we see that the weighted 9. Complete non-singular torus embeddings in dimension 3 73

link of n has weight 0 at n , hence n , n , n are collinear. The 8 4 { 2 8 3} collinearity for each vertex thus determined gives rise to a unique admissible double Z weighting with one integral parameter, which determines the admissible N-weighting [8 5 ] in Theorem 9.6, − ′ if we let n = n2, n′ = n5, n′′ = n6.

2. n , n , n , hence n , n , n and n , n , n are collinear. Look { 1 6 2} { 4 5 6} { 6 7 8} at the weighted link of n . We see that n , n , n are collinear. 2 { 5 2 7} Look then at the weighted link of n4 and n8. By symmetry and because of the fact that their weights at n cannot be 1, we have 3 − four possibilities:

(i) n , n , n are collinear, hence are on a great circle. Look at the { 1 4 8} weighted link of n . Its weights at n , n , n are necessarily. 2, 1 5 6 7 − 1, 2. By the compatibility of the Z weights around the 4-valent − − vertex n5, we have a contradiction.

(ii) n , n , n and n , n , n are collinear, hence n , n are antipodal. { 2 4 3} { 2 8 3} 2 3 We see then that n , n , n are collinear. Then the weighted link 93 { 3 1 6} of n1 leads us to a contradiction.

(iii) n , n , n and n , n , n are collinear, hence n , n are antipodal. { 3 4 5} { 3 8 7} 3 6 Again we see that n , n , n are collinear, a contradiction. { 3 1 6} (iv) n , n , n and n , n , n are collinear. Then by the compatibil- { 3 4 5} { 3 8 2} ity of the Zweights around the vertices n7 and n8, we see that the weighted link of n has weight 3, 1, 2, 1, 1, 0 at n , n , n , n , 1 − − − − 3 4 5 6 n , n , respectively, hence in particular n , n , n are collinear. 7 8 { 3 1 7} An admissible double Z -weighting is uniquely determined by these considerations and gives rise to the admissible N-weighting [8 5 ] of Thm 9.6. − ′′ Lemma 9.15. A weakly minimal N-weighting on the triangulation [8-8] is of the form in Theorem 9.6 up to isomorphism. 74 1. Torus embeddings

Proof. By symmetry, we may assume that n , n , n are collinear. { 4 1 5} Looking at the weighted link of n and of n , we see that n , n , n 2 3 { 1 2 6} and n , n , n are collinear. { 1 3 7}

Look then at the weighted link of n and of n . n , n , n are not 4 5 { 2 4 8} collinear, since otherwise they would be on a great circle, hence a contradiction for the weighted link of n2 would result. Similarly, we see that n , n , n are not collinear. By symmetry, we thus have three pos- { 3 5 8} sibilities:

94 1. n , n , n and n , n , n are collinear. Then n , n and n , n { 1 4 6} { 1 5 7} { 1 6} { 1 7} are pairwise antipodal, a contradiction.

2. n , n , n and n , n , n are collinear. Let the weighted link { 3 4 6} { 2 5 7} of n2 have weight b at n6. The collinearity we have so far then determines all the Z-weights around n2 in terms of b. But the compatibility for them leads to b = 1, a contradiction. − 3. n , n , n and n , n , n are collinear. Let the weighted link of { 3 4 6} { 1 5 7} n2 have weight b at n6. Then the compatibility of the Z weights around n shows that n , n , n are collinear. Hence the weighted 2 { 5 8 6} link of n necessarily has weights 2, 1 at n , n , n , respec- 8 − − 7 3 4 tively. In this way, an admissible double Z -weighting is uniquely determined and gives rise to the admissible N-weighting [8-8] of Thm 9.6.

9. Complete non-singular torus embeddings in dimension 3 75

Lemma 9.16. The triangulation [8-9] has no weakly minimal admissible N-weighting.

Proof. Suppose the contrary. Look at the weighted link of n1. By symmetry, it is enough to consider three cases.

1. n , n , n are collinear, hence on a great circle. Looking at the 95 { 2 1 3} weighted link of n2, n3 and n4, we immediately get a contradiction.

2. n , n , n are collinear. In this case, the weighted link of n has { 2 1 6} 1 weights 1 at n and n , a contradiction. − 7 8 3. n , n , n are collinear. The same argument applies to the other { 6 1 7} 6 valent vertices n2, n3 and n4. But for the weighted link of n2, n , n , n cannot be collinear, since otherwise we would easily { 5 2 8} conclude that n , n , n are also collinear hence on a great circle, { 3 2 4} a contradiction by (1) applied to n5 instead of n1. Taking into account the similar conclusion for n3, we need, by symmetry, to consider the following three cases:

(i) n , n , n and n , n , n are collinear. In this case, the consider- { 5 2 7} { 5 3 6} ation of the weighted link of n2 and n3 leads us to a contraction for the weighted link of n1 76 1. Torus embeddings

(ii) n , n , n and n , n , n are collinear. { 5 2 7} { 6 3 8} (iii) n , n , n and n , n , n are collinear. { 7 2 8} { 6 3 8} In cases (ii) and (iii), the Z- weights around n6, n7, n8 are necessarily 0, 0, 2 a contradiction for the weighted link of n . − 1 Lemma 9.17. A weakly minimal admissible N-weighting for the triangulation [8-10] is necessarily of the form in Theorem 9.6 up to isomorphism. Proof. Note ﬁrst that the triangulation can be written in a more symmetric form as on the right hand side.

96 Look at the weighted link of n . Then n , n , n . are collinear. 1 { 7 1 8} Look then at the weighted link of n . Then n , n , n are not collinear, 2 { 3 2 4} hence n , n , n are collinear. Indeed, if otherwise, n , n , n would { 5 2 6} { 3 2 4} be a great circle. Then looking at the weighted link of n3 and n4, we would get a contradiction for the weighted link of n1. If n , n , n or n , n , n were collinear, then n , n would be an- { 5 3 6} { 5 4 6} 5 6 tipodal, and we would get a contradiction for the weighted link of n1. If n , n , n or n , n , n were collinear, then they would be on a { 2 3 4} { 2 4 3} great circle, a contradiction again. Thus for the weighted link of n , either n , n , n or n , n , n are 3 { 2 3 7} { 1 3 6} collinear. We have a similar conclusion for the weighted link of n4. By symmetry, we need to consider the following three cases: 9. Complete non-singular torus embeddings in dimension 3 77

1. n , n , n and n , n , n are collinear. We get a contradiction 97 { 1 3 6} { 1 4 6} for the weighted link of n1.

2. n , n , n and n , n , n are collinear. In this case, we have a { 2 3 7} { 2 4 8} contradiction for the Z-weights around n7, n8, and n2.

3. n , n , n and n , n , n are collinear. In this case, we can de- { 1 3 6} { 2 4 8} termine all the double Z-weights uniquely from the collinearity conditions so far. We get the unique admissible N-weighting described in Theorem 9.6.

Lemma 9.18. A weakly minimal admissible N-weighting for the triangulation [8-11] is of the form in Theorem 9.6 up to isomorphism.

Proof. Look at the weighted link of n1 and n2. By symmetry, we need to consider the following three cases:

1. n , n , n and n , n , n are collinear, hence n , n , n , n are { 4 1 2} { 1 2 3} { 1 2 3 4} on a great circle. We easily get a contradiction for the weighted link of n5. 78 1. Torus embeddings

2. n , n , n and n , n , n are collinear. In this case, n , n , n { 4 1 2} { 5 2 6} 1 5 7 and n1, n6, n8 are necessarily collinear. The consideration of the 98 weighted link of n shows that n , n , n are collinear, hence 3 { 5 3 6} n5, n6 are antipodal.

3. n , n , n and n , n , n are collinear, hence n , n are again an- { 5 1 6} { 5 2 6} 5 6 tipodal.

Thus in cases (2) and (3), n and n are antipodal, hence n , n , n 5 6 { 5 3 6} and n , n , n are collinear. Look at the weighted link of n . Since { 5 4 6} 5 n , n , n cannot be on a great circle, and since the situation is sym- { 3 5 4} metric with respect to n and n , we may assume that n , n , n are 3 4 { 2 5 7} collinear. Looking at the weighted link of n4 and the Z-weights around n , n and n , we conclude that n , n , n are collinear. From what we 7 8 1 { 2 6 8} have seen so far, we get a unique admissible double Z-weighting with two integral parameters, which gives rise to the admissible N-weighting described in Theorem 9.6.

Lemma 9.19 (Nagaya). A weakly minimal admissible N-weighting on the triangulation [8-12] is of the form in Theorem 9.6 up to isomorphism.

Proof. The triangulation can again be written in a more symmetric form as on the right hand side.

= 9. Complete non-singular torus embeddings in dimension 3 79

99 By symmetry, we may assume that n , n , n are collinear. Look at { 3 1 7} the weighted link of n2 and of n3. We need to consider three cases:

1. If n , n , n are collinear, then n , n are antipodal, hence n , n , { 3 2 7} 3 7 { 3 5 n and n , n , n are collinear. Looking at the weighted link of 7} { 3 6 7} n , we see that n , n , n are collinear, a contradiction for the 5 { 5 4 8} weighted link of n6.

2. If n , n , n and n , n , n are collinear, then n , n are antipo- { 1 2 5} { 1 3 5} 1 5 dal, hence n , n , n , n , n , n and n , n , n are collinear. { 1 6 5} { 1 7 5} { 3 5 7} Looking at the weighted link of n , we see that n , n , n are 5 { 5 4 8} collinear, again a contradiction for the weighted link of n6.

3. Let n , n , n and n , n , n be collinear. If n , n , n were { 1 2 5} { 2 3 6} { 6 4 7} collinear hence on a great circle, the weighted link of n7 would have weight 2 at n . Moreover, the weighted link of n would − 2 5 have weight 1 at n , hence n , n , n would be collinear and 4 { 3 5 4} the weight at n would be 1. Since the weighted link of n 2 − 1 has weight 0 at n2, we would violate the compatibility for the Z-weights around n2.

Thus we conclude that n , n , n , n , n , n , n , n , n and n , { 3 1 7} { 1 2 5} { 2 3 6} { 5 n , n are collinear. We then have three possibilities for the weighted 4 8} link of n . If n , n , n were collinear, then n , n would be antipodal, a 5 { 3 5 7} 3 7 contradiction, since we are in case (1). If n , n , n were collinear, then { 2 5 6} n , n , n , n would be on a great circle, hence n , n would be antipo- 100 { 1 2 5 6} 2 6 dal. By symmetry, we are in case (1), a contradiction. Thus n , n , n { 3 5 4} are collinear. We then four possibilities for the weighted link of n . If n , n , n 6 { 3 6 4} were collinear, then n , n would be antipodal, hence n , n , n would 3 4 { 1 7 4} be collinear, a contradiction for the weighted link of n . If n , n , n 7 { 5 6 8} were collinear, then n , n would be antipodal, hence n , n , n would 5 8 { 5 7 8} be collinear. Looking at the weighted link of n6 and n7, we would get a contradiction for the Z- weights around n . If n , n , n were collinear 1 { 4 6 7} hence on a great circle, n , n , n would be collinear. Looking at the { 4 7 6} 80 1. Torus embeddings

weighted link of n6 and n7, we would again get a contradiction for the Z-weights around n1. Thus n , n , n are collinear. These collinearity conditions deter- { 1 6 8} mine a unique admissible double Z-weighting, which gives rise to the N-weighting [8-12] in Theorem 9.6.

Lemma 9.20. A weakly minimal admissible N-weighting on the triangulation [8-13] is either [8-13] or [8 13 ] of Theorem 9.6 up to iso- − ′′ morphism.

101 Proof. Look at the weighted link of n1 and n2. By symmetry, it suﬃces to consider three cases:

1. n , n , n and n , n , n are collinear. Look at the weighted { 2 1 6} { 1 2 7} link of n . By symmetry, we may assume that n , n , n , hence 5 { 2 5 6} n , n , n are collinear. Thus n , n are antipodal, and n , n , n { 4 3 5} 2 6 { 2 8 6} are collinear. Looking at the weighted link of n8, we see that n , n , n are collinear. We have two possibilities for the weigh- { 3 4 8} ted link of n , but by symmetry with respect to n , n , we may 6 { 3 4} assume that n , n , n are collinear. Thus n , n are antipodal, { 3 6 8} 3 8 hence n , n , n are collinear. These collinearity conditions de- { 3 7 8} termine a unique admissible double Z -weighting with two integral parameter which gives rise to the N-weighting [8 13 ] of − ′ Theorem 9.6. 9. Complete non-singular torus embeddings in dimension 3 81

2. n , n , n and n , n , n are collinear. In this case n , n are { 5 1 8} { 5 2 8} 5 8 antipodal, hence n , n , n and n , n , n are collinear. Look- { 5 6 8} { 5 7 8} ing at the weighted link of n and n , we see that n , n , n and 6 7 { 6 3 7} n , n , n are collinear, hence n , n are antipodal and n , n , n { 6 4 7} 6 7 { 6 5 7} and n , n , n are collinear. These collinearity conditions deter- { 6 8 7} mine a unique admissible double Z-weighting with four integral parameters, which gives rise to the N-weighting [8 13 ] of The- − ′′ orem 9.6. It remains to show that the following cannot happen: 3. n , n , n and n , n , n are collinear. { 5 1 8} { 1 2 7}

By symmetry with respect to n , n and our argument (1), (2) applied 102 { 3 4} to n , n instead of n , n , we may assume that n , n , n and n , n , n 3 4 1 2 { 6 3 7} { 3 4 8} are collinear. Looking at the weighted link of n5 and n6, we see then that n , n , n and n , n , n are collinear. Let a (resp b) be the Z-weight { 2 5 3} { 1 6 4} of the edge n3n5 at n3 (resp. n1n6 at n1). Then the Z-weights around 4- valent vertices n1, n2, n3, n4 are determined in terms of a and b. We see that a, b , 0, 1. Looking at the compatibility of the Z-weights around − n and n , we see that n , n , n and n , n , n are collinear. Then we 7 8 { 4 7 5} { 2 8 6} have a(b+1) = b(a+1) = 1, a contradiction, since a , 1 and b , 1. − − − Lemma 9.21. A weakly minimal admissible N-weighting on the triangulation [8-14] is either [8 14 ] or [8 14 ] of Theorem 9.6 up to − ′ − ′′ isomorphism. Proof. The triangulation can be written in a more symmetric from again.

= 82 1. Torus embeddings

103 First look at the 5-valent n , n , n around n (1). If n , n , n 3 4 5 1 { 3 4 5} are collinear, then they are on a great circle. Look at the weighted link of n . Because of the 3-valent vertex n , n , n , n and n , n , n are 6 2 { 5 6 8} { 4 6 7} not collinear n , n , n are not collinear, since otherwise they would { 7 6 8} be on a great circle disjoint from the other great circle n3, n4, n5, n3. Thus by symmetry we may assume that n , n , n are collinear. Look { 2 6 5} at the weighted link of n . If n , n , n were collinear, then n , n 7 { 2 7 5} 2 5 would be antipodal, a contradiction, since n2 is not on the great circle n3n4n5n3. For the same reason as above, we thus conclude that n , n , n and n , n , n are collinear. These collinearity condition { 2 7 3} { 2 8 4} determine a unique admissible double Z-weighting with two integral parameters, which gives rise to the N-weighting [8 14 ] of Theorem − ′′ 9.6. (2) By (1) and the symmetry around n1, we need to consider two cases: (i) n , n , n , n , n , n and n , n , n are collinear. Let b be the { 1 3 8} { 1 4 8} { 1 5 6} Z-weight around n1. Then the weighted link of n3 is completely determined in terms of b. By the compatibility of the Z-weights around n3, the weighted link of n has weight 0 at n , hence n , n , n are collinear, a contradiction. 8 3 { 4 8 7} 104 (ii) n , n , n , n , n , n and n , n , n are collinear. Again let b { 1 3 7} { 1 4 8} { 1 5 6} be the Z-weighted around n1. The collinearity condition so far determine some the the double Z weights. By our argument (1) applied − to n instead of n , we conclude easily that n , n , n , n , n , n and 2 1 { 2 8 4} { 2 6 5} n , n , n are collinear. We thus get a unique admissible double Z- { 2 7 3} weighting with an integral parameter, which gives to the N-weighting [8 14 ] of Theorem 9.6. − ′

We thus conclude the proof of Theorem 9.6.

Remark . By repeated application of Corollary 9.2 we determined all admissible N-weightings on the icosahedral triangulation of S 2. Ap- parently, there are 32 diﬀerent admissible N-weightings, some of which integral parameters, as in the table below, although some of them might be redundant because of the symmetric nature of the triangulation. 9. Complete non-singular torus embeddings in dimension 3 83

Some of them are projective and some others are not. The weak version of our conjecture at the beginning of this section holds for all of them. Since all the vertices are 5-valent, these admissible N-weightings are automatically weakly minimal. In the table below let n, n n be a { ′ ′′} Z-basis of N and a, b, c, dǫ, Z 84 1. Torus embeddings

105 ′′ n 1) + ′′ ′′ b ′′ bn ( n n + + − + ′ ′ ′′ ′ ′ ′′ n n n n n an + + + − ′′ ′ − − n n n n n n ′ ′ ′′ n n n n − − − − n − − ′′ ′′ n bn 1) + + ′′ ′ n c cn ( 1) + + ′′ + ′ ′ n ′′ ′′ a n n − ( an an ′ + − n n + + + ′′ ′ ′ ′ − − n n n n n n n ′ ′′ − n − − − n − − n cn n − n ′′ n 1) + c ′′ ( ′ n cn ′′ 1) + ′ an + ′′ + b bn n ′′ ( ′ ′′ ′ n + n n n + + ′ ′′ ′ + ′ − − + n n n n ′′ ′ ′ n n − n n − n − n − n 2 ′′ cn + ′′ ′ n cn 1) + ′′ ′′ + ′ n a bn ′′ − an ( ′′ ′ n + n n n + + ′′ ′ ′′ − n − + + n n n n ′′ ′ ′ ′ − − n n n n − n n − n n ′′ ′′ cn ′′ an ′′ + ′′ n n ′ + + − ′ an ′′ bn ′ ′ n ′ ′ n + n n 2 n n + ′ ′′ − − − − − − n n n ′ ′ ′′ − (1) (2) (3) (4) (5) − − n n n n n n n n n n 1 2 3 4 5 6 7 9 8 10 11 12 n n n n n n n n n n n 106 n 9. Complete non-singular torus embeddings in dimension 3 85 ′′ n ) c ′′ n + a 1) ( − ′′ ′′ + n c ′ ( cn ′′ n 1) + + 1) ′′ an + ′ ′ − + a an ′′ bn bn ( ′′ ′ b n + n n ( + + + ′′ ′ − − − + n n n n n ′ ′ ′′ n − n n n − n − n − n − ′′ n 1) ′′ + ′′ n c ′′ ( ′′ 1) n an + + + an + ′ ′′ ′ a + n ( n ′′ bn bn ′ ′ n + + + n n + + ′ ′ ′′ ′ − − − n n n n n n ′ ′ ′′ − − n − − n − n − n n n ′′ n ′′ 1) n ′′ − 1) c cn ( + ′′ + a + ′ ( an ′ ′′ + + bn n ′′ bn ′′ ′ ′ n + − n n n + ′ ′′ ′′ ′ − − − − n n n n n ′′ ′ ′ − n − n − n − n n − n ′′ n ) c + a ( + ′′ ′′ ′′ ′ n n n cn 1) 1) 1) ′′ + − + n − ′ ′′ a a − ′′ b ( ( bn ( n ′ ′′ an + + 2 n n + + ′′ ′ ′ + − + − n n n n n ′ ′′ ′ − (6) (7) (8) (9) n n − − − n n n n n − n 1 2 3 6 5 4 7 8 9 10 11 12 n n n n n n n n n n n n 107 86 1. Torus embeddings ′′ ′′ n ′′ n an − ′ ′′ ′ ′ ′′ + n n bn n n ′ − n + + + − ′′ ′ ′ − n n n n n n n n ′ ′′ − − − n − − − − n − n n ′′ ′′ ′′ n n bn ′′ 1) 1) + bn + + ′ b − b ′′ ( ( an ′′ ′ ′ ′ n n n n n + + ′ ′′ − + + − − n n n n ′′ ′ ′ n n − n n n − n n − − ′′ ′′ bn ′′ bn n + + ′ − n ′ ′ ′′ ′′ 2 n ′′ ′ n n n n n + + − ′′ − + − − n n n n n ′′ ′ ′ ′ − − n n − n n − n n an ′′ n ′′ 1) n ′′ ′′ + 1) bn b bn + ( + b + ′′ − ( n ′ ′ ′ ′′ 2 n n bn ′′ n n n + + + − − ′′ − − n n n n n n n ′ ′′ ′ − n n − n n − − − − an ′′ ′′ n cn 1) + ′′ + ′′ ′ n n c ( cn ′′ 1) 1) + + + ′′ an + ′ ′ a + b an ( ′′ ( bn bn ′′ ′ n + + n n + + + ′ ′′ ′ − − − n n n n n n ′ ′′ ′ − n n − n n − − − − n n n (10) (11) (12) (13) (14) 6 7 4 5 9 8 2 3 1 12 11 10 n n n n n n n n n n n 108 n 9. Complete non-singular torus embeddings in dimension 3 87 ′′ n ′′ 1) n − ′′ ′′ 1) n a an ( − 1) ′′ a + + ′′ n ( + ′ ′ n − − a 2 ′′ bn ( bn ′ ′ n − n n + + + ′ ′′ ′ − + − n n n n n n ′ ′ ′′ − n − − − − − n n n ) ′′ n + ′ n ( ′′ b ′′ + n an ′′ ′′ ′′ − + n ′′ n n ′ ′ n − n n + − ′ ′ ′′ + − + n n n n n n n ′ ′′ ′ − − − n − − − n n n n n ′′ n − ′ ′′ n n ′′ 1) − an − ′ ′′ ′ + b n ( n bn ′′ ′ − n n + − + ′ ′ ′′ + + n n n n n n n ′′ ′ − − − − − − − n n n n n ′′ n + ′′ ′ ′ ′′ ′′ n ′′ an bn n bn ′ ′ n + n n + − + ′ ′ ′′ + − + n n n n n n ′ ′′ ′ − − − n − − − n n n n ) ′ n + ′ n + n ( ′′ ′′ b n ′′ n an + − ′′ ′ ′′ ′ + + n n n n ′ ′ − n n + − − ′ ′′ ′ + − n n n n n n n ′ ′′ − n − n − − n − − − n n n n (15) (16) (17) (18) (19) 9 8 7 6 5 3 4 2 1 12 10 11 n n n n n n n n n n n n 109 88 1. Torus embeddings ) ′′ n − ′ n ′′ ( n c ′′ 1) + n + ′′ 1) b bn ( + ′′ d + + ( ′ ′ dn ′′ + n ′′ + an an ′′ ′ n + n n n + + ′′ ′ ′ − − − − n n n n n ′′ ′ ′ n n − − − n n n n − − ′′ ′′ ′′ n an an − ′ + n ′ ′ + ′ ′′ ′′ n n n ′′ ′ n n − 2 n n + − n ′′ − + + + + n n 2 n n n ′ ′′ ′ ′ − n − − − n n n n − n ′′ n 1) + ′′ ′′ a an ( n + + ′′ − ′′ ′ ′ ′ ′′ n ′′ an n n n ′ n 2 n 2 n + + − ′′ − − + + + n n n n ′ ′ ′′ ′ ′ n n n − − n n n n − − ′′ ′′ ′′ an n an + ′′ − + ′ ′′ ′ ′′ ′′ an n n n n ′ n n 2 n + + − − ′′ − + + + n n n n n ′ ′′ ′ ′ 2 n n − − n n n − n n n − ′′ n 1) ′′ + n ′′ a 1) ( ′′ n an − + ′′ ′′ ′ − − a ′′ n ( an n ′ ′ n n n − + + − ′′ ′ − ′ + − n n n n n ′ n ′ ′′ n − − 2 n − n n − n − ′′ ′′ n n ′′ − ′ + ′ ′ n ′ an ′′ n n 2 ′ n ′′ n + n n + − − + ′′ ′ − + + n n n n n ′ ′ n ′′ − n − n n − 2 n n n n n − (20) (21) (22) (23) (24) (25) − 9 8 6 7 5 4 3 1 2 12 11 10 n n n n n n n n n n n 110 n 9. Complete non-singular torus embeddings in dimension 3 89 ′′ ′′ ′′ n ′′ n ′′ n n 2 n − − ′′ ′ − − − ′ n n n ′ ′′ ′ ′ ′ ′′ − n n n n n n − + ′ − − − − + + n n ′ n ′′ n n n − n n n n 2 − n ′′ ′′ n ′′ n 2 n 2 + + + ′′ ′′ ′′ ′ ′ n ′ n n ′′ n n 2 n ′ ′′ n n n − − + − + − − ′ ′ − + n n n n n n ′ ′ ′′ 3 n n n 2 n − − n n n − − ′′ ′′ n n 2 − ′′ ′′ ′′ ′ + ′′ ′′ n ′′ n n n n ′ n ′′ 2 n n 2 n + − − + + + − + + n n n ′′ ′ n ′ ′ n n n − n 2 n n n − − ′′ n ′′ + ′′ n ′′ ′′ n ′ ′ ′′ ′ n n ′′ 2 n n n n ′′ 2 n + n + − − + − ′ + + + n n n n n n ′ ′′ ′ ′ n n n n n n − − − − − − ′′ n ′′ ′′ ′′ ′′ 2 n ′′ ′′ n n n ′′ n n + + 2 n − + ′ ′ ′ − + − − n n n n n n ′ ′ ′ n n − n n − − − n n − − ′′ ′′ n n ′′ n + − ′′ ′′ n ′ ′ − ′ n 2 n n n ′ n − + − + − ′ ′′ ′ − n n n n n n n ′′ ′ n n − 2 n n n − − n 2 − − − ′′ n ′′ 1) n ′′ + ′′ 1) n b bn ( + ′′ 1) d + + + dn ( ′′ ′ ′ c + + cn ( ′′ an an ′′ ′ ′ n + + n n n + + ′′ ′ ′ − − − − n n n n n ′ ′ ′′ − (26) (27) (28) (29) (30) (31) (32) n n − − − n n n n − n 1 5 4 6 3 2 7 8 9 11 10 12 n n n n n n n n n n n n

Chapter 2

Applications

Mumford et al. [63, Chap. 2] generalized the notion of tours embedding 111 to that of toroidal embeddings. Using this they proved, among other things, the important semi-stable reduction theorem is characteristic 0. Here we are concerned with more elementary applications of tours embedding. In this chapter we restrict ourselves to tours embedding over the ﬁeld C of complex numbers, although some of the result have interesting analogues over ﬁelds with non-archimedean rank one valuation.

10 Manifolds with corners associated to torus embeddings

Let U(1) = zǫC; z = 1 be the I-dimensional unitary group, i.e. { | | } I-dimensional real torus. Then the r-dimensional algebraic torus T = r r (C∗) has the r-dimensional real torus CT = (U(1)) as the maximal compact subgroup such that

r r T/CT (R>0) R

where R>0 is the multiplicative group of positive real numbers. Here is a coordinates-free description, which will be useful for torus embeddings. Recall that R0 is the set of non-negative real numbers. We

91 92 2. Applications

have the valuation log ord : C | | R0 − R −−−−−−−−→ −−−−→∼ ∪ {∞} 112 which induces a homomorphism

log | | − ord : C∗ R>0 R. −−−−−−−−→ −−−−→∼ Let N Zr be a free Z-module of rank r and let M be its dual. Then as in . 1 § T = N Z C∗ = Hom (M, C∗) N ⊗ gr is an r-dimensional algebraic torus over C.

Deﬁnition. We denote by CTN the compact (real) torus

CT = N Z U(1) = Hom (M, U(1)). N ⊗ gr The valuation ord applied to the second factor thus induces a surjec- tion

ord : T = Hom (M, C ) | | Hom (M, R> ) N gr ∗ −−−−−−−−→ gr 0 log − Homgr(M, R) = NR −−−−→∼ whose kernel is CTN, thus we have

ord : TN/CTN NR. −→∼ Let (N, ) be an r.p.p.decomposition. Then CT acts on the corre- △ N sponding tours embedding T emb( ) endowed with the classical N △ (Hausdorff) topology instead of the Zariski topology. We then adopt the following definition for the notion introduced by Mumford et al. [61, Chap. 1, . 1]. § Definition. We denote by

ord : T emb( ) Mc(N, ) = T emb( )/CT N △ → △ N △ N the quotient space with respect to the classical topology and call it the manifold with corners associated to T emb( ) N △ 10. Manifolds with corners associated to torus embeddings 93

113 It is indeed a manifold with corners in the usual sense (cf. Borel- Serre [4]) if T emb( ) is non-singular. By Theorem 4.2, T emb( ) is N △ N △ the union of aﬃne open sets

U(σ) = Hom . (σ ˇ M, C) unit semigr ∩ with σ running through , hence Mc(N, ) is the union of its open sets △ △

ord : U(σ)/CTN | | Homu.s.g(σ ˇ M, Ro) −−−−−−−−→∼ ∩ log − Homu.s.g(σ ˇ M, R ). −−−−→∼ ∩ ∪ {∞} s The second description shows that U(σ)/CTN is isomorphic to (Ro) r s × (R>o ) − for some s if σ is non-singular. Proposition 10.1. For an r.p.p.decomposition (N, ) the associated △ manifold with corners Mc(N, ) has an action of NR = R Z N and △ ⊗ is an equivariant partial compactiﬁcation of NR with the orbit decomposition. Mc(N, ) = orb(σ)/CTN △ aσǫ △ such that ord orb(σ)/CTN NR/(R subspace generated by σ). −−→∼ − In particular

dimC orb(σ) = dimR orb(σ)/CT = rankN dim σ. N − Proof. The ﬁrst part is obvious, since Mc(N, ) has an action of △ T /CT = NR. Since orb(σ) = Hom (σ M, C ) by Theorem 4.2, N N gr ⊥ ∩ ∗ we see that

orb d orb(σ)/CTN Homgr(σ⊥ M, R) = NR/(R subspace gen by σ). −−→∼ ∩ − A map h:(N, ) (N , ) of r.p.p. decompositions obviously gives 114 △ → ′ △′ rise to an equivariant continuous map

Mc(h) : Mc(N, ) Mc(N′, ′) △ → △ 94 2. Applications

Example (Mumford). Consider the projective line P1(C), which is the 2-sphere in the classical topology. It corresponds to the r.p.p. decomposition (N, ) = (Z R , 0 , R ). △ ·{ ◦ { } − ◦} CTN = U(1), and the orbits under it are the circles of the same latitude. The quotient Mc(N, ) is the closed interval, a compactiﬁcation △ of the open interval isomorphic to NR = R.

orb

ord (N, ) P (C) Mc(N, ) △ 1 −−→ △ Example. For a 2-dimensional (N, ), Mc(N, ) looks like this △ △

115 We encounter many other examples later in . 11, 13, 14, 15. Since §§ 10. Manifolds with corners associated to torus embeddings 95 the decomposition of Mc(N, ) into NR-orbits faithfully reﬂects the in- △ cidence among the T -orbits of T emb ( ) the pictures we can draw of N N △ Mc(N, ) in dimensions 2 and 3 will give us a good geometric insight. △ Remark. Over a ﬁeld k with a non-archimedean rank one valuation

ord : k R , → ∪ {∞} we have an obvious analogue of Mc(N, ). △ Although the following description of the fundamental group has nothing to do with the manifolds with corners, we state it here, since it will be very convenient later.

Proposition 10.2 (Mumford). Let (N, ) be an r.p.p.decomposition and △ let T emb( ) be the corresponding torus embedding over C endowed N △ with the classical topology. Then we have the following canonical description of the fundamental group.

d π1(TN emb( )) = N/(the subgroup gen (σ N)). △ σ[ ∩ ∈△ Proof. For simplicity, let T = T and X = T emb( ). We have a N N △ well-known isomorphism

N ∼ Hom (U(1), T) = π (T) −→ gr 1 which sends n N to the loop U(1) u u Hom (M, C ) = T, ∈ ∋ → ∈ gr ∗ where , : M N Z is the dual pairing as before. By Theorem 4.2, X 116 h i × → is covered by U(σ) = Hom . . (σ ˇ M, C) T with σ running through . u s g ∩ ⊃ △ By generalized van Kampen’s theorem in Olum [48], it is thus enough to show that the canonical homomorphism

N = π (T) π (U(σ)) 1 → 1 is surjective with the subgroup N’ generated by σ N as the kernel. ∩ Note that N′ is a direct summand of N. 96 2. Applications

Since U(σ) is non-canonically isomorphic to the product of the algebraic torus (N/N ) Z C and the lower dimensional T -embedding ′ ⊗ ∗ N′ corresponding to σ considered as a cone in NR′ , it is obviously enough to show that U(σ) is simply connected if σ generates NR. First of all, N = π (T) is mapped surjectively onto π (U(σ)), since U(σ) T is of 1 1 − real codimension 2 in U(σ), by the so-called general position argu- ≥ ment. On the other hand, for n σ N and 0 ε 1, the map ∈ ∩ ≤ ≤ U(1) u (εu) Hom . . (σ ˇ M, C) = Uσ ∋ 7→ ∈ u s g ∩

establishes a homotopy which kills n in π1(U(σ)) as ε goes to 0, since < m, n > 0 for m σˇ M. Since σ N generates N as a group, we ≥ ∈ ∩ ∩ are done. For illustrative examples, we refer the reader to . 13, 14. §§

11 Complex tori

117 The following “multiplicative” formulation of complex tori and polar- izations was given by Tate in his lecture in July 1967. It was used eﬀec- tively by Mumford et al. [61]. In this way, we can also formulate their analogues in the non-archimedean case, or even in the ideal-adic case. (cf. Morikawa [36], McCabe [33], Mumford [38] and Gerritzen [11].) Given a g-dimensional complex torus X, there exists a symmetric g g matrix τ = (τ ) with the positive deﬁnite imaginary part im(τ) × jk such that X = Cg/(Zg + Zτ + + Zτ ) 1 ··· g where τ1,...,τg are the row vectors of τ. We have a surjective homomorphism g g C (C∗) → sending (z1,..., zg) to (exp(2πiz1),..., exp(2πizg)). Let Γ be the image of the lattice, which is a free abelian subgroup of g (C∗) of rank g such that

g X = (C∗) /Γ. 11. Complex tori 97

This description is very eﬃcient, since it gets rid of the redundancy Zg. It also ﬁts in nicely with torus embeddings and will give us good insight into the construction of degenerating families. Here is the coordinate- free approach: Let N Zg be a free abelian group of rank g with the dual M. As 118 before, let T = TN = Homgr(M, C∗) be the algebraic torus.

Deﬁnition. A period for T = Homgr(M, C∗) is a homomorphism

q : Γ T, γ qγ → 7→ from a free Z-module Γ of rank g, which is injective with compact cokernel X = T/qΓ, a complex torus. Composed with the valuation ord(?) = log ? : C R, q gives − | | ∗ → rise to a homomorphism

ord q : Γ NR = Mc(N, 0 ). ◦ → { } The injectivity and the compactness of the cokernel of q are equivalent to those of ord q. ◦ For γ Γ and m M, let Q(γ, m) = qγ(m) C . Thus we have a ∈ ∈ ∈ ∗ non-degenerate biadditive pairing

Q : Γ M C∗. × → Deﬁnition. For a period q : Γ T = Hom (M, C ), let → gr ∗

qˆ : M Tˆ = Hom (Γ, C∗) → gr be the period given byq ˆm(γ) = qγ(m). We call Xˆ = Tˆ/qˆM the dual complex torus. Recall that m M gives rise to the character T t e(m)(t) C ∈ ∋ 7→ ∈ ∗ (cf. . 1). Consider the formal Laurent series §

θ(t) = ame(m)(t) mXM ∈ for t T and a C. The following is well-known: ∈ m ∈ 98 2. Applications

Proposition 11.1. The ring Ho1(T) of holomorphic functions consists 119 of the Laurent series which are convergent everywhere on T. The units in Ho1(T) are of the form

ae(m) for a C∗ and m M. ∈ ∈ It is also well-known that the positive divisors on the complex torus X = T/qΓ are in one to one correspondence with the Γ-invariant positive divisors on T, which are of the form div(θ) for 0 , θ Ho1(T). From ∈ what we saw above, the Γ-invariance means that there exist maps

c : Γ C∗ and Γ M → → such that θ(qγt)c(γ)tα(γ) = θ(t) for all γ Γ. (*) ∈ Such θ(t) is called a theta function. We see immediately the following:

Lemma 11.2. The pair (α, c) satisﬁes the following conditions and is called a theta type for the period q.

(i) α : Γ M is a Q-symmetric homomorphism, i.e. →

α(γ + γ′) = α(γ) α(γ′) and Q(γ′,α(γ)) = Q(γ,α(γ′)). ·

(ii) c : Γ C is a quadratic character with respect to Q(?,α(?)), i.e. → ∗

c(γ + γ′)/c(γ)c(γ′) = Q(γ,α(γ′)).

Given theta functions θ(t) and θ′(t) with the theta types (α, c) and 120 (α′, c′), respectively, we see that θ(t)θ′(t) is of the theta type (α+α′, cc′). Moreover, div(θ) and div (θ′) give rise to linearly equivalent divisors on X if and only if there exists m M such that ∈

α′ = α and c′ = c Q(?, m). · Thus we conclude: 11. Complex tori 99

Proposition 11.3. We have an isomorphism

Pic(X) = theta types (α, c) / (0, Q(?, m)); m M { } { ∈ } and an exact sequence

0 Xˆ Pic (X) HomQ symm(Γ, M) 0. → → → − → Given a theta type (α, c), let Γ(α, c) denote the corresponding line bundle on X. Proposition 11.4. Given a theta type (α, c) for q, we have a homomorphism Λ(L(α, c)) : X Xˆ → induced by α : T = Hom (M, C ) Tˆ = Hom (Γ, C ). ◦ gr ∗ → gr ∗ (1) α : Γ M is injective if and only if ord Q(?,α(?)) : Γ Γ R → ◦ × → is non-degenerate. In this case, Λ(α, c)) is an isogeny whose degree is equal to the order deg(α) of coker [α : Γ M], and L(α, c) is called → non-degenerate. (2) If, moreover, ord Q(?,α(?)) : Γ Γ R is positive deﬁnite, then ◦ × → L(α′, c) is an ample line bundle, and α is called a polarization. In this case, the theta functions of the theta type (α, c) form a C-vector space of dimension deg(α). The polarization is principal if and only if 121

α : Γ ∼ M, i.e. Λ(L(α, c)) : X ∼ Xˆ. → → When we have a principal polarization α on X, we can identify Γ with M via α. Thus a principally polarized g-dimensional complex torus X is determined by a symmetric biadditive map

Q : M M C∗ × → with P = ord Q : M M R positive definite so that ◦ × → X = T/ Q(m, ?); m M . { ∈ } Let us fix M = Zg with the Z-basis m ,..., m . Let { 1 g} symmetric biadditive Q : M M C∗ PdSym(M M, C∗) = × → × with ord Q positive definite ◦ 100 2. Applications

We then have a versal family over PdSym(M M, C ) of principally × ∗ polarized g-dimensional complex tori, whose total space is the quotient of T PdSym(M M, C ), with T = Hom (M, C ), by the action of M × × ∗ gr ∗ deﬁned by (t, Q) (Q(m, ?)t, Q) for m M. 7→ ∈ Pdsym(M M, C ) is an open subset of × ∗ Sym (M M, C = symmetric biadditive Q : M M C , which, × ∗ { × → ∗} by component wise multiplication, is a g(g+1)/2-dimensional algebraic torus over C with the character group 122 S 2(M) = the symmetric product of M of degree 2 and the group of one parameter subgroups

Sym(M M, Z) = symmetric bilinear mapsM M Z . × { × → }

As usual, let S g be the Siegel upper half plane of complex symmetric gxg matrices τ = (τ jk) with the positive deﬁnite imaginary part Im(τ) > 0. Then an element AB CD! of the Siegel modular group S p (Z) acts on S via τ (Aτ + B)(Cτ + g g 7→ D) 1. The map τ Q deﬁned by − 7→

Q(m j, mk) = exp(2πiτ jk)

establishes an embedding

1 B S g/ S pg(Z) ∼ PdSym(M M, C∗) S ym(M M, C∗). { 0 1! ∈ } −→ × ⊂ × The first step in compactifying the moduli space of principally polarized complex tori in Mumford et al. [61] is to choose a nice S ym(M × M, C∗)-embedding corresponding to an admissible r.p.p.decomposition (S ym(M M, Z, ) which is invariant under the canonical action of × △ Aut(M) GL (Z) with /Aut(M) finite and fills up the convex hull g △ in S ym(M M, R) of the set of positive semi-definite integral forms. × The Delony-Voronoi decomposition is such a decomposition and was used by Namikawa [46] for his Delony-Voroni compactification of 11. Complex tori 101

123 the moduli space, over which he could even construct a family of what he calls stable quasi-abelian varieties. Here is a brief coordinate-free account of the part relevant to us: Let P : M M R be a positive semi-definite bilinear form. P × → 1/2 defines a pseudo-norm x = P(x, x) on MR. For x MR let || ||p ∈ w(x, P) = m M; x m P = min x m′ P . { ∈ || − || m M || − || } ′∈ The convex hull D(x, P) in MR of w(x, P) is an (unbounded if P is not definite) polyhedron in MR and is called a Delong cell. The set Del(MR, P) of Delony cells is the Delony decomposition and is invariant under the translation action of M. On the other hand,

V(x, P) = y NR; m′, y + P(m′, m′ + 2m) { ∈ h i ≥ for all m′ M and allm w(x, P) ∈ ∈ } is a polyhedron in NR and is called a Voronoi cell. Let P : MR NR ∗ → be the canonical linear map defined by x P(x , ?). Then V(x, P) is ′ 7→ ′ the image under 2P∗ of x′ MR; x′ m P = min x′ m′ p for − { ∈ || − || m M || − || ′∈ all m w(x, P) . The set Vor(NR, P) of Voronoi cells is the Voronoi ∈ } decomposition of the image of P : MR NR and is invariant under the ∗ → translation action of 2P∗(M). This is the original definition by Voronoi and is different from the usual one, for instance in Oda-Seshadri [49].

Definition . Positive semi-definite bilinear forms P, P : M M R 124 ′ × → are equivalent if Del(MR, P) = Del(MR, P′). An equivalence class of positive semi-definite forms is a polyhedral cone, a Delony-VoronoiP cone, in S ym(M M, R). Let Del-Vor × be the set of such equivalence classes. Then we have an admissible r.p.p.decomposition (Sym(M M, Z, Del-Vor), × the Delony-Voronoi decomposition, which is Aut(M)-invariant with Del- Vor/Aut(M) finite and which fills up the convex hull of the set of positive semi-definite integral forms. 102 2. Applications

On the other hand, for a Delony-Voronoi cone and a Delony cell D of the Delony decomposition common to forms inP , P σ( , D) = (y, P) NR ; m′, y + P(m′, m′ + 2m) 0 X ∈ × X h i ≥ for all m′ M and all m M D ∈ ∈ ∩

is a polyhedral cone in NR S ym(M M, R). The set Mixed of these ⊕ × cones, mixed cones, gives rise to an r.p.p.decomposition

(N Sym(M M, Z), Mixed) ⊕ × called the mixed decomposition. The second projection induces a map of r.p.p.decompositions

p : (N Sym (M M, Z), Mixed) (S ym(M M, Z), Del vor). 2 ⊕ × → × − 125 Thus we have an equivariant morphism of torus embeddings

ρ : TN Sym (M M, Z emb(Mixed) TSym (M M,Zemb(Del Vor) ⊕ × → × − X|| || B a “family of semi-universal coverings” of Namikawa [46, . 8]. § For various reasons (cf. ibid. (13.8)), it is more convenient to consider the level 2ν action (ν 1) of the lattice M on . It is induced by ≥ B the action of M on N S ym(M M, Z) by ⊕ × (y, P) (y + P(2νm, ?), P) for m M 7→ ∈ which preserves the mixed decomposition. On T PdSym (M M, C) × × ⊂ , it coincides with the action B (t, Q) (Q(2νm, ?)t, Q) for m M. 7→ ∈ Let X X be the interior of the closure of PdSym (M M, C ). ◦ ⊂ × ∗ Then = ρ 1(X ) is the inverse image under ord of the open set in B◦ − ◦ Mc(N Sym (M M, Z), Mixed) consisting of NR PdSym (M M, R) ⊕ × × × and the boundary at inﬁnity. 11. Complex tori 103

The level 2ν action of M induced on the open set, hence also its action on , is properly discontinuous and fixed point free. Thus we B◦ have a family (ν) (ν) ω : a = B◦/(level 2ν action of M) X ◦ → of “stable quasi-abelian varieties of dimension g and of degree ν” of Namikawa [46, . 13]. § We now look more specifically at one parameter degenerations of 126 principally polarized g-dimensional complex tori which was studied in detail by Mumford [38], Nakamura [44], [45] and Namikawa [46, . 17]. § See also Namikawa [47] for toroidal degenerations. Definition. We denote by U = λ C; λ < 1 the open unit disk and { ∈ | | } U = U 0 the punctured disk. ∗ −{ } It will be more convenient later to think of U∗ and U as open sets in TZℓ emb( R ℓ, 0 ) = C for a free Z-module Zℓ of rank one with the { 0 { }} base ℓ with l U = ord− (R> ℓ ) 0 ∐ {∞} l U∗ = ord− (R>0ℓ) where Mc(Zℓ, Roℓ, 0 ) = Rℓ . { { }} ∐g {∞} As before, we fix M Z and its dual N and let T = TN = Hom (M, C ). Given a family π : Y U of principally polarized gr ∗ ∗ ∗ → ∗ g-dimensional complex tori, we are interested in extending it to a nice family π : Y U which is proper and flat. → The semi-stable reduction theorem allows us to replace U by its covering ramified at 0 and reduces the problem to the case of unipotent monodromy. (For details see Namikawa [46, . 17]). We may thus § assume that there exists a homomorphic map U λ Qλ (the ∋ 7−→0 ∈ closure of PdSym(M M, C in Sym(M M, C )) with ord Qλ positive × ∗ × ∗ definite for λ , 0, and a positive semi-definite B S ym(M M, Z) such 127 ∈ × that the restriction of ord0 Q to x M ; B(x, x ) = 0 for all x M 0 { ∈ R ′ ′ ∈ R} is positive definite and that Y is the quotient of T U under the action ∗ × ∗ of M defined for m M by ∈ B(m,?) f˜m(t, λ) = (Qλ(m, ?)λ , t, λ). 104 2. Applications

Let N˜ = N Zℓ and consider the action of M on N˜ deﬁned for ⊕ m M by ∈ h˜ (n) = n for n N m ∈ h˜m(ℓ) = ℓ + B(m, ?).

Then f˜ is the composite of the translation by (Qλ(m, ?), 1) T ˜ = m ∈ N T C and the automorphism of T ˜ as an algebraic group induced by × ∗ N h˜m. To make sure that there is a nice open set V below, let (N˜ , ˜ ) be an △ r.p.p.decomposition invariant under h˜ for all m M such that the union m ∈ of cones in ˜ coincides with △

0 B∗(MR) R> ℓ , { }∪{ × 0 }

where B : MR NR is the linear map deﬁned by B (x) = B(x, ?). ∗ → ∗ Such ˜ can be most easily described as the join of 0 with a polyhedral △ decomposition

˜ (B∗(MR) + ℓ) △ ∩ of the affine subspace B∗(MR)+ℓ by bounded polyhedral invariant under the translation by the lattice B(m, ?); m M . It is a special case of { ∈ } 128 torus embeddings over a discrete valuation ring in Mumford et al. [63, Chap.IV, . 3]. § Let V T ˜ emb( ˜ ) be the inverse image under ord of the union of ⊂ N △ NR R> ℓ and the boundary at infinity of Mc(N˜ , ˜ ). Then there exists × 0 △ a unique extension of the action of M onto V, denoted again by f˜m for m M, which is properly discontinuous and fixed point free. Then ∈ π : Y = V/ f˜ ; m M U { m ∈ }→ is the sought-for extension of the family π : Y U . ∗ ∗ → ∗ An example of such ˜ was obtained by Namikawa [46] and Naka- △ mura [44], which is of the following form:

˜ = σ˜ (D); D Del(MR, B) , △ { ∈ } 11. Complex tori 105 i.e. of the Voronoi type, where

2 m′, y + wB(m′, m′ + 2m) 0 σ˜ (D) = (y, wℓ N˜ R = N˜ R Rℓ; h i ≥ . { ∈ ⊕ for all m M and m M D ′ ∈ ∈ ∩ For the relation of all these to the compactiﬁcation of the generalized Jacobian varieties, we refer the reader to Namikawa [46, .18] and § Nakamura [44] in the C case and to Oda-Seshadri [49] and Ishida [26] in the general case.

Example 11.1. (1) Elliptic curves (g = 1) In this case the versal family is already a one-parameter family. With the canonical principal polarization, an elliptic curve X over C is of the following form: Let N = M = T = Z. A period

q : Z C∗ = T → is determined by the image λ = ql exp(2πiτ) C with 0 < λ < 1, i.e. 129 ∈ ∗ | | im(τ) > 0. Hence Z X = Xλ = C∗/λ where λZ = λa; a Z . Dividing everything out by CT = U(1), we get { ∈ }

ord q : Z R = NR. ◦ → As before, let U = λC; λ < 1 and U = U 0 . Then we have a { | | } ∗ −{ } versal family Z π∗ : Y∗ = (C∗ U∗)/ f˜ U∗ × → 1 of elliptic curves Xλ = (π ) (λ), where f˜ is the automorphism of C ∗ − ∗ × U∗ deﬁned by f˜(t, λ) = (λt, λ). This can be extended to a family

π : Y U → whose fiber over λ = 0 is the rational curve with a node obtained by identifying 0 and of P . Indeed, let N˜ = N Zℓ with the Z-basis ∞ 1 ⊕ n,ℓ . C U is the inverse image under ord of the upper half plane { } ∗ × ∗ R R> ℓ of Mc(N˜ , 0 ) = N˜R. Let X˜ = T ˜ emb( ˜ ), where ˜ consists of × 0 { } N △ △ the faces of σ˜ ν = Ro(n + νℓ) + Ro(n + (ν + 1)ℓ) 106 2. Applications with ν running through Z. Let h˜ be the automorphism of N˜ defined by h˜(n) = n and h˜(ℓ) = ℓ + n. Obviously h˜ preserves ˜ , hence defines an 130 △ automorphism f˜ of X˜. Mc(N˜ , ) looks like the picture below and Mc(h˜) △ (i) acts as the translation by wn for points of NR + wℓ and (ii) transforms the boundary components orb(σ ˜ ν)/CTN˜ and orb(Ro(n + νℓ))/CTN˜ at infinity to the next ones corresponding to ν + 1. Since the group generated by Mc(h˜) thus acts properly discontinuously without fixed point on the “upper half”, the group generated by f˜ also acts properly discontinuously without fixed point on the open set V X˜ which is the inverse image under ord of the upper half of ⊂ Mc(N˜ , ˜ ). Then the horizontal projection induces △

π : Y = V/ f˜Z U. →

1 Note that π− (0)/CT is the circle obtained by identifying the two end points of a closed interval. We refer the reader to Mumford et al. [61, Chap. 1. . 4] for the § certiﬁcation of universal elliptic curves.

0 0 11. Complex tori 107

131 (2) Principally polarized 2-dimensional complex tori Let M = Z2 with a Z basis m , m and let N the dual of M − { 1 2} with the dual basis n , n . Then the Delony-Voronoi decomposition { 1 2} of Sym(M M, R) consists of the Aut(M)-translates of ×

= R ℓ1 + R ℓ2 + R ℓ3 ◦ ◦ ◦ X3 and its faces 2 = R ℓ1 + R ℓ2, 1 = R ℓ2 and 0 = 0, where ◦ ◦ ◦ ℓ1, ell2,ℓ3 SymP M M, Z are deﬁnedP as follows: P ∈ ×

ℓ1(m1, m1) = 1, ℓ1(m2, m2) = 0, ℓ1(m1, m2) = 0,

ℓ2(m1, m1) = 0, ℓ1(m2, m2) = 0, ℓ2(m1, m2) = 0, ℓ (m , m ) = 1, ℓ (m , m ) = 1, ℓ (m , m )= 1, 3 1 1 1 2 2 3 1 2 − (cf. Namikawa [, 2.7)]). In S 2(M), m2 + m m , m2 + m m , m m 132 { 1 1 2 2 1 2 − 1 2} form the Z-basis dual to ℓ ,ℓ ,ℓ . Thus the parameter space X of the { 1 2 3} ◦ family of semi-universal coverings is covered by the Aut(M)-translates of the open set isomorphic to the interior of the closure in A of (λ , λ , 3 { 1 2 108 2. Applications

λ ) A ; log λ log λ + log λ log λ + log λ log λ > 0, and 3 ∈ 3 | 1| | 2| | 2| | 3| | 3| | 1| λ λ < 1 . | 1 3| } We now look at semi-stable one parameter degenerations of 2 - dimensional principally polarized complex tori. It is well-known that, up to the Aut(M)-equivalence, positive semi-deﬁnite B S ym(M M, Z) ∈ × is of the following from:

(1) B = 0

(2) B(m1, m1) = 0, B(m2, m2) = b, B(m1, m2) = 0

(3) B(m1, m1) = a, B(m2, m2) = b, B(m1, m2) = 0 (4) B(m , m ) = a + c, B(m , m ) = b + c, B(m , m ) = c 1 1 2 2 1 2 −

˜ ˜ ˜ ˜ for positive integers a, b, c. For simplicity, let h1 = hm1 and h2 = hm2 . They are the identity on NR. In case (1), a decomposition ˜ satisfying our requirements is neces- △ sarily of the form ˜ = 0 , R ℓ . △ {{ } o } Hence V = T U and π 1(0) is the complex torus × −

T/ Q (m, ?); m M . { 0 ∈ }

In case (2). h˜ 1(ℓ) = ℓ and h˜2(ℓ) = ℓ + bn2. For any divisor b′ 133 of b, the decomposition of Rn2 + ℓ into “intervals” of length b′ as in picture below gives rise to ∆˜ satisfying our requirements. Consider the Z P1-bundle W over the elliptic curve C∗/Q0(m1, m1) obtained by dividing C P out by the group generated by the automorphism (t , t ) ∗ × 1 1 2 7→ (Q0(m1, m1)t1, Q0(m1, m2)t2). Then by Theorem 4.2 (iii), we see easily 1 that π− (0) is a cycle of b/b′ copies of W obtained by identifying the 0-section of one W with the -section of the next. If Y is required to ∞ be non-singular, take b′ = 1 and get the Voronoi type degeneration of Namikawa and Nakamura. 11. Complex tori 109

The boundary at infinity of

In case (3), h˜ 1(ℓ) = ℓ+an1 and h˜ 2(ℓ) = ℓ+bn2. Again for divisors a′ and b′ of a and b, take the decomposition of Rn1 + Rn2 + ℓ into “rectan- gles” of size a′ b′ as in the picture below (or a suitable sub division of × 1 it, if Y is required to be non-singular). Without any subdivision, π− (0) consists of (a/a )(b/b ) copies of P P glued along 0 P , P , ′ ′ 1 × 1 × 1 ∞× 1 P 0, P ’s like a “toroidal net”, again by Theorem 4.2 (iii). If 1 × 1 ×∞ a′ = b′ = 1, we again get the Voronoi type degeneration 134

The boundary at infinity of

In case (4), h˜ (ℓ) = ℓ +(a+c)n cn and h˜ (ℓ) = ℓ cn +(b+c)n . 1 1 − 2 2 − 1 2 There are many diﬀerent decompositions of Rn1 + Rn2 + ℓ which give rise to allowable ∆˜ ’s. The “Namikawa decompositions” in Oda-Seshadri 110 2. Applications

[49], among which is the Voronoi type decomposition, are examples. For instance, let a = b = c = 1, and consider the two decompositions below.

The boundary at infinity of

1 It is the Voronoi type decomposition and π− (0) consists of two copies of P2 glued along the three coordinates axes, by Theorem 4.2 (iii) and (7.4).

The boundary at infinity of 135 1 In this case, π− (0) is obtained by gluing together, as in the picture above, two copies of P2 and one W obtained by blowing P2 up along the three coordinates vertices, by Theorem 4.2 (iii) and Proposition 6.7. Note that Y is non-singular in this case and each of the six rational curves on W has the self-intersection number 1. This is Deligne’s example − described in Mumford [38]. 12. Compact complex surfaces of class VII 111

12 Compact complex surfaces of class VII

Kodaira [30] classified non-singular compact complex surfaces into 136 seven classes. Among then, the last class VII has been the least un- derstood until recently. Definition. (Kodaira [30, II, Thm 26]) A non-singular compact complex surface X is said to be of class VII (resp. of class VII0) if the first Betti number b1(X) = 1 (resp. b1(X) = 1 and X is minimal, i.e. without exceptional curves of the first kind). As usual, let i bi(X) = dimC H (X, C) p,q q p h (X) = dimC H (ΩX) and let c1(x) and c2(X) be the Chern classes of X. + Moreover, let b and b− be the number of positive and negative eigen- values of the cup products on H2(X, R). Then Kodaira [30, I, Thm 3] showed that a class VII surface X has the following numerical characters: h0,1 = 1, h1,0 = h2,0 = 0 + b1 = 1, b = 0 2 c = c = b = b− − 1 2 2

Deﬁnition. An, r-dimensional compact complex manifolds X is called a Hopf manifold if its universal covering manifold X˜ is isomorphic to Cr − 0 . If, moreover, π (X) Z, then X is called a primary Hopf manifold. { } 1 (See e.g. [1] for a generalization, the Calabi-Eckmann manifold.)

Kodaira [30, III, Thm 41] gave a topological characterization of 2- 137 dimensional Hopf manifolds, i.e. Hopf surfaces : b2 = 0 and π1(X) contains an inﬁnite cyclic subgroup of ﬁnite index. Again according to Kodaira [30, II, Thm 30], a primary Hopf surface X is of the following form : X = (C2 0 )/γZ, −{ } 112 2. Applications

where γZ is the group of automorphisms generated by γ of the form

b γ(z, z′) = (λz + µ(z′) , λ′z′)

with a positive integer b and λ, λ , µ C satisfying ′ ∈

0 < λ λ′ < 1 | | ≤ | | b (λ (λ′) )µ = 0. −

We have the following characterizations of some of the class VII0 surfaces:

Theorem . (Kodaira [K3, II, . 9& . 10]) X is a class VII surface § § 0 with b2 = 0 and containing at least one curve if and only if X is either

a Hopf surface, or

a certain elliptic surface over P1 with at most non-reduced non- singular ﬁbers, which is obtained from the product of P1 and an elliptic curve by a ﬁnite succession of “logarithmic transforma- tions”.

138 Theorem (Inoue [22]). X is a class VII0 surface with b2 = 0 containing no curve, and with a line bundle L such that H0(Ω1 L) , 0 if and only X ⊗ if X is isomorphic to one of the surfaces

+ S M, S N,p,q,r and S −p,q,r

constructed by Inoue.

All these class VII0 surface satisfy b2 = 0. Recently, Inoue [23], [24] and Kato [29] constructed series of example with b2 , 0. As we see in . 14 and . 15, these can most easily be described in terms of § § torus embeddings. In this connection, the following result of Kato [29] is of utmost importance: 13. Hopf surfaces and their degeneration 113

Deﬁnition. A global spherical shell in an r-dimensional compact complex manifold X is an open submanifold isomorphic to

z = (z ,..., z )εCr; 1 ε< z < 1 + ε { 1 r − | | } with 0 <ε< 1 and z 2 = z 2 + + z 2, such that the complement in | | | 1| ··· | r| X is connected.

An elliptic curve, for instance, contains global spherical shells.

Theorem. (Kato [29]) Let r 2. If an r-dimensional compact complex 139 ≥ manifold X contains a global spherical shell, then X is a deformation of (hence is diffeomorphic to) the blowing up of a primary Hopf manifold along a finite number of points. In particular, X is non-Kähler and π1(X) = Z. As we see in . 14 and . 15, Kato could also show that all the § § examples of Inoue with b , 0 dealt with in . 14 and . 15 contain 2 § § global spherical shells, construct a versal family of deformations for the Inoue surfaces in . 14 (cf. Theorem 14.2), and construct may new class § VII0 surfaces with b2 , 0 which contain global spherical shells (cf. Remark after Thm. 14.1).

13 Hopf surfaces and their degeneration

In this section, we deal with those Hopf surfaces which can be described in terms of torus embeddings. Although we restrict ourselves to the complex case, the non-archimedean analogue might be interesting to formulate. (cf Gerritzen-Grauert [15]). Let us ﬁrst look at the primary Hopf surface

X=˙ (C2 0 )/γZ −{ } 114 2. Applications

where γ(z, z ) = (λz, λ z ) for (z, z ) C2 0 and λ, λ εC with 0 < λ < ′ ′ ′ ′ ∈ −{ } ′ | | 1, 0 < λ < 1. | ′| Since γ is deﬁned in terms of monomials, X can be described by 140 means of torus embeddings as follows: Let N Z2 with a Z-basis n, n , { ′} and consider the r.p.p.decomposition

= R n, R n′, 0 . △ { o o { }} Then obviously T emb( ) C2 0 , which is simply connected N △ −{ } for instance by Proposition 10.2. γ is the translation action by (λ, λ′) = ?,n ?,n λ λ h ′i T . Thus it induces the translation actionγ ¯ by ( log λ ) h i ′ ∈ N − | | n + ( log λ )n N on the manifold with corners Mc(N, ). As in the − | ′| ′ ∈ R △ picture below, the action ofγ ¯ Z on Mc(N, ) is properly discontinuous △ and ﬁxed point free, and has a compact fundamental domain. Thus the action of γZ on T emb( ) has the same properties, and we conclude N △ that X = T emb( )/γZ N △ is a non-singular compact complex surface with π1(X) = Z. Moreover, X contains mutually disjoint elliptic curves

Z Z E = orb(Ron′)/γ C∗/λ Z Z E′ = orb(Ron′)/γ C∗/λ′ .

By Proposition 6.6, the canonical bundle of X is obviously

ω = 0 ( E E′). X X − − Since E E = 0 and, furthermore, E2 = E 2 = 0, as we see below, we · ′ ′ have b = c2 = 0. Indeed, the vertical projection N Zn induces 2 − 1 → orb(R n ) T emb( ) orb(R n) / C o ′ ⊂ N △ − o ∗ C || ≀ C ∗ × Z 141 on which γ acts by γ(z, z′) = (λz, λ′z′). Dividing these out by λ , we get

Z E X E C∗/λ . ⊂ − −→ 13. Hopf surfaces and their degeneration 115

X E′ is obviously the total space V(L) of a line bundle L of degree 0 − Z on the elliptic curve C∗/λ and E is its 0-section. Thus

E2 = deg L = 0. − X is thus a rather curious compactiﬁcation of V(L). X has a global spherical shell indicated in the picture.

fund. domain

For a positive integer a, let F¯a be the non-projective algebraic sur- 142 face with an ordinary double curve obtained by identifying the 0-section and the -section of the rational ruled surface ∞

F = P(0P 0P (a)) a 1 ⊗ 1 116 2. Applications

by means of the map (z, 0) (φ(z), ) for φ Aut (P ). Assuming 7→ ∞ ∈ 1 that the coeﬃcients of φ as a linear fractional transformation are small, Kodaira [30, III, Thm 45] showed:

Theorem. There exists a complex analytic family over the unit disk

π(a) : Y(a) U = λ C; λ < 1 −→ { ∈ | | } such that 1 π(a)− (0) = F¯a 1 , and π(a)(−λ) is a Hopf surface for λ 0. Using torus embeddings, we prove this theorem in the special case where φ is the identity. The proof can be modiﬁed so that it works for φ(z) = cz , c C . ± ∈ ∗ For this purpose, let us ﬁrst consider the following Hopf surface: As before, let N Z2 with a basis n, n . For a positive integer a, let { ′}

(a) = R n, R ( n + an′), 0 . △ { o o − { }} Then by Proposition 10.2, we see that

π (T emb( (a)) Z/aZ. 1 N △ 143 In fact the universal covering space is T emb( (a)) C2 0 , where N′ △ −{ } N N is the subgroup generated by n and n + an . By (7.6 ), we ′ ⊂ − ′ ′ also see that T emb( (a)) is the complement of the 0-section and the N △ -section of F . ∞ a For λ in the punctured unit disk U∗, consider the translation action γλ on T emb( (a)) of N △ λ T . ∈ N Then the induced actionγ ¯λ on Mc(N, (a)) is the translation by △ ( log λ )n , hence is properly discontinuous, ﬁxed point free with a − | | ′ compact fundamental domain. Thus we see that

Z Xλ(a) = T emb( (a))/γ N △ λ 13. Hopf surfaces and their degeneration 117 is a Hopf surface with

π1(Xλ(a)) Z x Z/aZ.

We then consider N˜ = N Zℓ and its automorphismγ ˜ deﬁned by ⊕

h˜(n) = n, h˜(n′) = n′, h˜(ℓ) = ℓ + n′.

Let (N˜ , ˜ (a)) be the r.p.p.decomposition consisting of the faces of 144 △ σν, σν′ with ν running through Z, where

σν = R n + R (ℓ + νn′) + R (ℓ + (ν 1)n′) o o o − σ′ = R ( n + an′) + R (ℓ + νn′) + R (ℓ + (ν 1)n′). ν o − o o −

Thus ˜ (a) NR = (a) and ˜ (a) induces on the aﬃne subspace NR + ℓ △ ∩ △ △ the polyhedral decomposition ˜ (a) (NR + ℓ) as in the picture below. △ ∩ h˜ preserves ˜ (a) and gives rise to an automorphismγ ¯ of non-singular △ T ˜ emb( ˜ (a)). N △ Note that the boundary at inﬁnity of Mc(N˜ , ˜ (a)) consists of two “walls” △ orb(R n)/CT ˜ and orb(R ( n + an ))/CT ˜ as well as the “roof” as in o N o − ′ N the picture. The horizontal projection induces

π˜(a) : T ˜ emb( ˜ (a)) TZℓ( R ℓ, 0 ) = C. N △ → { o { }} Let V be the inverse image of the unit disk U byπ ˜(a). Thus ord(V) is the upper half of Mc(N˜ , ˜ (a)). △ 118 2. Applications

The action ofγ ¯ Z on V is properly discontinuous and ﬁxed point free, 1 γ˜ coincides with γλ above onπ ˜(a) (λ) = T emb( (a))x λ . Hence − N △ { }

π(a) : Y(a) = V/γ˜ Z U →

is the sougth-for family with

1 1 π(a)− (0) = F¯a and π(a)− (λ) = Xλ(a) for λ , 0

by Theorem 4.2 (iii) and (7.6′).

0 14. Inoue’s examples of class VII0 surfaces... 119

0 0

145

14 Inoue’s examples of class VII0 surfaces with b2 = 1

Torus embeddings are very convenient to describe Inoue’s examples of class VII0 surfaces with b2 , 0. In this section we describe the ﬁrst series of his examples in [23]. Besides, we will be able to describe their degeneration easily in terms of torus embeddings. We also sketch Kato’s generalization of Inoue’s construction which provides us with many new 146 examples. As before, let N Z2 with a basis n, n . Consider the r.p.p. de- { ′} composition (N, ), where △

= R n′ and the faces of σν for ν Z △ { o ∈ } σν = R (n + νn′) + R (n + (ν 1)n′). o o − By Proposition 10.2, T emb( ) is simply connected. Let h be the auto- N △ 120 2. Applications

morphism of N deﬁned by

h(n) = n + n′, h(n′) = n′.

h obviously preserves and gives rise to the automorphism h of TN △ ∗ emb( ). △ For λ U = λ C; 0 < λ < 1 , consider the automorphism ∈ ∗ { ∈ | | } ?,n γλ = (the translation by λh i) h ◦ ∗ of T emb( ). For (z, z ) T , we see that n △ ′ ∈ N γλ(z, z′) = (λz, zz′), hence ν ν ν(ν 1)/2 ν γ (z, z′) = (λ z, λ − z z′) for ν Z. λ ∈ γλ induces an automorphismγ ¯λ of Mc(N, ) △ γ¯λ = (the translation by ( log λ )n) Mc(h). − | | ◦ Z Z As we see easily in the picture, the action ofγ ¯λ , hence that of γλ , is properly discontinuous, ﬁxed point free and with a compact fundamental domain. Thus Z Xλ = T emb( )/γ N △ λ 147 is a non-singular compact complex surface with π1(Xλ) = Z, in particular b1 = 1.

0 14. Inoue’s examples of class VII0 surfaces... 121

Theorem 14.1 (Inoue [23]). For λ U , the surface Xλ is of class VII ∈ ∗ 0 with b = 1 Xλ has exactly two irreducible curves, an elliptic curve E 2 · and a rational curve C with a node, such that E2 = 1,C2 = 0,E C = 0 and C homologous to zero 2 − · H (Xλ, Q) is 1-dimensional and is generated by E. Z Z Proof. orb(Ron′) = C∗ is certainly γλ -invariant and E = orb(Ron′)/γλ Z C∗/λ is an elliptic curve. On the other hand, the closure of orb(Ro(n + νn ) is P for ν Z, and their union C is γZ-invariant. C = C /γZ is a ′ 1 ∈ ∗ λ ∗ λ rational curve obtained from P by identifying 0 and . Since C is the 1 ∞ ∗ ﬁber of the equivariant morphism

T emb( ) TZ emb( R n, 0 ) N △ −→ n { o { }} induced by the vertical projection, we see that C2 = 0. Certainly, C and E are disjoint, hence C E = 0. Let us now look at 148 · / orb(R n ) T emb( R n , 0 ) TZn emb( 0 ) = C∗ o ′ ⊂ N { o ′ { }} { } || C C ∗ × Z ν ν ν(ν 1)/2 ν on which γλ acts via γλ(z, z′) = (λ z, λ − z z′). Dividing these out, we have Z E Xλ C C∗/λ , ⊂ − −→ which makes Xλ C the total space V(L) of a line bundle L of degree 1 − over C /λZ and E its 0-section. Hence E2 = deg L = 1. ∗ − − By Proposition 6.6, we see that the canonical bundle is

ω = 0 ( E C), Xλ Xλ − − 2 2 2 hence b = c = ( E C) = 1. Thus H (Xλ, Q) is necessarily 2 − 1 − − − spanned by E and C is homologous to zero.

Suppose there were an irreducible curve D , E, C. Since C D = 0, D would be contained in Xλ C = V(L). Thus E D, · − · which is non-negative by assumption, would be the degree of the pull- 1 Z back of L by D V(L) C /λ , a contradiction. Xλ is minimal, − → → ∗ since it thus has no exceptional curve of the ﬁrst kind. 122 2. Applications

1 Remark . Kato [29] looks at Xλ this way: it is obtained, from ord− of the shaded area in Mc(N, ), by identifying via γλ the two spherical 149 1 △ shells which is ord− of the two thick arcs in the picture.

In this way, we need to consider not TN emb(∆) but the simpler torus embedding

Z = TN emb( R n, R (n + n′), R n′, 0 { ◦ ◦ ◦ { }} 2 2 obt ained by blowing up C along 0. Hence Z = V1 V2 with V1 = C 2 ∪ and V2 = C with coordinates (z, ζ) and (z′, ζ′), respectively, glued along 2 (C∗) via 1 z′ = zζ and ζ′ = ζ− . 2 Let φλ : C V Zbe deﬁned by → 1 ⊂

φλ(z, z′) = (λz, z′/λ).

Consider, for ε> 0 small, the spherical shell

2 2 2 1/2 = (z, z′) C ; 1 ε< ( z + z′ ) < 1 + ε X { ∈ − | | | | } 2 and its inverse image ′ by the blowing up z C . Let ′′ = φλ( ). → λ Then ′ and λ′′ areP spherical shells whose images underP ord areP the P P 14. Inoue’s examples of class VII0 surfaces... 123 thick arcs in the picture. Thus φλ induces an isomorphism ψλ : ′ ∼ −→ λ′′. P P

Then Xλ is obtained from the inverse image in Z of the shaded area 150 between the thick arcs ord( ) and ord( ) by identifying and ′ λ′′ ′ λ′′ via ψλ. The common imageP of ′ and Pλ′′ in Xλ is obviouslyP a globalP spherical shell. P P This process was generalized by Kato [29] to produce new class VII0 surfaces with global spherical shells. Instead of Z, he takes Z′ obtained from C2 by a ﬁnite succession of blowing ups each time along a point over the previous center. Kato’s formulation also enabled him to construct a versal family of deformations of Inoue’s example Xλ. As before, let U∗ be the punctured unit disk and let B C2 be a small open ball centered at 0 with the ⊂ coordinate (τ, τ′). Theorem 14.2 (Kato [29]). There exists a versal family of deformations

X = Xλ,τ,τ ; (λ, τ, τ′) U∗ B U∗ B { ′ ∈ × } −→ × and families of curves

C = Cλ,τ,τ and E = Eλ,τ,τ { ′ } { ′ } 124 2. Applications

in X such that

(i) Xλ,0,0 = Xλ is Inoue’s example with Cλ,0,0 = C and Eλ,0,0 = E of Theorem 14.1.

, (ii) For τ′ 0,Eλ,0,τ′ is empty and Xλ,0,τ′ is a new class VII0 surface

with a unique irreducible curve Cλ,0,τ′ rational with a node of self- intersection number 0. , 151 For diﬀerent τ′ 0′ s, Xλ,0,τ′ are all isomorphic.

, (iii) For τ 0,Xλ,τ,τ′ is blowing up of a primary Hopf surface along a point.

Here is a sketch of the construction: Let Z = V V and ′ be as 1 ∪ 2 in the Remark above. Consider P

2 φλ,τ,τ : C V Z ′ −→ 1 ⊂ = + deﬁned by φλ,τ,τ′ (z, z′) (λ(zτ), (zζ τ′)/λ). It again induces an isomorphism

′ ′′ ′ ψλ,τ,τ : ˜ = φλ,τ,τ ( ). ′ −→ ′ X λ,τ,τX′ X

1 Then take ord− of the shaded area in the picture of Z/CTN. Let Xλ,τ,τ′ be the manifold obtained from it by identifying and via ψ . ′ λ,τ,τ′ λ,τ,τ′ We then let C and E be the images in X of P P

(z′, ζ′, λ, τ, τ′) V U∗ B : z′ζ′ + λ′ + λτ/(λ 1) = 0 { ∈ 2 × × − } and

(z, ζ′, λ, τ, τ′) V U∗ B; ζ = τ′ = 0 , { ∈ 1 × × } respectively. 14. Inoue’s examples of class VII0 surfaces... 125

the blowing up of primary Hopf surfaces Inoue's surface 0

new surface

152 We now look at the degeneration of Inoue’s surface Xλ as λ tends to 0.

Theorem 14.3. There exists a proper ﬂat family

Π : λ U = λ C; λ < 1 −→ { ∈ | | } 1 with Y non-singular such that for λ , 0 Π− (λ) is Inoue’s surface 1 Xλ, and that Π− (0) is a non-normal surface obtained by identifying two 1′ s on the 2-dimensional complete non-singular torus embedding W as in the picture. 126 2. Applications

Proof. Let N = N ℓ and consider the automorphism h of N deﬁned ⊗ by e e e h(n) = n + n′, h(n′), h(ℓ) = ℓ + n. 153 Let (N, ∆) be thee r.p.p.decomposition,e wheree ∆ consists of the faces of

ν ν+1 ν 1 e e oh (ℓ) + oh (ℓ) + ohe− (n) ν+1 ν 1 ν oeh (ℓ) + eoh − (n) + eoh (n) ν ν+1 oeh (ℓ) + oh e (ℓ) + on′e e e with ν running through . Then ∆ N coincides with ∆ of Theorem ∩ 14.1, and ∆ inducers the polyhedral decomposition ∆ (N + ℓ) as in e ∩ the picture below. Note that e e ν h (ℓ) = ℓ + νn + (ν(ν 1)/2)n′. − e Obviously, h preserves ∆ and gives rise to an automorphism γ of TN emb(∆). The horizontal projection induces Π : T emb(∆) Tℓ e e N −→ e emb( ℓ, 0 ) = C. Again let V be the inverse image of the unite disk e{ o { }} e e e U under π. Thus ord (V) is the “upper half” of Mc(N, ∆). The action of γ on V is properly discontinuous and ﬁxed point free. Again by e e Theoreme 4.2 (iii) and Proposition 6.7, we see that e π : Y = V/γ U −→ is the sought-for family e 15. Hilbert modular surfaces and class VII0 surfaces 127

154

15 Hilbert modular surfaces and class VII0 surfaces

Inoue [24] constructed another series of examples of class VII0 surfaces with b2 , 0, using the minimal desingularization by Hirzebruch [19] 128 2. Applications

of neighborhoods of cusps of the Hilbert modular surfaces. Again torus embeddings are very convenient to describe them. We first describe the relevant part of Hirzebruch’s theory in terms of torus embeddings. (See also Cohn [6].) For the complete description of compactified Hilbert modular surfaces, we refer the reader to Mumford et al. [61, Chap. 1, . 5]. See also Rapoport [54] for the case of totally § real fields in general. 155 Let K = Q( √d) be a real quadratic field. Then we have two embeddings of K into

K ξ ξ and K ξ ξ′ ∋ 7→ ∈ ∋ 7→ ∈ so that we have a canonical isomorphism of -algebras

2 Q K a ξ (aξ, aξ′) ⊗ ∋ ⊗ 7→ ∈ with which we identify them from now on. Deﬁnition. For a lattice N in K, let − U = positive units u of k such that u = N N { N } U+ = totally positive units u of k such that uN = N . N { } + Then it is known (see, for instance Hirzebruch [19]) that UN and UN are inﬁnite cyclic groups with

+ [UN : UN] = 1 or 2. We have the canonical identiﬁcation

2 N = Q K = ⊗ with N lying irrationally in 2 with respect to the coordinate system of 2 R . Consider the convex hull ′N of P 2 N (> ) ∩ 0

and the inﬁnite r.p.p.decomposition (N, ∆′N ), where ∆′N is the decompo- 2 sition of the ﬁrst quadrant (>0) into sectors by rays joining 0 and the points in N ∂ ′ . ∩ N P 15. Hilbert modular surfaces and class VII0 surfaces 129

Thus Mc(N, ∆′N) is the union of Nand the infinite chain of intervals at infinity. + 156 The action of UN on N by multiplication certainly preserves ∆′N, + 1 hence we have an action of UN on TN emb(∆′N ). Let VN′ be ord− of 2 the union of the first quadrant (R>0) and the boundary at infinity of + Mc(N, ∆′N). Then the action of UN on ord(VN′ ), hence that on VN′ itself, is properly discontinuous and fixed point free. Thus the quotient

+ WN′ = VN′ /UN

+ (or, more generally, the quotient by a subgroup of UN of ﬁnite index) is non-singular, and contains a ﬁnite cycle of P1′ s. Such WN′ appears as a neighborhood of the minimal resolution of a cusp of a Hilbert modular surface.

0 0

fund. domain

Inoue [24] obtains a new class VII0 surface by gluing two appropri- 157 ate such WN′ ’s together. In our formulation, this amounts to applying the process above to the ﬁrst and the fourth quadrant simultaneously: 130 2. Applications

Theorem 15.1 (Inoue). Let N be a Z-lattice in a real quadratic field K. 2 Let ′ (resp. ′′) be the convex hull of N (R> ) (resp. N (R> N N ∩ 0 ∩ 0 × R<0))P. ConsiderP the infinite r.p.p.decomposition (N, ∆N ), where ∆N is the decomposition of R> R into sectors by rays joining 0 and 0 × ′ ′′ (N ∂ ) (N ∂ ). ∩ ∪ ∩ XN XN 1 Let V be ord− of the union of R> R and the boundary at infinity of N 0 × Mc(N, ). Then the following hold: △N + (i) XN = VN/UN is a non-singular compact surface of class VII0 with exactly b2 , 0 irreducible curves, which form two disjoint cycles of rational curves.

+ (ii) If [UN : UN] = 2, then

Xˆ N = VN/UN

is also a non-singular compact complex surface of class VII0 with exactly. b2 , 0 irreducible curves, which form a cycle of rational curves. Here is a sketch of the proof: + 158 From the picture below, the action of UN (or UN in case (ii)) on VN is easily seen to be properly discontinuous, fixed point free and with a compact fundamental domain. Note that in case (ii), the multiplication by an element of UN not in + UN interchanges the first and the fourth quadrant, hence ∆N is in a sense “symmetric”. 1 ord− of the two infinite chains of the boundary gives rise to two disjoint cycles

C = C0 + C1 + + Cr 1 ··· − C = D0 + D1 + + Dr 1 ··· − of rational curves in XN. In case (ii), we have r = s and the images of C and D in Xˆ N coincide. We thus have a cycle of rational curves

Cˆ = Cˆ0 + Cˆ1 + Cˆr 1 ··· − 15. Hilbert modular surfaces and class VII0 surfaces 131

0 0 fund. domain

in case (ii)

We refer reader to Inoue [24] for the proof of the facts: 159

H1(XN, Z) Z (r+s) H2(XN, Z) Z⊕

C0,..., Cr 1 and D0,..., Ds 1 are the only irreducible curves in XN. In − − particular, XN and Xˆ N are minimal. Note that from the picture, we easily 1 see that ord− of the positive half of the abscissa in Mc(N, ∆N) gives rise to a CT -bundle over S 1 in X , and that X C D is homeomorphic N N N − − to the product of R and the bundle.

Remark. As Kato [29] has shown, XN contains a global spherical shell, hence is deformation of the blowing up of a primary Hopf surface along 1 a finite number of points. Indeed, ord− of a tubular neighborhood of the boundary of our fundamental domain in the picture gives rise to one ˆ To study WN′ , XN and XN in more detail, the “modified” continued fraction expansion is very convenient as in Hirzebruch [19] and Inoue[24]. Definition. Let ξ be a real number. Then the modified continued fraction expansion ξ = [[e0, e1,...]] 132 2. Applications

is deﬁned as follows: For non-negative integers ν determine ξν and eν inductively by

ξ0 = ξ

eν is the integer with eν 1 <ξν eν − ≤ ξν = eν 1/ξν+ . − 1 160

We call ξν′ s the intermediate terms of the expansion of ξ.

Then eν 2 for ν > 0 with no infinite consecutive equality eν = ≥ 2 allowed when the expansion is infinite. As in the theory of usual continued fractions, we can show that ξ is a irrational quadratic number if and only if its expansion is eventually periodic, i.e. periodic from certain ν on. (cf. Perron [51]). As before, we identify an element ω K with its canonical image 2 ∈ ω,ω ) via K ֒ NR = R . Then the continued fraction expansion of) ′ → ω and the convex geometry in R2 have a very close relationship. To be able to describe degenerations of Inoue’s surfaces XN, we need the following slight amplification of the results pointed out by Hirzebruch [19] and Cohn [6]. Proposition 15.2. (1) Up to the multiplication of a totally positive element of K, a Z-lattice N is of the form N = Z + Zω

with ω> 1 >ω′ > 0, i.e. ω reduced.

161 (2) ω K is reduced if and only if it has a purely periodic continued ∈ fraction expansion

ω = [[a0, a1,..., ar 1]] − (with the smallest period r).

(3) For reduced ω, let ων be the intermediate terms of the expansion of ω with the smallest period r. Then u = 1/(ω ω ω ) 1 2 ··· r 15. Hilbert modular surfaces and class VII0 surfaces 133

+ is a generator of UZ+Z , and nν ν Z are the consecutive elements ω { }{ ∈ } of (Z + Zω) ∂Σ′ , ∩ Z+Zω where

q+1 n + = (ω + ω )u for q Z and 0 j < r qr j j 1 ··· r ∈ ≤ and nν 1 + nν+1 = aνnν. − (4) For ω reduced, 1/ω has the expansion of the form

1/ω = [[1, e, a0∗,..., a∗s 1]], − where s is the smallest period and e 1 < ω/(ω 1) < e. Let − −

ω∗ = [[a0∗,..., a∗s 1]], i.e.ω/(ω 1) < e 1/ω∗ − − −

and let ων∗ be the intermediate terms of the expansion of ω∗.

Then u = 1/(ω1ω2 ωr) = 1/(ω∗ω∗ ω∗s), and nν∗ ν Z are the con- ··· 1 2 ··· { } ∈ secutive elements of (Z + Zω) ∂ ′′ , where ∩ Z+Zω P q n∗ = u (ω 1)/(ω∗ ...ω∗) for q Z and 0 j < s qs+ j − 0 j ∈ ≤ and

nν∗ 1 + nν∗+1 = aν∗nν. −

Corollary 15.3. For N = Z+Zω with ω reduced, the decomposition ∆N 162 of R> R consists of the faces of 0 ×

σν = Ronν + Ronν+1

σν∗ = Ronν∗ + Ronν∗+1 for ν Z, with nν and n as in Proposition 15.2. ∈ ν∗ Combined with Proposition 6.7, this implies the following: 134 2. Applications

Proposition 15.4. Let N = Z + Zω with ω reduced and let ω∗ be as in Proposition 15.2 (4) with

ω = [[a0,..., ar 1]] − ω∗ = [[a0∗,..., a∗s 1]] − in the smallest periods r and s. Then Inoue’s surface XN has two cycles of rational curves

C = C0 + C1 + + Cr 1 ··· − C = D0 + D1 + + Ds 1 ··· −

and Xˆ N a cycle of rational curves

Cˆ = Cˆ0 + Cˆ1 + + Cˆr 1 ··· − with the following properties: (i) If r = 1 or s = 1, then C, Cˆ or D is an irreducible rational curve with one node with

2 2 2 C = a + 2, Cˆ = a , +2, D = a∗ + 2. − 0 − 0 − 0 (ii) If r 2 or s 2, then C , Cˆ or D are non-singular rational curves ≥ ≥ i i i with

C2 = a , Cˆ2 = a i = 0,..., r 1 i − i i − i − 2 D = a∗ i = 0,..., s 1. i − i − 163 To illustrate the relationship between the continued fraction expansion of ω and the r.p.p.decomposition (N, ∆N ) with N = Z+Zω, we now sketch the proof of Proposition 15.2 except fo the eventual periodicity of the expansion and the fact that

u = 1/(ω ω ) = 1/(ω∗ ω∗) 1 ··· r 1 ··· s + generates UN. Hopefully, it will explain the way we number the element of N (∂ ′ ∂ ′′ ). ∩ N ∪ N The proofP of theP following inducted step is obvious. 15. Hilbert modular surfaces and class VII0 surfaces 135

Lemma 15.5. Let ξ K with ξ >ξ > 0. For the integer e with ∈ ′ e 1 <ξ< e, let ξ = e 1/η. Then η> 1, η > η > 0 and 1/η Z + Zξ. − − ′ ∈ If ξ>ξ′ > 1, then ξ is in the interior of the convex hull of (Z + Zξ) 2 ∩ (R>0) . If ξ > 1 > ξ > 0, i.e. ξ is reduced, then e 2 and η is also reduced. ′ ≥ Moreover, ξ, 1 and 1/η are consecutive elements in the intersection of 2 Z + Zξ with the boundary of the convex hull ′ of (Z + Zξ) (R> ) . Z+Zξ ∩ 0 By repeated application of Lemma 15.5 toP the intermediate terms of the expansion of ω, we get: Lemma 15.6. Let ω K with ω>ω > 0 with the expansion ∈ ′

ω = [[a0, a1,...... ]] and the intermediate term ωv. Let t be the smallest integer such that 164 1 > ωt′. Then (i) the expansion is periodic from v = t on. Let r be the smallest period. Thus ω + = ω and a + = aν for v t. Let v r v v 1 ≥

ζ 1 = 1 and ζv = 1/(ω0ω1 ωv) for v 0. − ··· ≥

Then (ii) ζv v t 1 are consecutive elements of the intersection of { } ≥ − 2 Z(1/ω)+Z with the boundary of the convex hull of (Z(1/ω)+Z) (R> ) . ∩ 0 Moreover, ζv+1 + ζv 1 = avζv for v t. − ≥ (iii) 1/(ω + ω + ) belongs to U . t 1 ··· t r Z∗+Zω In particular, we get (1), (2) and a part of (3) of Proposition 15.2, since ω(Z(1/ω) + Z) = Z + Zω.

Let us now prove (4). Let ξ = 1/ω with the intermediate terms ξv and with the expansion ξ = [[e0, e1,...]]. Since ω is reduced, hence ξ > 1 > ξ > 0 by assumption, we have e = 1 and 1/ξ R> R< . ′ 0 0 1 ∈ 0 × 0 Since 0 < e ξ = 1/ξ < 1 and 1/ξ = e ξ > e 2, we see that 1 − 1 2 2′ 1 − 1′ 1 ≥ ξ = ω/(ω 1), ξ = ω and e = e of (4). We now apply (3) to ω , 1 − 2 ∗ 1 ∗ taking u = 1/(ω ω ) for granted. Thus 1∗ ··· ∗s q+1 (ω∗ ω∗)/(w∗ ω∗) = ω∗n∗ /(ω 1) j+1 ··· s 1 ··· s qs+ j − 136 2. Applications

for q Z and 0 j < s are the consecutive elements of ∈ ≤ (Z + Zω∗) ∂Σ′ . ∩ Z+Zω∗ 165 On the other hand, we have

Z + Zω = ω(Z + Z(1/ω)) = ω(Z + Z(1/ξ1))

= ω(Z(1/ξ ) + Z(1/ξ ξ )) = ((ω 1)/ω∗) Z + Zω∗ . 1 1 2 − { }

Since (ω 1) > 0 > (ω′ 1), we see that nv∗ v Z are the consecutive − − { } ∈ elements of

(Z + Zω) ∂ΣZ′′+Zω = (ω 1/ω∗) (Z + Zω∗) ∂Σ′Z + Zω∗ . ∩ − { ∩ }

We now describe degenerations of Inoue’s surface XN. Proposition 15.7 (Makio). For reduced ω K, let N = Z + Zω. Then there exists a proper and ﬂat family

π : Y C → 1 1 with Y normal such that π− (λ) = XN for all λ , 0 and that π− (0) is a non-normal surface obtained by identifying two P1′ s on the 2 - dimensional complete normal TN-embedding Z as in the picture, where nv and nv∗ are as in Prop. 15.2.

0 15. Hilbert modular surfaces and class VII0 surfaces 137

166 Remark . Unlike Theorem 14.3, Y and Z may be singular in general. They are non-singular if r = 1. For each speciﬁc N, we can certainly 1 replace Y by a blow up to obtained non-singular Y and π− (0) consisting of non-singular components crossing normally. As is obvious from the construction of Y below, there, many other choices for Y.

Proof. As before, let N˜ = N Zℓ. Left the action of U on N to N˜ by ⊕ N∗ letting it act as the identity for ℓ. By Proposition 15.2, we have

n0 = 1, n 1 = ω, n∗ 1 = ω 1. − − −

˜ ˜ This fact guarantees that (N, ∆) is a UN∗ -invariant r.p.p.decomposition, where ∆˜ consists of the faces of

Ro(nir + ℓ) + Ronir+v 1 + Ronir+v i Z, 0 v < r − ∈ ≤ R (n + ℓ) + R n∗ + R n∗ i Z, 0 v < s o ir o is+v o is+v+1 ∈ ≤ Ro(nir + ℓ) + Ro(n(i+1)r + ℓ) + Ron(i+1)r 1 i Z − ∈ R (n + ℓ) + R (n + + ℓ) + R n∗ i Z. o ir o (i 1)r o (i+1)s ∈

Seen from above the north pole, the decomposition induced by ∆˜ on a sphere in N˜R centered at 0 looks like the picture below.

1 Let B be ord− of the union in Mc(N˜ , ∆˜ ) of (R>0 R) Rℓ and the + × × boundary at inﬁnity. Then B is UN-invariant and the action is properly discontinuous without ﬁxed point. By Theorem 4.2 (iii), we are done. 138 2. Applications

167

Examples. (1) If ω = 2 + √3 = [[4]],¯ then ω∗ = 1 + √3/3 = [[2, 3]]. Hence XN is as in the picture. We have

n1 = ω + 4, n0 = 1, n 1 = ω, n 2 = 4ω 1 − − − − n2∗ = 2ω 7, n1∗ = ω 3, n0∗ = ω 2, n∗ 1 = ω 1, n∗ 2 = 2ω 1. − − − − − − − + Since r = 1, Y and Z of Prop. 15.7 are non-singular. Taking the UN- translates of Ro(n0+ℓ)+Rono+Ron1 and Ro(n0+ℓ)+Ro(n1+ℓ)+Ron1 instead of Ro(n0 +ℓ)+Ro(n1 +ℓ)+Ron0 and Ro(n0 +ℓ)+Ron 1 +Ron0, − we get a diﬀerent degeneration Z′

2 1

0 0 15. Hilbert modular surfaces and class VII0 surfaces 139

(2) Let ω = (3 + √5)/2 = [[3]]¯ = ω∗. Then UN is generated by 168 ( 1 + √5)/2. Kato showed that the only possible conﬁgurations of − curves for a minimal surface with b2 = 1 and with a global spherical shell are that of XˆN here as well as those in Theorem 14.2 (i) and (ii). In this case, we have

n1 = 3 ω, n0 = 1, n 1 = ω − − n1∗ = 2ω 5, n0∗ = ω 2, n∗ 1 = ω 1. − − − − Again Y and Z of Prop. 15.7 are non-singular in this case. By a modiﬁcation as in (1) above, we get a diﬀerent Z′.

2 1 0 0

(3) If ω = (3+ √7)/2 = [[3,¯6]], then ω∗ = (5+ √7)/3 = [[3, 3, 2¯, 2, 2]].

6 3

3 3 2 2 2

Bibliography

[1] K. Akao On prehomogeneous compact Kahler manifolds, in 169 Manifolds-Tokyo 1973 (Hattori, ed.), Univ.of Tokyo Press, 1975, 365-371

[2] K.Akao, Complex structure on S 2p+1 S 2q+1 with algebraic codi- × mension 1, in Complex Analysis and Algebraic Geometry (Baily and Shioda, eds.), Iwanami Shoten and Cambridge Univ.Press, 1977, 205-225

[3] A. Borel, Linear algebraic groups, Benjamin, New York, 1969

[4] A.Borel and J.-P.Serre, Corners and arithmetic groups, Comm.Math.Helv. 48 (1973), 436-491

[5] P.Cartier, Questions de ratinalité des diviseurs en géométrie algébrique, Bull.Soc.Math.France, 86 (1958), 177-251 Appendice

[6] H.Cohn, Support polygon and the resolution of modular functional singularities, Acta Arithmetica, 24 (1973), 261-278

[7] C.Delorme, Sous-mono¨ıdes d’intersection compl´ete de N, Ann.Sci.Ecole Norm.Sup., 9 (1976), 145-154

[8] M. Demazure, Sous-groupes alg´ebriques de rang maximum du groupe de Cremona, Ann.Sci.Ecole Norm.Sup. 3 (1970), 507-588

[9] P.Deligne and D.Mumford, The irreduciblility of the space of curves of given genus, Publ.Math.IHES, 36 (1969), 75-110 170

141 142 BIBLIOGRAPHY

[10] A. Grothendieck and J.Dieudonné, Elément´ de géométrie alge- brique, Publ.Math.IHES, 4, 8, 11, 17, 20, 24, 28, 32 (1960-1967)

[11] L.Gerritzen, On non-archimedean representations of abelian varieties, Math.Ann. 196 (1972), 323-346

[12] G.Gonzalez-Sprinberg, Eventails en dimension 2 et transforme de Nash, Centre de Math. Ecole Norm.Sup. Equipe de Rech.Assoc. au CNRS No.589, Fev.1977.

[13] B.Gr¨unbaum, Convex polytopes, Interscience, New York, 1967

[14] B.Gr¨unbaum, Polytopes, graphs, and complexes, Bull. Amer. Math. Soc., 76 (1970), 1131-1201

[15] L.Gerritzen and H.Grauert, Die Azyklizit¨at der aﬃnoiden Uberdeckung, in Global Analysis (Iyanaga and Spencer, eds.), Univ.of Tokyo Press and Princeton Univ.Press, 1969, 159-184

[16] R.Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Math. 156, Springer-Verlag, 1970

[17] J.Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970), 175-193

171 [18] H.Hironaka, Resolution of singularities of an algebraic variety over a ﬁeld of characteristic zero, I-II, Ann.of Math. 79 (1964), 109-326

[19] F.Hirzebruch, Hilbert modular surface, L’enseignement Math. 21 (1973), 183-282

[20] M.Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann.of Math. 96 (1972), 318-337

[21] J.Herzog and E.Kunz, Die Wertehalbgruppe eines lokalen Rings der Dimension 1, Sitzungsberichte der Heidelberger Akad.der Wiss., 2.Abh., Springer-Verlag, 1971 BIBLIOGRAPHY 143

[22] M.Inoue, On surface of class VII0, Inventiones math. 24 (1974), 269-310

[23] M.Inoue, New surfaces with no meromorphic functions, in Pro- ceedings of the Intern. Congress of Math., Vancouver 1974, vol.1, 423-426

[24] M.Inoue, New surfaces with no meromorhic functions, II, in Com- plex Analysis and Algebraic Geometry (Baily and Shioda, eds.), Iwanami Shoten and Cambridge Univ.Press, 1977, 91-106

[25] M.Ishida, Graded factorial rings of diemension 3 of a mesticted bype, to appear in J.Math.Kyoto Univ.

[26] M.Ishida, Compactiﬁcations of a family of generalized Jacobian varieties, to appear in the Proceedings of the Intern.Symp.on Alg.Geom., Kyoto 1977

[27] B.Iversen, A ﬁxed point formula for action of tori on algebraic 172 varieties, Inventiones math. 16 (1972), 229-236

[28] T.Kaneyama, On equivariant vector bundles on an almost homogeneous variety, Nogoya Math.J. 57 (1975), 65-86

[29] Ma.Kato, Complex manifolds containing “global” spherical shell, I, to appear in the Proceedings of the Intern.Symp.on Alg.Geom., Kyoto 1977

[30] K.Kodaira, On the structure of compact complex analytic surfaces I Amer. J. Math. 86 (1964), 751-798 II ibid. 88 (1966), 682-721 III ibid. 90 (1968), 55-83 IV ibid. 1048-1066 See also Kunihiko Kodaira Collected Works, Vol.III Iwanami Shoten and Princeton Univ. Press, 1975

[31] K.Kodaira, Holomorphic mappings of polydiscs into compact complex manifolds, J.Diﬀ. Geom. 6 (1971), 33-46 144 BIBLIOGRAPHY

[32] T.Mabuchi, Almost homogeneous torus actions on varieties with ample tangent bundle, to appear in Tohoku Math. J.

[33] J.McCabe, p-adic theta functions, thesis Harvard Univ.1968

[34] S.Mori, On a generalization of complete intersections, J. Math. Kyoto Univ. 15 (1975), 619-646

[35] S.Mori, Graded factorial domains, to appear in J.Math. Kyoto Univ.

173 [36] H.Morikawa, Theta functions and abelian varieties over valuation ﬁelds of rank one, I-II, Nagoya Math.J. 20(1962), 1-27 and 21(1962), 231-250

[37] J.Morrow, Minimal normal compactiﬁcations of C2, Rice Univ.Studies 59 (1973), 97-112

[38] D.Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239-272

[39] D.Mumford, Introduction to algebraic geometry, mimeographed notes, Harvard Univ.

[40] K.Miyake and T.Oda, Almost homogeneous algebraic varieties under algebraic torus action, in Manifolds-Tokyo 1973 (Hattori,ed.), Univ.of Tokyo Press, 1975, 373-381

[41] S.Mori and H.Sumihiro, On Hartshorne’s conjecture, to appear in J.Math.Kyoto Univ.

[42] M.Nagata, On rational surfaces, I-II, Memoirs Coll.Sci. Univ.Kyoto, 32 (1960), 351-370 and 33 (1960), 271-293

[43] O.Nagaya, Classiﬁcation of 3-dimensional complete non-singular torus embeddings, Master’s thesis, Nagoya Univ. 1976

[44] I.Nakamura, On moduli of stable quasi-abelian varieties, Nagoya Math.J. 58(1975), 149-214 BIBLIOGRAPHY 145

[45] I.Nakamura, Relative compactification of the Neron model and its 174 applications, in Complex Analysis and Algebraic Geometry (Baily and Shioda,eds.),Iwanami Shoten and Cambridge Univ.Press, 1977,205-225 [46] Y.Namikawa, A new compactification of the Siegel space and the degeneration of abelian varieties, I-II, Math.Ann. 221 (1976), 97- 141 and 201-241 [47] Y.Namikawa, Toroidal degeneration of abelian varieties, in Com- plex Analysis and Algebraic Geometry (Baily and Shioda, eds.), Iwanami Shoten and Cambridge Univ.Press, 1977, 227-237 [48] P.Olum, Non-abelian cohomology and van Kampen’s theorem, Ann.of Math. 68 (1958), 658-668 [49] T.Oda and C.S.Seshadri, Compactifications of the generalized Ja- cobian variety, to appear in Transactions Amer.Math.Soc.

[50] P.Orlik and P.Wagreich, Algebraic surfaces with k∗-action, to appear [51] O.Perron, Die Lehre von Kettenbrünchen, B.G.Teubner, Leipzig und Berlin, 1913 [52] V.L.Popov, Quasi-homogeneous affine algebraic varieties of the group SL(2), Izv.Akad.Nauk SSSR 37 (1973), 792-832 = Math.USSR Izv. 7 (1973), 793-831 [53] V.L.Popov, Classification of 3-dimensional affine algebraic varieties that are quasi-homogeneous with respect to an algebraic group, Izv.Akad.Nauk.SSSR 39 (1975), 566-609 = Math.USSR Izv. 9 (1975), 535-576

[54] M.Rapoport, Compactiﬁcations de 1′ espace de modules de 175 Hilbert-Blumenthal,thesis Universite de Paris-Sud, 1976. [55] M.Raynaud, Faiscaux amples sur les schemas en groupes et les espaces homogenes, Lecture Notes in Math. 119, Springer-Verlag, 1970 146 BIBLIOGRAPHY

[56] R.T.Rockafellar, Convex analysis, Princeton Univ.Press 1970

[57] I.Satake, On the arithmetic of tube domains, Bull.Amer.Math. Soc. 79 (1973), 1076-1094

[58] R.R.Simha, Algebraic varieties biholomorphic to C C , to appear ∗× ∗ in Tohoku Math.J.

[59] H.Sumihiro, Equivariant completion, I-II, J.Math.Kyoto Univ.14 (1974), 1-28 and 15 (1975), 573-605

[60] T.Suma, On hyperelliptic surfaces, J.Fac.Sci.Univ.Tokyo, 16 (1970), 469-476

[61] A.Ash, D.Mumford, M.Rapoport and Y.Tai,Smooth compcatiﬁca- tion of locally symmetric varieties, Lie Groups: History Frontiers and Applications IV, Math.Sci.Press, Brookline, Mass. 1975

[62] H.Tsuchihashi, Degenerations of hyperelliptic surfacces, Master’s thesis, Tohoku Univ., 1978

[63] G.Kempf, F.Knudsen, D.Mumford and B.Saint-Donat, Toroidal embeddings I, Lecture Notes in Math. 339, Springer 1973

[64] T.Ueda, to appear