Distribution of Rational Points on Algebraic Varieties

Distribution of rational points on algebraic varieties

Sho Tanimoto

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy Department of Mathematics New York University May 2012

Professor Yuri Tschinkel

To my wife, Mei.

iii Acknowledgements

First and foremost it is my pleasure to thank my advisor Yuri Tschinkel for his patience and guidance throughout my Ph.D.. He has influenced my development from an ordinary math student to a professional researcher in mathematics. I learned from him, his patience, work ethic, creativity, and broad outlook in mathematics, and these have shaped me as a professional mathematician. I would also like to thank my second advisor Fedor Bogomolov. His passion, intuition, creativity, and the view of mathematics have inspired me in my research. His generosity has supported my study and life, and without it, I would not have completed this dissertation. I would like to thank Sylvain Cappell for his warmest support which he has given me since I was a first year student at Courant. Brendan Hassett is a great inspiration for my research, and I also benefited from conversations and discus- sions with Benjamin Bakker, David Harvey, Sonal Jain, Ilya Karzhemanov, Brian Lehmann, and Tony Varilly. Many thanks to my fellow students in our algebraic geometry and number theory group, Michael Burr, Edgar Costa, Lukas Koehler, Fedor Soloviev, Jordan Thomas, and Pankaj Vishe, for their insights, questions, help, and friendship. In particular, Michael and Jordan helped me to improve the exposition of this thesis, and Lukas did a fantastic job to activate our algebraic geometry seminar. I would also like to thank friends with whom I shared time as a graduate student at Courant, especially Andrea, Chaitu, Dingyu, Felix, Hantaek, Ivan, Jong Ho, Jungwoon, Kela, Nengli, Shoshana, and Sukbin. I owe a special debt of gratitude to Tamar Arnon for her warmest support throughout my Ph.D.. Going back to Japan, I would like to thank my teachers at my college, Takao

iv Fujita, Shihoko Ishii, and Nobushige Kurokawa for their support and encourage- ment. I learned a great deal of algebraic geometry from my advisor Takao Fujita when I was an undergraduate student at Tokyo Institute of Technology. Ryoichi Aoki, my physics teacher at my high school, is one who let me consider a career path in academia. He always kept his door open for me, and he shared time with me to discuss my questions in physics. Finally, I thank my family: my parents, Shin and Mariko, my brother Ryo, and our lovely cats, Chao and Mie have supported me via skype for the last ﬁve years and provided me an ideal atmosphere when I was in Japan. And my wife Mei, for me, she is everything. She has supported and encouraged me all the time with her love via skype, phone, and shipments, and without her support, I could not have achieved any accomplishment which today I have. I thank her with all my love, and I wish the best luck for our continuing journey.

v Abstract

We study the asymptotic formulas for the number of rational points of bounded height on equivariant compactifications of solvable linear algebraic groups over a number field, and we introduce the notion of balanced line bundles. In the first part, we discuss Height zeta functions of equivariant compactifications of solvable linear algebraic groups and show that the main term of a Height zeta function coincides with the prediction of Manin’s conjecture by using geometric integration techniques developed by Chambert-Loir and Tschinkel. Then we study Height zeta functions of equivariant compactifications of the ax + b-group and prove Manin’s conjecture under some mild geometric conditions. In the second part, we introduce the notion of balanced line bundles and give a systematic study of this notion in terms of birational geometry. We identify balanced big line bundles for many cases, e.g., del Pezzo surfaces, Fano 3-folds of Picard rank one, flag varieties, toric varieties, and equivariant compactifications of homogeneous spaces. The last result has important arithmetic applications in [CLT02] and [GTBT11].

vi Contents

Dedication ...... iii Acknowledgements ...... iv Abstract ...... vi Introduction ...... 1

I Height zeta functions 8

1 Height functions 9 1.1 Metrics and Heights ...... 9 1.2 Measures ...... 16

2 Manin’s conjecture and Height zeta functions 22 2.1 Manin’s conjecture ...... 23 2.2 Height zeta functions ...... 30

3 Height zeta functions of equivariant compactifications of semi- direct products of algebraic groups 37 3.1 Geometry of equivariant compactifications ...... 38 3.2 The main terms of Height zeta functions of equivariant compactifications of solvable algebraic groups ...... 46

vii 3.3 Height zeta functions of equivariant compactiﬁcations of the ax + b- group ...... 64

II Balanced line bundles 99

4 Balanced line bundles 100 4.1 Generalities ...... 100 4.2 Balanced line bundles ...... 112

5 Fano varieties 115 5.1 Del Pezzo surfaces ...... 115 5.2 Del Pezzo surface ﬁbrations ...... 120 5.3 Fano threefolds ...... 122

6 Equivariant geometry 125 6.1 Generalized ﬂag varieties ...... 125 6.2 Equivariant compactiﬁcations of homogeneous spaces ...... 129 6.3 Toric varieties ...... 137

Bibliography 146

viii Introduction

One of the classical problems in mathematics is to study integral or rational solutions to Diophantine equations, i.e., integral or rational solutions to polynomial equations with integer coeﬃcients. The most famous Diophantine equations are the Fermat equations : xn + yn = zn, where n is a positive integer. There are many traditional questions concerning Diophantine equations, which include:

1. (Existence) Do integral or rational solutions exist?

2. (Finiteness) If solutions exist, are there ﬁnitely or inﬁnitely many?

3. (Distribution) If there are inﬁnitely many solutions, how are they distributed? For example, one could determine their density in various topologies or asymptotic formulae for the counting functions of solutions with respect to appropriate size functions.

4. (Construction) Construct some or all solutions eﬀectively.

The solutions to each of these problems have lead to intensive and long studies, and they are still active areas in modern mathematics. Although there are many elementary and ad hoc approaches to these questions, modern research in this ﬁeld focuses on intrinsic properties of Diophantine equations, e.g., their geometry.

1 Diophantine geometry

Note that answers to the above questions are stable under changes of coordinates. Perhaps this is the first observation to support the philosophy that “Geom- etry determines Arithmetic”, appearing in [HS00]. The subject which studies the phenomena realizing this philosophy is called Diophantine geometry, and is one of main topics in current mathematics. One of the most important invariants in Diophantine geometry is the canonical line bundle KX of a smooth projective variety X, and we believe that this geometric invariant roughly determines the behavior of the distribution of rational points, at least after taking a finite extension. To illustrate this idea, consider the case of curves: Let F be a number field and C a smooth projective curve defined over F . The behavior of the distribution of rational points is determined by one geometric invariant, the genus g of C:

• When g = 0, −KC is ample and C(F ) is either empty or inﬁnite;

• When g = 1, KC is trivial and C(F ) is either empty or a ﬁnitely generated abelian group. C(F ) is inﬁnite if and only if the Mordell-Weil rank, the rank of C(F ), is positive.

• When g ≥ 2, KC is ample and C(F ) is ﬁnite. (Mordell-Faltings theorem)

In particular, after taking a ﬁnite extension of F , C(F ) is inﬁnite when −KX is ample or trivial. Based on this evidence, we believe in the following principle which is the most important guiding principle in Diophantine geometry:

The positivity of the canonical line bundle (or the anticanonical line bundle) determines the distribution of rational points on X.

2 We hope that the principle applies to higher dimensional cases as well, and thus we consider the following classiﬁcation of smooth projective varieties:

n • (Fano-type): −KX is ample, e.g., P , del Pezzo surfaces, and Fano manifolds;

• (Intermediate type): KX is trivial, e.g., abelian varieties and K3 surfaces;

• (General type): KX is ample, e.g., projective manifolds of general type.

My research interest lies in Fano and intermediate type varieties which are expected to have inﬁnitely many rational points, at least after taking a ﬁnite extension. More precisely, I am interested in the distribution of rational points on these varieties.

Distribution of rational points

There are many problems involving the distribution of rational points on algebraic varieties, but here we would like to mention two problems: Let X be a smooth projective variety deﬁned over a number ﬁeld F . One can ask the following two questions:

• (Potential density): Is X(F ) dense in the Zariski topology, possibly after taking a ﬁnite extension of F ?

• (Asymptotic formulae): Let L = (L, k · k) be a big, adelically metrized

line bundle on X and HL the associated height (see Deﬁnitions 4 and 6 in Subsection 1.1.2). What is the asymptotic formula of the counting function of rational points of bounded height? In other words, determine

◦ ◦ N(X , L, B) := #{x ∈ X (F ) | HL(x) 6 B} ∼ ???

as B → ∞, for a suitable Zariski open subset X◦.

3 The answer to the potential density problem is believed to be affirmative for Fano and Intermediate type varieties. It is possible that X(F ) is empty due to the arithmetic obstructions, e.g., the Hasse principle and the Brauer-Manin obstructions (see, for example, [Man86]). To focus on the effect of geometry, it is essential to allow one to take a finite extension. [HT00] and [BT99] provided the first results in this direction, and [Cam04] and [Abr09] describe a general framework of this problem. An answer for the second question should be viewed as a strong, quantitative version of the density of rational points, and this thesis mainly discusses this problem.

Manin’s conjecture

Next, we describe the second problem in more detail. Let X ⊂ Pn be a smooth projective variety defined over the field of rational numbers Q with given embedding into a projective space. A height function H on Pn is defined as follows:

q n n+1 2 2 H : P (Q) = Zprim/ ± 1 3 (x0 : ··· : xn) 7→ x0 + ··· + xn ∈ R>0, (1)

where (x0, ··· , xn) is a primitive integral vector, i.e., xi ∈ Z and gcdi(xi) = 1. Consider the following function which counts the number of rational points of bounded height:

◦ ◦ N(X , H, B) = #{ P ∈ X (Q) | H(P ) ≤ B }. where X◦ ⊂ X is a Zariski open subset and B is a positive real number. Manin’s program, initiated in [FMT89], relates the asymptotics of N(X◦, H, B), for a suitable

4 Zariski open subset X◦, to certain global geometric invariants of the underlying variety X. The following is the proposed conjecture:

Conjecture 1 (Manin’s conjecture). Let X be a smooth Fano variety defined over a number field F . Let L be a big, adelically metrized line bundle on X. Then, after taking a finite extension, there exist a Zariski open subset X◦ ⊂ X and a positive constant c(X, L) > 0 such that

N(X◦, L, B) = c(X, L)Ba(X,L) log(B)b(X,L)−1(1 + o(1)), B → ∞,

Here a(X,L) and b(X,L) are geometric constants which were introduced in this context in [FMT89] and [BM90] (and recalled in Chapter 2, Subsection 2.1.1) and c(X, L) is a Tamagawa-type number deﬁned in [Pey95], [BT98b].

Although Manin’s conjecture admits counterexamples, found by Batyrev and Tschinkel in [BT96b] (see Example 19 in Subsection 2.1.2), it has turned out to be quite inﬂuential. Over the last 20 years, it lead to many important theorems and stimulated the development of several new research directions and methods. The following methods have been intensively studied by various researchers:

• universal torsor method combined with lattice point counts;

• variants of the circle method;

• ergodic theory and mixing;

• and Height zeta functions and spectral theory on adelic groups.

The universal torsor method has been successful in the treatment of singular del

Pezzo surfaces over Q. The circle method was used to prove Manin’s conjecture

5 for low degree hypersurfaces in high dimensional projective spaces. Ergodic theory and mixing settled many new cases of equivariant compactiﬁcations of reductive groups. The Height zeta functions method has the advantage that it is able to analyze the second term of the counting function. For recent surveys highlighting these methods, see [Tsc09], [Bro07], [Bro09], [CL10], and [Oh10]. In this thesis, we discuss two diﬀerent aspects of Manin’s program: One is the theory of Height zeta functions, and another is the notion of balanced line bundles.

Height zeta functions

In part I, we discuss the theory of Height zeta functions. A Height zeta function is a certain Dirichlet series deﬁned by

◦ X −s Z(X , L, s) = H(P ) , s ∈ C. P ∈X◦(F )

This series absolutely and uniformly converges when <(s) 0, for a suitable Zariski open subset X◦ ⊂ X, so that Z(X◦, L, s) defines a holomorphic function for <(s) 0. The connection between Manin’s conjecture and Height zeta functions is given by the Tauberian theorem (see Theorem 27). Hence, our task here is to obtain the meromorphic continuation of the Height zeta function and to identify the location of the pole and its order. In general, the study of meromorphic continuations of Height zeta functions is extremely difficult, and it has been studied intensively when X is an equivariant compactification of a linear algebraic group

G by using Spectral theory on an adelic space G(AF ). In this part, ﬁrst we review the basic background of heights, Manin’s conjecture, and Height zeta functions, then we discuss results in [TT12], e.g., Height zeta functions of equivariant com-

6 pactifications of semi-direct products. We discuss the main term of a Height zeta function of an equivariant compactification of a solvable group, and we confirm that the main term coincides with the prediction of Manin’s conjecture. Then we study meromorphic continuations of Height zeta functions of the ax+b-group, and produce new examples of rational surfaces satisfying Manin’s conjecture.

Balanced line bundles

In part II, we introduce and study the notion of balanced line bundles. This part consists of results from [HTT12]. Implicitly, Manin’s conjecture asserts that in the case when L = −KX , any proper subvariety Y of X, which is not contained in the exceptional set X \ X◦, does not contribute a positive proportion to the main term of the asymptotic formula, i.e.,

N(Y ◦, L| , B) Y → 0 as B → ∞. N(X◦, L, B)

Internal consistency would imply that for all such Y we have

(a(Y,L|Y )), b(Y,L|Y )) < (a(X,L), b(X,L)), in the lexicographic ordering. We say that L is balanced with respect to Y when this strict inequality holds. One can ask the following question: “How can this inequality be checked by geometry?” This question arose from the study of ergodic theory in [GTBT11], where they needed a version of this statement to prove that the asymptotic volume of a height ball approximates the number of rational points of bounded height. In this part, we give a systematic study of balanced line bundles in terms of birational geometry, and explore the geometric meaning of this notion.

7 Part I

Height zeta functions

8 Chapter 1

Height functions

In this chapter, we review the theory of height functions. Height functions measure arithmetic and geometric complexities of solutions of Diophantine equations, and they provide us appropriate quantification of the size of solutions. This enables us to count the number of rational points on projective varieties with in- finitely many rational points. Height functions were initially developed by André Weil and D. G. Northcott, and a very nice exposition of the theory of the Weil height machine can be found in [HS00]. Here we discuss height theory which was developed in the context of Manin’s conjecture. The main references of this chapter are [CLT10], [Sal98], and [HS00]. We closely follow the exposition of [CLT10]

1.1 Metrics and Heights

Let F be a number field and oF its ring of integers. Let Val(F ) be the set of equivalence classes of absolute values of F . For each v ∈ Val(F ), we say v is non-archimedean if v defines the p-adic topology on Q, and v is archimedean if it defines the real topology on Q. We denote the corresponding completion of F with

9 respect to v by Fv and the ring of integers by ov when v is non-archimedean. If v is non-archimedean, we identify v with the discrete valuation of ov. Let | · |v be the associated norm on Fv, normalized by

  ordinal absolute value |x| if F = ,  v R  2 |x|v = square of ordinal absolute value |x| if Fv = C,   −v(x)  (Npv) if v is non-archimedean,

where pv is the unique maximal ideal of ov and Npv is the size of the ﬁnite residue

field ov/pv. With these norms, any additive Haar measure µ on the local field Fv satisfies

µ(xΩ) = |x|vµ(Ω). (1.1)

Furthermore, the product formula

Y |x|v = 1, (1.2) v∈Val(F ) holds for any x ∈ F × := F \{0}.

1.1.1 Metrics on local ﬁelds

Let F be a local field, i.e., R, C or a finite extension of Qp. Let X be a smooth variety defined over F . The set of F -valued points X(F ) naturally becomes a F -analytic manifold (see [Sal98, Section1] for definitions). Let U ⊂ X(F ) be an analytic open set. A complex valued function f : U → C is smooth if it is C∞ when F = R or C and it is locally constant when F is a p-adic field. For any non-vanishing analytic function f, P 7→ |f(P )| is a smooth function.

10 Let L be a line bundle on X. L(F ) is a F -analytic line bundle on X(F ). We denote a ﬁber of L at P ∈ X by LP :

Deﬁnition 2 (Smooth metrics). A smooth metric on L is a collection of functions

LP (F ) → R, for all P ∈ X(F ), denoted by l 7→ klk such that

• for any l ∈ LP (F ) \{0}, klk > 0;

• for any l ∈ LP (F ) and x ∈ F , kxlk = |x|klk;

• for any analytic open subset U ⊂ X(F ) and any non-vanishing analytic

section s ∈ Γ(U, L(F )), a function U 3 P 7→ ks(P )k ∈ R is smooth.

Note that k · k is smooth if and only if there exists a Zariski open covering {Ui} of

X and non-vanishing sections si ∈ Γ(Ui,L) such that Ui(F ) 3 P 7→ ksi(P )k ∈ R is smooth. The trivial line bundle OX admits a canonical metric deﬁned by k1k = 1. Let L, M be two metrized line bundles on X. We can deﬁne smooth metrics on L ⊗ M and L−1 by

|φ(l)| kl ⊗ mk = klkkmk, kφk = , klk

−1 where P ∈ X(F ), l ∈ LP (F ), m ∈ MP (F ), and φ ∈ LP (F ). An important construction of metrics is given by an integral model of X. Here we assume that F is non-archimedean. Let oF be a ring of integers of F . Suppose that we have a ﬂat oF -scheme X and a line bundle L on X extending X and L, respectively. The set of integral points X (oF ) ⊂ X(F ) is an open compact set in the analytic topology. Let U ⊂ X be a Zariski open subset trivializing L by a

0 ∗ section s ∈ H (U, L). For any integral point σ ∈ U(oF ) extending P ∈ X(F ), σ L

11 deﬁnes a oF -lattice in a F -vector space LP (F ). We deﬁne a metric on LP (F ) by

∗ for any l ∈ LP (F ), klk ≤ 1 if and only if l ∈ σ L(oF ).

It follows that kσ∗sk = 1 so that this metric is smooth. Since analytic open sets of the form U(oF ) cover X (oF ), we induced a smooth metric on X (oF ). In particular, when X is a projective model, this construction provides a smooth metric on X(F ).

n Example 3. Let X = PF = Proj(F [x0, ··· , xn]) be a projective space over F .

Consider a standard integral model X = Proj(oF [x0, ··· , xn]) → Spec(oF ). Let

L = OX (1) be the dual of the tautological bundle, which is extending L = OX (1). The induced metric from this integral model is given by

|l(x)| klk = , max{|x0|, ··· , |xn|}

where [x] = [x0 : ··· : xn] ∈ X(F ) and l ∈ L[x](F ).

1.1.2 Adelic metrics and heights

Let F be a number ﬁeld and Val(F ) the set of normalized norms of F .

Definition 4 (Adelic metrics). Let X be a smooth projective variety defined over F and L a line bundle on X. An adelic metric on L is a collection of v-adic smooth metrics k · kv on the associated analytic line bundles L(Fv) on the Fv- analytic varieties X(Fv), for all v ∈ Val(F ) such that there exist a Zariski open subset U ⊂ Spec(oF ), a projective flat morphism X → U, and a line bundle L on X , extending X and L respectively, and satisfying that for any non-archimedean place v ∈ U, k · kv is the induced metric from (X , L).

12 Remark 5. It is well-known that two projective ﬂat models are isomorphic at almost all places, so they deﬁne the same metrics at these places.

Deﬁnition 6 (Heights). Let X be a smooth projective variety deﬁned over F and L = (L, k · k) an adelically metrized line bundle on X. Let P ∈ X(F ) and l ∈ LP (F ). For almost all v ∈ Val(F ), we have klkv = 1 so that

Y klkv, v∈Val(F ) absolutely converges. We deﬁne a height function HL : X(F ) → R>0 by

Y −1 HL(P ) = klkv . v∈Val(F )

This function is well-deﬁned because of the product formula (1.2).

Example 7. Let F = Q, and we consider Example 3. We deﬁne a smooth metric at the archimedean place ∞ by

|l(x)|∞ klk∞ = , p 2 2 x0 + ··· + xn where [x] = [x0 : ··· : xn] ∈ X(Q) and l ∈ L[x](Q). It follows from the product formula that

q Y 2 2 HL([x]) = max{|x0|p, ··· , |xn|p}· x0 + ··· + xn. p:prime

This coincides with a height introduced in Introduction. See the equation (1).

Let X be a smooth projective variety deﬁned over F and L an adelically

13 metrized line bundle on X. A logarithmic function

log HL : X(F ) → R, agrees with Weil’s classical height machine up to O(1). See [HS00, Theorem B.3.2 and Theorem B.10.7]. In particular, they share some important properties:

Theorem 8. Let X be a smooth projective variety deﬁned over F and L = (L, k·k) an adelically metrized line bundle on X. Then we have

1. let B be the base locus of the linear system |L|. Then

log HL(P ) ≥ O(1),

for any P ∈ (X \ B)(F );

2. when L is ample, the set

{P ∈ X(F ) | HL(P ) ≤ B}

is ﬁnite where B > 0 is any positive real number.

For a proof, see [HS00, Theorem B.3.2].

1.1.3 Adelic heights

Let F be a number ﬁeld. The ring of adeles AF of F is the restricted product of all local ﬁelds Fv for all v ∈ Val(F ), with respect to ov for non-archimedean places v. It is a locally compact topological ring. Also it is known that F ⊂ AF is discrete and AF /F is compact. See [Tat67] for more details.

14 Let X be a smooth variety deﬁned over F . A locally compact topological space

X(AF ), which is called the space of adelic points on X, is deﬁned as follows: Choose an integral model X of X over a Zariski open subset U ⊂ Spec(oF ). Then, the set Q X(AF ) consists of all (Pv) ∈ v∈Val(F ) X(Fv) such that Pv ∈ X (ov) for almost all v ∈ U. In other words, X(AF ) is the restricted product of X(Fv) with respect to

X (ov) for all v ∈ U. Since such integral models are isomorphic at almost all places, the set X(AF ) does not depend on the choice of integral models. Moreover, since

X (ov) is open and compact for all v ∈ U, we can endow the restricted product Q topology on X(AF ). In particular, when X is projective, then X(AF ) = v X(Fv) and the topology is just given by the product topology. See, for example, [Sal98, Section 4] for more details.

Deﬁnition 9 (Adelic heights). Let X be a smooth projective variety deﬁned over F and L = (L, k · k) an adelically metrized line bundle on X. Choose a non-zero global section s ∈ H0(X,L), and consider a Zariski open subset X◦ := X \ div(s).

◦ ◦ We deﬁne an adelic height HL, s : X (AF ) → R>0 on an adelic space X (AF ) by

Y ◦ HL, s(P ) := HL, s,v(Pv), for P = (Pv) ∈ X (AF ). v∈Val(F )

−1 where HL, s,v(Pv) := ks(Pv)kv , which is called the local height associated to L

0 ◦ and s ∈ H (X,L). For any P = (Pv) ∈ X (AF ), ks(Pv)kv = 1 for almost all non-archimedean places v so that this adelic height is well-deﬁned. Note that the deﬁnition does depend on the choice of sections s ∈ H0(X,L).

Adelic heights satisfy some important properties which are analogous to properties in Theorem 8:

Lemma 10. Let X◦ = X \ div(s). Then,

15 ◦ 1. A function HL, s : X (AF ) → R>0 is continuous;

2. A logarithmic function is bounded from below, i.e.,

log HL, s(P ) ≥ O(1),

◦ for any P ∈ X (AF );

3. when L is ample, the set

◦ { P ∈ X (AF ) | HL, s(P ) ≤ B },

is compact in the adelic topology.

For proofs, see [CLT10, Lemma 2.3.2].

1.2 Measures

Let F be a number ﬁeld. For v ∈ Val(F ), Fv is a locally compact topological

ﬁeld, and we have an additive Haar measure µv on Fv. We ﬁx µv so that µv(ov) = 1 for almost all non-archimedean places v ∈ Val(F ).

1.2.1 Measures on local ﬁelds

Let F be a local ﬁeld and X a smooth variety deﬁned over F . Let n =

0 dim X and ω ∈ H (X,KX ) a n-form. Consider a F -analytic manifold X(F ).

Let x1, ··· , xn be F -analytic coordinates on an analytic open subset U ⊂ X(F ).

16 Then ω has the following form:

ω = f(x)dx1 ∧ · · · ∧ dxn, where f(x) is an analytic function. The product measure µn determines a measure on U, and we denote this measure by |dx1| · · · |dxn|. We deﬁne a measure |ω| by

|ω| = |f(x)||dx1| · · · |dxn|.

This measure does not depend on the choice of analytic coordinates. Thus, by using a partition of unity, we can extend a measure |ω| to X(F ). Slightly diﬀerent construction of |ω| can be found in [Sal98, Example 1.13].

Deﬁnition 11 (Measures induced from metrics). Let X be a smooth variety de-

fined over F . Choose a metrization k · k on the canonical line bundle KX . For any local non-vanishing top form ω, we define a measure by |ω|/kωk. Since this construction does not depend on the choice of ω, we can patch these measures and define a global measure on X(F ). We denote this measure by τX . For more details, see [CLT10, Section 2] or [Sal98, Section 1].

The following formula is due to Weil [Wei82]:

Theorem 12. Let F be a p-adic ﬁeld. Let X be a smooth variety deﬁned over F and X a smooth oF -model of X. We consider a smooth metric on KX which is induced from X . Suppose that µ(oF ) = 1. Then, we have

Z − dim X τX = q |X (k)|, X (oF )

where k is the residue ﬁeld of oF and q is the size of k.

17 For a proof, see [Wei82] or [Sal98, Corollary 2.15].

1.2.2 Tamagawa measures on adelic spaces

Let F be a number field. For any non-archimedean place v, we denote the residue field of ov by kv and the size of kv by qv. Let X be a smooth projective variety defined over F . In this subsection, we always assume that

1 2 H (X, OX ) = H (X, OX ) = 0.

This holds when X is a smooth rationally connected variety. Also note that the dimensions of these cohomology spaces are birationally invariant (see [CLT10, Re- mark 2.4.8]). The assumption guarantees that Pic(X)Q = NS(X)Q. We consider an adelically metrized canonical line bundle KX = (KX , k · k). One might hope Q to deﬁne a ﬁnite measure on a compact topological space X(AF ) = v X(Fv) by considering Y τX,v, v∈Val(F ) where τX,v is a measure on X(Fv), induced by k · kv. However, Theorem 12 claims that for almost all non-archimedean places v ∈ Val(F ),

|X (kv)| τX,v(X(Fv)) = dim X , qv

where X is a ﬂat projective oF -model of X, and the inﬁnite product

Y τX,v(X(Fv)), v∈Val(F )

18 never converges absolutely. Indeed, by Deligne’s theory of the Weil conjecture, one can conclude that for almost all non-archimedean places v,

! |X (k )| 1 1 v = 1 + Tr(Fr | Pic(X ) ) + O , (1.3) dim X v kv Q 3/2 qv qv qv where Fr is the geometric Frobenius acting on Pic(X ) . See [Pey95, p. 117]. v kv Q Thus, we need to introduce a regularization to obtain convergence:

Deﬁnition 13 (the Artin L-functions). Let X be a smooth projective model of

X over a Zariski open subset U ⊂ Spec(oF ) such that every ﬁber is geometrically connected. For any v ∈ U, we deﬁne the local L-function at v ∈ U by

L (s, Pic(X ) ) := det(1 − q−sFr | Pic(X ) )−1, v kv Q v v kv Q and we deﬁne the global L-function by

Y LU (s, Pic(X ) ) := L (s, Pic(X ) ). F Q v kv Q v∈U

This inﬁnite product converges absolutely and uniformly for <(s) > 1. Moreover,

U L (s, Pic(XF )Q) has a meromorphic continuation to the entire plane with a pole of order r = Pic(X) at s = 1. See [Pey95, Lemma 2.2.5].

[CLT10, Lemma 2.4.5] claims that L (σ, Pic(X ) ) is a positive real number v kv Q for any σ > 0. We deﬁne convergence factors λv by

  1/L (1, Pic(X ) ) if v ∈ U,  v kv Q λv =  1 if v∈ / U.

19 Then we have

1 2 Theorem 14. Assume that H (X, OX ) = H (X, OX ) = 0. The inﬁnite product

Y λvτX,v(X(Fv)), v∈Val(F ) converges absolutely.

Proof. Since we have,

det(1 − q−1Fr | Pic(X ) ) = 1 − q−1Tr(Fr | Pic(X ) ) + O(q−3/2), v v kv Q v v kv Q v it follows from 1.3 that

−3/2 λvτX,v(X(Fv)) = 1 + O(qv ).

Thus, our assertion follows from this. For more details, see [CLT10, Theorem 2.4.7].

Let

U r U L∗ (1, Pic(XF ) ) := lim(s − 1) L (s, Pic(XF ) ) > 0. Q s→1 Q

Deﬁnition 15 (Tamagawa measure). Let X be a smooth projective variety deﬁned over F such that

1 2 H (X, OX ) = H (X, OX ) = 0.

The Tamagawa measure on the adelic space X(AF ) is deﬁned as follows:

U Y τX := L∗ (1, Pic(XF )Q) λvτX,v. v∈Val(F )

20 Note that this definition does not depend on the choice of the model X . For more information regarding this measure, see [Sal98, Definition 6.12]. [CLT10] provides generalizations of Tamagawa measures for an open variety U with a smooth compactification X satisfying the above assumptions. See [CLT10, Definition 2.4.11].

21 Chapter 2

Manin’s conjecture and Height zeta functions

Manin’s conjecture was proposed in [FMT89] and [BM90], and more precise re- finement concerning the leading constant of the asymptotic formula was introduced in [Pey95] and [BT98b]. The conjecture has been confirmed to be true for large classes of varieties, including flag varieties, complete intersections of small degree, toric varieties, certain del Pezzo surfaces, and other equivariant compactifications of algebraic groups, and these results are based on several different methods. One of them is the Height zeta functions method which was introduced in [FMT89]. In this chapter, we describe the general picture of Manin’s conjecture, and we introduce Height zeta functions. Then we will discuss the strategy to obtain meromorphic continuations of Height zeta functions of equivariant compactifications of algebraic groups by using spectral theory on adelic groups. The main references of this chapter are [Tsc09] and [CL10].

22 2.1 Manin’s conjecture

2.1.1 Geometric invariants

In this subsection, we introduce global geometric invariants which play central roles in the context of Manin’s conjecture. Here we assume that the ground field is an algebraically closed field of characteristic zero. Let X be a smooth projective variety. We denote its Picard group by Pic(X). Let Pic0(X) be the subgroup consisting of isomorphism classes of line bundles algebraically equivalent to the trivial bundle OX . The Néron-Severi group NS(X) is defined by NS(X) = Pic(X)/Pic0(X).

It is well-known that NS(X) is a finitely generated abelian group, and its rank is called Picard rank, denoted by ρ(X). The Néron-Severi group is unchanged by algebraically closed base extension. See [MP09, Proposition 3.1]. We denote the real Néron-Severi group NS(X) ⊗ R by NS(X)R. We use

Λeff (X) ⊂ NS(X)R, to denote the cone of pseudo-effective divisors, i.e., the closure of the cone of effective Q-divisors on X in the real Néron-Severi group NS(X)R. This cone plays a central role in the Minimal Model Program. See, for example, [HK00].

Suppose that X is a smooth Fano variety, i.e., −KX is ample. Kodaira vanish-

1 ing theorem implies that H (OX ) = 0, so we have Pic(X)Q = NS(X)Q. The recent development of the Minimal Model Program [BCHM10, Corollary 1.3.1] concludes that X is a Mori dream space (see [HK00] for a deﬁnition) so that in particular, the

23 cone of pseudo-effective divisors Λeff (X) is finitely generated by effective divisors. Let L be a big line bundle, i.e., the numerical class [L] is in the interior of

Λeff (X) (see [Laz04a, Section 2.2] for a definition and other characterizations of big line bundles). We define invariants a(L) and b(L) as follows:

a(L) := min{a ∈ R | a[L] + [KX ] ∈ Λeﬀ (X)},

b(L) := the codimension of the minimal face of Λeﬀ (X) containing a(L)L + KX .

Since Λeff (X) is finitely generated by effective divisors, a(L) is a rational number. We will explore some birational geometric aspects of these invariants in Part II.

2.1.2 Manin’s conjecture

Let F be a number field. Let X be a smooth projective variety defined over F and L = (L, k · k) an adelically metrized big line bundle on X. Every big line bundle is linearly equivalent to a sum of an ample Q-divisor A and a effective Q-divisor E (see [Laz04a, Corollary 2.2.7]). Let B be the stable base locus of E (see [Laz04a, Definition 2.1.20]). Note that we can choose A and E to be defined over F so that B is also defined over F . Theorem 8 ensures that for any Zariski open subset X◦ ⊂ X \B, the following counting function of the number of rational points of bounded height is finite:

◦ ◦ N(X , L, B) := #{ P ∈ X (F ) | HL(P ) ≤ B } < ∞, where B > 0 is any positive real number. Note that in general, it is possible that N(X, L, B) = ∞ unless L is ample, so it is important to consider a suﬃciently

24 small Zariski open subset X◦. By the general principle of Diophantine geometry, one hopes to relate the asymptotic formula for the counting function N(X◦, L, B) as B → ∞, for a suit-

◦ able Zariski open subset X , to global geometric invariants of X when −KX is suﬃciently positive, e.g., −KX is ample. Manin and others proposed the following conjecture in [FMT89] and [BM90]:

Conjecture 16. Let X be a smooth Fano variety deﬁned over F and L an adelically metrized big line bundle on X. Then, after passing to a ﬁnite extension, there exists a Zariski open subset X◦ ⊂ X and a positive constant c(L) > 0 such that N(X◦, L, B) = c(L)Ba(L) log(B)b(L)−1(1 + o(1)), as B → +∞ where a(L) and b(L) were introduced in Subsecton 2.1.1. An adelic interpretation of the constant c(L) will be discussed in the next subsection 2.1.3.

n Example 17. Let F be a number ﬁeld and X = PF . We consider the following height function associated to O(1)X :

Y H([x]) := max{ |x0|v, ··· , |xn|v }, v∈Val(F )

where [x] = [x0 : ··· : xn] ∈ X(F ). Note that this height function is slightly diﬀerent from the height deﬁned in Example 7. The following formula is due to Schanuel [Sch79]: N(X, H, B) = c(F, n)Bn+1(1 + o(1)).

25 The constant c(F, n) is given by

n+1 hR/ω 2r1 (2π)r2 c(F, n) = √ (n + 1)r1+r2−1, ζF (n + 1) DF where each invariant is given by

h : class number of F ,

R : regulator of F ,

ω : number of roots of unity in F ,

ζF : zeta function of F ,

r1 : number of real embeddings of F ,

r2 : number of complex embeddings of F ,

DF : absolute value of the discriminant of F/Q.

See [Sch79] or [HS00, Section B.6].

Remark 18. It is essential to take a suﬃciently small Zariski open subset X◦ ⊂ X even when L is ample. For example, let X be a smooth cubic surface in P3. The

3 anticanonical line bundle L = −KX gives us an embedding into P . Manin’s conjecture predicts that, at least after taking a ﬁnite extension, the counting function N(X◦, L, B) behaves like B log(B)6 for a suitable Zariski open subset X◦. On the other hand, there are 27 lines on X, and the number of rational points on these lines behaves like B2. Thus rational points will be accumulating on these lines, and we need to remove all lines. Conjecturely, we believe that removing all lines will suﬃce. One of the concrete examples, for which Manin’s conjecture has been

26 conﬁrmed, is the singular cubic surface X ⊂ P3 deﬁned by

xyz = w3.

This is a singular cubic surface with three isolated singularities of type A2, which contains three lines {x = w = 0}, {y = w = 0}, {z = w = 0}. Moreover, X is a toric variety, so we can apply results of [BT98a] and [BT96a]. Let X◦ be the complement of three lines. [BT98a] and [BT96a] conclude that for the anticanonical line bundle L = −KX , the counting function behaves like

N(X◦, L, B) ∼ cB log(B)6, which coincides with the prediction of Manin’s conjecture. However, the number of rational points on three lines behaves like B2, so rational points are more accumulating on these lines than the complement.

Example 19. Conjecture 16 admits counter examples found by Batyrev and

3 3 Tschinkel in [BT96b]. Let X be a hypersurface in Px × Py deﬁned by

3 X 3 xiyi = 0. i=0

3 X is a smooth Fano 5-fold of Picard rank 2, and the ﬁrst projection π1 to Px provides us a diagonal cubic surfaces-ﬁbration on X. Manin’s conjecture asserts

◦ that for L = −KX , the counting function N(X , L, B) behaves like B log(B) for a

◦ suitable Zariski open subset X ⊂ X. On the other hand, a general ﬁber Xx of √ π1 is a diagonal cubic surface. Suppose that F contains Q( −3). Then the rank of Pic(Xx) is 7 whenever xi’s are cubic in F , and a more precise conjecture for

27 del Pezzo surfaces predicts that the number of F -rational points on these ﬁbers behaves like B log(B)6. Thus Conjecture 16 is not consistent. Actually Batyrev and Tschinkel proved that X cannot satisfy Conjecture 16. This failure is also related to failure of the balance property, which we will discuss in Part II.

2.1.3 Tamagawa number

Let F be a number ﬁeld and X a smooth Fano variety deﬁned over F . Consider an adelically metrized canonical line bundle KX = (KX , k · k). In this subsection, we discuss an adelic interpretation of the constant c(−KX ) in Conjecture 16, which was proposed by Peyre in [Pey95].

For any v ∈ Val(F ), we normalize a Haar measure µv in the following way:

  ordinal Lebesgue measure dx if F = ,  v R  µv = 2dudv where x = u + iv if Fv = C,   − 1  a measure normalized by µv(ov) = (Ndv) 2 if v is non-archimedean,

where dv is the local absolute diﬀerent. See [Tat67].

Let τX be a Tamagawa measure introduced in Subsection 1.2.2.

Deﬁnition 20. Let X(F ) ⊂ X(AF ) be the closure in the adelic topology. We deﬁne the Tamagawa number τ(−KX ) by

Z τ(−KX ) := τX . X(F )

Let (A, Λ) be a pair consisting of a lattice and a strictly convex closed cone Λ ⊂

∗ ∗ ∗ AR, i.e., Λ ∩ −Λ = {0}. Let (A , Λ ) be the dual of (A, Λ), i.e., A = Hom(A, Z)

28 and Λ∗ := {a∗ ∈ A∗ | (a∗, a) ≥ 0 for any a ∈ Λ}. R

We normalize a Haar measure da∗ on A∗ by vol(A∗ /A∗) = 1. R R

Deﬁnition 21. The X -function of Λ is deﬁned as the integral:

Z −(s,a∗) ∗ XΛ(s) = e da . Λ∗

This integral converges absolutely and uniformly when <(s) is contained in any compact subset in Λ◦ where Λ◦ is the interior of Λ. See [BT98a, Section 5] for more details.

Deﬁnition 22. We deﬁne α(X) as follows:

α(X) = XΛeﬀ (X)(−KX ).

Conjecture 23. The leading constant c(−KX ) in Conjecture 16 is given by

α(X)β(X)τ(−K ) c(−K ) = X , X (rk Pic(X) − 1)! where β(X) = #Br(X)/Br(F ). Here Br(X) and Br(F ) denote Brauer groups of X and F respectively. See [Man86] for more details.

n Example 24. [Pey95, Proposition 6.1.1] Let X = PF and consider the height

29 introduced in Example 17. Then, α(X) = 1/(n + 1) and β(X) = 1. We have

  2n(n + 1) if F = ,  v R  n τX,v(X(Fv)) = (2π) (n + 1) if Fv = C,   qn+1−1  (Nd )−n/2q−n v if v is non-archimedean.  v v qv−1

The Artin L-function is given by

−s −1 Lv(s, Pic(Xk¯v )) = (1 − qv ) , L(s, Pic(XF¯)) = ζF (s).

Since X(F ) = X(AF ), we conclude that

n+1 hR/ω 2r1 (2π)r2 r1+r2−1 c(−KX ) = √ (n + 1) , ζF (n + 1) DF which coincides with the constant in Example 17.

Generalization of the constant c(L) for arbitrary big line bundles is obtained by Batyrev and Tschinkel. See [BT98b].

2.2 Height zeta functions

2.2.1 Height zeta functions

Let F be a number ﬁeld and X a smooth projective variety deﬁned over F . Let L = (L, k · k) be an adelically metrized big line bundle on X. Inspired by Conjecture 16, we consider the following Dirichlet series, which is called the Height zeta function:

◦ X −s Z(X , L, s) = HL(P ) , P ∈X◦(F )

30 where X◦ is a Zariski open subset and s ∈ C.

Deﬁnition 25 (Abscissa of convergence). The abscissa of convergence αX◦ (L) is

 

 X −s  αX◦ (L) := inf σ ∈ R HL(P ) absolutely converges for <(s) > σ. .

 P ∈X◦(F ) 

When the above set is empty, we let αX◦ (L) = +∞. Since HL(P ) is a positive real

◦ number, αX◦ (L) is also equal to the inﬁmum of σ such that Z(X , L, s) converges for <(s) > σ. It could be that 0 ≤ αX◦ (L) ≤ +∞ or αX◦ (L) = −∞. The latter

◦ happens exactly when X (F ) is ﬁnite. Also note that αX◦ (L) does not depend on the choice of metrizations. Z(X◦, L, s) is a holomorphic function in the right half

◦ plane <(s) > αX◦ (L), and Z(X , L, s) cannot extend holomorphically to any larger right half plane <(s) > αX◦ (L) − for any > 0. See [HR64] for more details.

Lemma 26. We have

◦ 1. if L is ample, then αX◦ (L) < +∞ for any Zariski open set X ;

◦ 2. if L is big, then αX◦ (L) < +∞ for a suﬃciently small Zariski open set X .

Proof. When L is ample, Example 17 implies that the counting function N(X◦, L, B) grows at most polynomially. [HR64, Theorem 8] concludes that αX◦ (L) < +∞. Suppose that L is big. A big line bundle L can be expressed as a sum of an ample

Q-divisor A and a eﬀective Q-divisor E. Let B be the stable base locus of E and X◦ ⊂ X \ B a Zariski open subset. We ﬁx adelic metrizations A = (A, k · k) and E = (E, k · k). It follows from Theorem 8 that

HL(P ) HA(P ),

31 for any P ∈ X◦(F ). Our assertion follows from this.

The connection between Manin’s conjecture and Height zeta functions is given by a Tauberian theorem:

Theorem 27 (Tauberian theorem). Let {hn} be an increasing sequence of positive real numbers such that limn→∞ hn = +∞. Let {an} be another sequence of positive real numbers and consider +∞ X an Z(s) := . hs n=1 n Assume that this series converges absolutely to an analytic function for <(s) > a where a is some positive real number. Furthermore suppose that Z(s) has the following meromorphic continuation:

f(s) Z(s) = , (s − a)b where f(s) is an analytic function in the right half plane {<(s) > a − }, for some

> 0, b ∈ N, and f(a) = c > 0. Then,

X c N(B) := a = Ba log(B)b−1(1 + o(1)), n a(b − 1)! hn≤B as B → ∞.

See [Ten95, Section II.7, Theorem 15]. Thus our goal here is to obtain the meromorphic continuation of a Height zeta function. This task is extremely hard in general, and we will discuss how one can attack this problem when X is an equivariant compactiﬁcation of a linear algebraic group. The following theorem is also important in the analysis of Height zeta functions.

32 For U ⊂ Rn, we deﬁne a tube domain:

n TU := {s ∈ C | <(s) ∈ U}.

Then we have

Theorem 28 (Convexity principle). Let U ⊂ Rn be a connected open domain. ¯ Suppose that f is a holomorphic function on TU . Let U be the convex hull of U. Then f has a holomorphic extension to U¯.

See [FG02, Exercises in Section II-7].

2.2.2 Height zeta functions of equivariant compactiﬁca-

tions of algebraic groups

In this subsection, we explain the general strategy, which is described in [Tsc09, Subsection 6.3], to obtain a meromorphic continuation of a Height zeta function of an equivariant compactification of a connected linear algebraic group by using spectral theory on an adelic group. Let F be a number field and G a connected linear algebraic group defined over F . Suppose that X is a smooth projective equivariant compactification of G defined over F , i.e, X is a smooth projective variety with a group action by G such that the group action admits the open dense orbit which is isomorphic to G. Since

X is a rational variety, we have Pic(X)Q = NS(X)Q. We assume that G is acting on X from the right.

Choose a basis of Pic(X)Q consisting of ample line bundles L1, ··· ,Lr and ﬁx

33 adelic metrizations Li = (Li, k · k). For each Li, we have a height function

HLi : X(F ) → R>0,

and we deﬁne a height system H : Pic(X)C × X(F ) → C by extending linearly, i.e.,

! X Y si H siLi,P = HLi (P ) . i i

We are interested in a meromorphic continuation of a Height zeta function

X Z(s) = H(s, γ)−1. γ∈G(F )

This inﬁnite series converges absolutely and uniformly to a holomorphic function when <(si) 0 for any i. Step 1 : One deﬁnes an adelic height pairing

Y H = Hv : Pic(X)C × G(AF ) → C, v whose restriction to Pic(X) × G(F ) → R>0 descends to the above height system. One of important properties of the height pairing is K-invariance. The fact that the right action extends to X implies that the local height

Hv : Pic(X)C × G(Fv) → C,

is invariant under the right action of a compact subgroup Kv ⊂ G(Fv) for any non-archimedean place v. Moreover, Kv = G(ov) for almost all non-archimedean places v. This property plays a signiﬁcant role in the analysis of a Height zeta

34 Q function. Let K := v ﬁnite Kv. Step 2 : We consider a Height zeta function on an adelic space:

X Z(s, g) = H(s, γg)−1, γ∈G(F )

where s ∈ Pic(X)C and g ∈ G(AF ). One can prove that this inﬁnite series converges to a function which is continuous in g and is holomorphic in s when <(s) is suﬃciently large. Moreover, one can prove that

2 K Z(s, g) ∈ L (G(F )\G(AF )) , for a suﬃciently large <(s). Thus, we can apply spectral theory, i.e., the decomposition of L2-space into unitary irreducible representations, and we obtain

X Z(s, g) = Zπ(s, g), (2.1) π

2 K where π runs over all unitary irreducible representations occurring in L (G(F )\G(AF )) . Step 3 : Analysis of each term in (2.1) can be done by using geometric integral techniques developed in [CLT10]. For example, when π is trivial, then the trivial part is given by

Z Z −1 Y −1 Z0(s, g) = H(s, g) dg = Hv(s, gv) dgv, G(AF ) v G(Fv)

where dgv is the right invariant Haar measure on G(Fv). The study of this type of Height integrals has been carried out in complete generality in [CLT10], and in many examples, this trivial part coincides with the main term of a Height zeta

35 function. Other terms also can be studied by using techniques in [CLT10], and one hopes to obtain a meromorphic continuation from this spectral decomposition with geometric integration techniques. This strategy has been carried out and lead to a proof of Manin’s conjecture for the following varieties:

• toric varieties [BT95], [BT98b], [BT96a];

n • equivariant compactiﬁcations of additive groups Ga [CLT02];

• equivariant compactiﬁcations of unipotent groups [ST04], [ST];

• wonderful compactiﬁcations of semi-simple groups of adjoint type [STBT07].

Moreover, applications of Langland’s theory of Eisenstein series allowed proofs of Manin’s conjecture for ﬂag varieties [FMT89], their twisted products [Str01], and horospherical varieties [ST99], [CLT01]. In the next chapter, we apply this strategy to a relatively simple group, e.g., the ax + b-group. However, our results contain new ingredients. In previous work, the proofs of a suitable meromorphic continuation of such spectral decomposition relied on the fact that both the left and right action of G on itself extended to actions on X. This ensured that the sum in (2.1) was actually over a discrete set. In the noncommutative situation, this does not hold, generally. In [TT12], we conducted analysis that relied on tight control of local integrals with rapidly oscillating phase. Applying analytic techniques from [CLT10], we succeeded in ob- taining a meromorphic continuation of a Height zeta function for one-sided actions under some geometric conditions.

36 Chapter 3

Height zeta functions of equivariant compactiﬁcations of semi-direct products of algebraic groups

In this chapter, we discuss results of [TT12]. First, we review general geometric facts concerning equivariant compactifications of solvable algebraic groups. Then we discuss the main terms of Height zeta functions of equivariant compactifications of solvable algebraic groups and confirm that the main terms coincide with the prediction of Manin’s conjecture. In the last section, we discuss Height zeta functions of equivariant compactifications of the ax + b-group and prove Manin’s conjecture under some geometric conditions. We closely follow the exposition of [TT12].

37 3.1 Geometry of equivariant compactiﬁcations

In this section, we review geometric facts concerning equivariant compactifications of connected solvable linear algebraic groups. Here we assume that the ground field is an algebraically closed field of characteristic zero. This section is extracted from [TT12, Section 1].

3.1.1 Equivariant compactiﬁcations

Let G be a connected linear algebraic group. In dimension 1, the only examples are the additive group Ga := Spec(k[x]) and the multiplicative group Gm :=

−1 ∗ Spec(k[x, x ]). Let X (G) := Hom(G, Gm), the group of algebraic characters of G. For any connected linear algebraic group G, this is a torsion-free Z-module of ﬁnite rank. Let X be a projective equivariant compactiﬁcation of G. When X is normal, then it follows from Hartog’s theorem (see [Har77, Proposition 6.3A]) that the boundary D := X \ G, is a Weil divisor. Moreover, after applying equivariant resolution of singularities, if necessary, we may assume that X is smooth and that the boundary

D = ∪ιDι,

is a divisor with normal crossings. Here Dι are irreducible components of D. Let PicG(X) be the group of equivalence classes of G-linearized line bundles on X. (See [MFK94, Chapter 1, Section 3] for a deﬁnition.)

38 Generally, we will identify divisors, associated line bundles, and their classes in Pic(X), resp. PicG(X). The following proposition is taken from [TT12, Proposition 1.1]:

Proposition 29. Let X be a smooth and proper equivariant compactiﬁcation of a connected solvable linear algebraic group G. Then,

1. we have an exact sequence

0 → X∗(G) → PicG(X) → Pic(X) → 0,

G 2. Pic (X) = ⊕ι∈I ZDι, and

3. the closed cone of pseudo-eﬀective divisors of X is spanned by the boundary components: X Λeﬀ(X) = R≥0Dι. ι∈I

Proof. The first claim follows from the proof of [MFK94, Proposition 1.5]. The crucial point is to show that the Picard group of G is trivial. As an algebraic variety, a connected solvable group is a product of an algebraic torus and an affine space. The second and third assertions hold since every finite-dimensional representation of a solvable group has a fixed vector.

The following proposition is taken from [TT12, Proposition 1.2]:

Proposition 30. Let X be a smooth and proper equivariant compactiﬁcation for the left action of a connected linear algebraic group. Then the right invariant top

39 degree diﬀerential form ω on X◦ := G ⊂ X satisﬁes

X −div(ω) = dιDι, ι∈I

where dι > 0. The same result holds for the right action and the left invariant form.

Proof. This fact was proved in [HT99, Theorem 2.7] or [CLT02, Lemma 2.4]. Sup- pose that X has the left action. Let g be the Lie algebra of G. For any ∂ ∈ g, the global vector ﬁeld ∂X on X is deﬁned by

X ∂ (f)(x) = ∂gf(g · x)|g=1,

where f ∈ OX (U) and U is a Zariski open subset of X. Note that this is a right

◦ invariant vector ﬁeld on X = G. Let ∂1, ··· , ∂n be a basis for g. Consider a global section of det TX ,

X X δ := ∂1 ∧ · · · ∧ ∂n , which is the dual of ω on X◦. The proof of [CLT02, Lemma 2.4] implies that δ vanishes along the boundary. Thus our assertion follows.

The following proposition is taken from [TT12, Proposition 1.3]:

Proposition 31. Let X be a smooth and proper equivariant compactiﬁcation of a connected linear algebraic group. Let f : X → Y be a birational morphism to a normal projective variety Y . Then Y is an equivariant compactiﬁcation of G such that the contraction map f is a G-morphism.

Proof. This fact was proved in [HT99, Corollary 2.4]. Choose an embedding Y,→

40 PN , and let L be the pullback of O(1) on X. Since Y is normal, Zariski’s main theorem implies that the image of the complete linear series |L| is isomorphic to Y . According to [MFK94, Corollary 1.6], after replacing L by a multiple of L, if necessary, we may assume that L carries G-linearizations. Fix one G-linearization of L. This deﬁnes the action of G on H0(X,L) and so on P(H0(X,L)∗). Now note that the morphism

0 ∗ Φ|L| : X → P(H (X,L) ), is a G-morphism with respect to this action. Thus our assertion follows.

3.1.2 Semi-direct products of Ga and Gm

The simplest solvable groups are Ga and Gm, as well as their products. New examples arise as semi-direct products. For example, let

ϕd : Gm → Gm = GL1, a 7→ ad and put

Gd := Ga oϕd Gm, where the group law is given by

(x, a) · (y, b) = (x + ϕd(a)y, ab).

Note that Gd ' G−d. Now we collect several results illustrating speciﬁc phenomena connected with noncommutativity of Gd and with the necessity to distinguish actions on the left, on the right, or on both sides. These play a role in the analysis

41 of height zeta functions in following sections. The following lemma is taken from [TT12, Lemma 1.4]:

Lemma 32. Let X be a biequivariant compactiﬁcation of a semi-direct product

GoH of linear algebraic groups. Then X is a one-sided (left- or right-) equivariant compactiﬁcation of G × H.

Proof. Fix one section s : H → G o H. Deﬁne a left action by

(g, h) · x = g · x · s(h)−1, for any g ∈ G, h ∈ H, and x ∈ X.

In particular, there is no need to invoke noncommutative harmonic analysis in the treatment of height zeta functions of biequivariant compactiﬁcations of general solvable groups since such groups are semi-direct products of tori with unipotent groups and the lemma reduces the problem to a one-sided action of the direct product. Height zeta functions of direct products of additive groups and tori can be treated by combining the methods of [BT98a] and [BT96a] with [CLT02], see Theorem 48. However, Manin’s conjectures are still open for one-sided actions of unipotent groups, even for the Heisenberg group.

The next observation is that the projective plane P2 is an equivariant compact- iﬁcation of Gd, for any d. Indeed, the embedding

2 (x, a) 7→ (x : a : 1) ∈ P deﬁnes a left-sided equivariant compactiﬁcation, with boundary a union of two

42 lines. The left action is given by

d (x, a) · (x0 : x1 : x2) 7→ (a x0 + xx2 : ax1 : x2).

The following proposition is taken from [TT12, Proposition 1.5]:

Proposition 33. If d 6= 1, 0, or −1, then P2 is not a biequivariant compactiﬁcation of Gd.

Proof. Assume otherwise. Proposition 29 implies that the boundary must consist of two irreducible components. Let D1 and D2 be the two irreducible boundary

2 ∼ components. Since O(KP ) = O(−3), it follows from Proposition 30 that either both components D1 and D2 are lines or one of them is a line and the other a conic. Let ω be a right invariant top degree diﬀerential form. Then ω/ϕd(a) is a left invariant diﬀerential form. If one of D1 and D2 is a conic, then the divisor of ω takes the form

−div(ω) = −div(ω/ϕd(a)) = D1 + D2,

but this is a contradiction. If D1 and D2 are lines, then without loss of generality, we can assume that

−div(ω) = 2D1 + D2 and − div(ω/ϕd(a)) = D1 + 2D2.

However, div(a) is a multiple of D1 − D2, which is also a contradiction.

Combining this result with Proposition 31, we conclude that a del Pezzo surface is not a biequivariant compactiﬁcation of Gd, for d 6= 1, 0, or, −1. Another sample result in this direction is the following proposition which is taken from [TT12, Proposition 1.6]:

43 Proposition 34. Let S be the singular quartic del Pezzo surface of type A3 + A1 deﬁned by

2 2 x0 + x0x3 + x2x4 = x1x3 − x2 = 0

Then S is a one-sided equivariant compactiﬁcation of G1, but not a biequivariant compactiﬁcation of Gd if d 6= 0.

Proof. For the ﬁrst assertion, see [DL10, Section 5]. Assume that S is a biequivariant compactiﬁcation of Gd. Let π : Se → S be its minimal desingularization. Then

Se is also a biequivariant compactiﬁcation of Gd because the action of Gd must ﬁx the singular locus of S. See [DL10, Lemma 4]. It has three (−1)-curves L1, L2, and L3, which are the strict transforms of

{x0 = x1 = x2 = 0}, {x0 + x3 = x1 = x2 = 0}, and {x0 = x2 = x3 = 0},

respectively, and has four (−2)-curves R1, R2, R3, and R4. The nonzero intersection numbers are given by:

L1.R1 = L2.R1 = R1.R2 = R2.R3 = R3.L3 = L3.R4 = 1.

Since the cone of curves is generated by the components of the boundary, these negative curves must be in the boundary because each generates an extremal ray.

Since the Picard group of Se has rank six, it follows from Proposition 29 that the number of boundary components is seven. Thus, the boundary is equal to the union of these negative curves.

2 Let f : Se → P be the birational morphism which contracts L1, L2, L3, R2, and

R3. According to Proposition 31, this induces a biequivariant compactiﬁcation on

44 2 −1 2 P . The birational map f ◦ π : S 99K P is given by

2 S 3 (x0 : x1 : x2 : x3 : x4) 7→ (x2 : x0 : x3) ∈ P .

The images of R1 and R4 are {y0 = 0} and {y2 = 0} and we denote them by D0 and D2, respectively. The images of L1 and L2 are (0 : 0 : 1) and (0 : 1 : −1), respectively; so that the induced group action on P2 must fix (0 : 0 : 1), (0 : 1 : −1), and D0 ∩ D2 = (0 : 1 : 0). Thus, the group action must fix the line D0, and this fact implies that all left and right invariant vector fields vanish along D0. It follows that

−div(ω) = −div(ω/ϕd(a)) = 2D0 + D2, which contradicts d 6= 0.

The following examples are taken from [TT12, Examples 1.7 and 1.8]:

∗ 1 Example 35. Let l ≥ d ≥ 0. The Hirzebruch surface Fl = PP ((O ⊕ O(l)) ) is a biequivariant compactiﬁcation of Gd. Indeed, we may take the embedding

Gd ,→ Fl

l (x, a) 7→ ((a : 1), [1 ⊕ xσ1]),

1 where σ1 is a section of the line bundle O(1) on P such that

div(σ1) = (1 : 0).

1 1 Let π : Fl → P be the P -ﬁbration. The right action is given by

l d l ((x0 : x1), [y0 ⊕ y1σ1]) 7→ ((ax0 : x1), [y0 ⊕ (y1 + (x0/x1) xy0)σ1]),

45 −1 1 on π (U0 = P \{(1 : 0)}) and

l l l−d l ((x0 : x1), [y0 ⊕ y1σ0]) 7→ ((ax0 : x1), [a y0 ⊕ (y1 + (x1/x0) xy0)σ0]),

−1 1 on U1 = π (P \{(0 : 1)}). Similarly, one deﬁnes the left action. The boundary

−1 −1 consists of three components: two ﬁbers f0 = π ((0 : 1)), f1 = π ((0 : 1)) and the special section D characterized by D2 = −l.

Example 36. Consider the right actions in Examples 35. When l > d > 0, these actions fix the fiber f0 and act multiplicatively, i.e., with two fixed points, on the

ﬁber f1. Let X be the blowup of two points (or more) on f0 and of one ﬁxed point

P on f1 \ D. Then X is an equivariant compactiﬁcation of Gd which is neither a

2 toric variety nor a Ga-variety. Indeed, there are no equivariant compactiﬁcations

2 2 of Gm on Fl ﬁxing f0, so X cannot be toric. Also, if X were a Ga-variety, we

2 would obtain an induced Ga-action on Fl ﬁxing f0 and P . However, the boundary consists of two irreducible components and must contain f0, D, and P because D is a negative curve. This is a contradiction. In Section 3.3, we prove Manin’s conjecture for X with l ≥ 3.

3.2 The main terms of Height zeta functions of

equivariant compactiﬁcations of solvable al-

gebraic groups

In this section, we discuss the main term of a Height zeta function of an equivariant compactiﬁcation of a solvable group. This subsection is based on results in

46 [TT12, Section 2]. Let F be a number field and G a connected solvable linear algebraic group defined over F . (See [Bor85] for a definition.) We may fix a flat group scheme G over Spec(oF ) whose the generic fiber is isomorphic to G. For example, choose an embedding G,→ GLn and take the Zariski closure of G in GLn over Spec(oF ). We frequently denote the set of integral points G(ov) by G(ov) for a non-archimedean place v. For connected solvable linear algebraic groups, the maximal tori are all conjugate to each other. Let T be a maximal torus. For simplicity, we assume

∼ r that T is split, i.e., T = Gm over F . Let U be the set of unipotent elements in G. It is known that U is a closed, connected, nilpotent, normal subgroup of G, and G/U is isomorphic to T so that G is a semi-direct product of T and U. For more information, see [Spr81, Subsection 6.3]. From now on, we always assume that U is unipotent and hence U is a closed subgroup of the group of upper triangular matrices whose diagonal entries are 1. Let X be a smooth projective equivariant compactiﬁcation of G deﬁned over F . We assume that G is acting on X from the right. Consider the boundary divisor D = X \ G and its irreducible decomposition

D = ∪ι∈I Dι.

We assume that the irreducible components are geometrically irreducible and a P boundary divisor D = ι Dι is a divisor with strict normal crossings, i.e., components are geometrically meeting transversely.

47 3.2.1 Height pairings and Height zeta functions

Let Lι = O(Dι) and fι a F -section corresponding to an eﬀective divisor Dι.

Fix adelic metrizations Lι = (Lι, k · k). We consider the local height functions deﬁned in Subsection 1.1.3

HLι,fι,v : G(Fv) → R>0, for v ∈ Val(F ). According to Proposition 29, there is a canonical isomorphism

G ∼ Pic (X) = ⊕ι∈I ZDι.

With this identiﬁcation, we deﬁne the local height pairing

G Hv : Pic (X)C × G(Fv) → C and the adelic height pairing

G H : Pic (X)C × G(AF ) → C as follows:

Y sι Y Hv(s, gv) = HLι,fι,v(gv) , H(s, g) = Hv(s, gv), ι v∈Val(F )

P where s = ι sιDι and g = (gv)v∈Val(F ) ∈ G(AF ). By adjusting adelic metrizations, we may assume that the following property holds: For any f ∈ X∗(G), let div(f) =

48 P ι dιDι. Then

Hv(s, gv)|f(gv)|v = Hv(s − d, gv).

G Due to the product formula, the restriction of H to Pic (X)Q × G(F ) descends to the height system on Pic(X)Q × G(F ), which is introduced in Subsection 2.2.2. The following lemma plays a central role in the study of Height zeta functions:

Lemma 37. For any non-archimedean place v ∈ Val(F ), there exists a compact open subgroup Kv ⊂ G(ov) such that Hv(s, gv) is invariant under the right action of Kv. Moreover, Kv = G(ov) for almost all non-archimedean places v ∈ Val(F ).

For a proof, see [CLT02, Lemma 3.2 and Proposition 3.3]. We deﬁne

Y K = Kv. v:ﬁnite

We consider the Height zeta function on an adelic space:

X Z(s, g) = H(s, γg), γ∈G(F )

G where s ∈ Pic (X)C and g ∈ G(AF ).

Lemma 38. Z(s, g) converges absolutely and uniformly to a function which is holomorphic in s and is continuous in g when <(s) is suﬃciently large.

P Proof. Consider an ample divisor L = ι tιDι where tι ∈ Z>0. We may assume that the abscissa of convergence αG(L) < 1 so that

X H(L, γ)−1, γ∈G(F )

49 converges absolutely. Let g = (gv)v ∈ G(AF ). For any archimedean place v, let Kv ⊂ G(Fv) be a suﬃciently small open neighborhood containing e and let Q U = K × v|∞ Kv. Then gU ⊂ G(AF ) is an open neighborhood of g. Since the

∗ right action extends, rhfι is also a section corresponding to Dι where rh is the right multiplication map by h ∈ G. This implies that

log kfι(γgvkv)kv = log kfι(γ)kv + O(1),

where γ ∈ G(F ) and kv ∈ Kv. Moreover, for almost all non-archimedean places v, we have

kfι(γgvkv)kv = kfι(γ)kv.

Hence we can conclude that

X X H(L, γg0)−1 H(L, γ)−1 < ∞, γ∈G(F ) γ∈G(F ) where g0 ∈ gU.

In general, let <(sι) > tι. According to Lemma 10, there exists a positive constant Cι > 0 such that

HLι,fι (g) ≥ Cι, for any g ∈ G(AF ). Thus we conclude that

X −1 X −1 Y −(<(sι)−tι) H(<(s), gu) ≤ H(L, γgu) Cι , γ∈G(F ) γ∈G(F ) ι where u ∈ U. Our assertion follows from this.

Let duv be a Haar measure on U(Fv). Since every unipotent group has no

50 algebraic characters, duv is both left and right invariant. We normalize them by

Z duv = 1, U(ov)

Q for almost all non-archimedean places v. Let du = v duv. This is a Haar measure on U(AF ). Since U(F )\U(AF ) is compact, we may assume that

Z du = 1. U(F )\U(AF )

× Let dav be a multiplicative Haar measure on Gm(Fv) which is normalized by

  dx /|x | if v is archimedean, ×  v v v dav =  Npv dx /|x | if v is non-archimedean,  Npv−1 v v v

where dxv is a normalized additive Haar measure which is introduced in Subsec-

× tion 2.1.3. Let dtv be a Haar measure on T (Fv) which is a product of dav , and Q let dt = v dtv which is a Haar measure on T (AF ). For a semi-direct product G = U o T , we consider the following Haar measures:

Y dgv = duvdtv, dg = dgv. v

They are right invariant Haar measures, but not left invariant.

Proposition 39. If <(s) is suﬃciently large, then

1 2 K Z(s, g) ∈ L (G(F )\G(AF ), dg) ∩ L (G(F )\G(AF ), dg) .

Proof. A proof here is a generalization of the proof in [TT12, Lemma 5.2].

51 Integrability follows from [CLT10, Proposition 4.3.4] since

Z Z X |Z(s, g)| dg ≤ H(<(s), γg)−1 dg G(F )\G(AF ) G(F )\G(AF ) γ∈G(F ) Z = H(<(s), g)−1 dg. G(AF )

∞ To prove that Z(s, g) is square-integrable, we prove that Z(s, g) ∈ L (G(F )\G(AF )). Consider a toric variety T ⊂ Y = (P1)r and an equivariant projective compactiﬁ- cation Z of U. Let π : X˜ → X be a resolution of indeterminacy such that rational maps X 99K Y mapping ut 7→ t and X 99K Z mapping ut 7→ u are honest mor- phisms where u ∈ U and t ∈ T . Note that the right action of G does not extend to ˜ X since a rational map X 99K Z mapping ut 7→ u is not a G-rational map unless

G is a product group of U and T . Let H1 and H2 be pull backs of ample boundary divisors on Y and Z respectively. Fix positive real numbers c, d. It follows from Lemma 10 that

H(<(s), g) H(cH1, g) H(dH2, g), assuming that <(s) is suﬃciently large and is contained in a compact subset of

G Pic (X)R. Write g = ut ∈ G(AF ) where u ∈ U(AF ) and t ∈ T (AF ).

X −1 X −1 −1 H(<(s), γg) H(cH1, γg) H(dH2, γg) γ∈G(F ) γ∈G(F )

X −1 −1 = H(cH1, αt) H(dH2, βϕαu) α∈T (F ), β∈U(F )

X −1 X −1 = H(cH1, αt) H(dH2, βϕαu) , α∈T (F ) β∈U(F ) where ϕ : T → Aut(U) is a homomorphism given by conjugation. Note that the

52 following Dirichlet series

X −1 Z2(d, u) = H(dH2, βu) , β∈U(F ) is a Height zeta function of an equivariant compactiﬁcation Z, and Lemma 38 implies that it is continuous in u ∈ U(AF ) when d is suﬃciently large. Moreover,

U(F )\U(AF ) is compact, so it follows that Z2(d, u) is bounded. Hence we need to bound

X −1 Z1(c, t) = H(cH1, αt) , α∈T (F ) but this is a Height zeta function of Y . For our purpose, we only need to bound it for a speciﬁc height, so we may assume that

r Y Y −1 H(H1, t) = max{|ti,v|v, |ti,v|v }, i=1 v

r where t = (t1, ··· , tr) ∈ Gm(AF ). Then Z1(c, t) is a product of a Height zeta function of P1, so we may assume that r = 1. Note that

X −1 Z1(c, t) ≤ Hﬁn(cH1, αt) , α∈T (F )

Q −1 where Hfin(H1, t) = v:finite max{|tv|v, |tv|v }. The later series converges absolutely when c > 1. Since the later series is T (F )-periodic, finiteness of the class group of F implies that it takes only finitely many values. Thus our assertion follows.

53 3.2.2 Poisson formula and the main term of a Height zeta

function

Poisson formula was used in the analysis of Height zeta functions of additive groups and toric varieties (see [BT96a], [BT98a], and [CLT02]):

Theorem 40. [DE09, Theorem 3.6.3] Let G be a locally compact abelian group with a Haar measure dg, H ⊂ G a closed subgroup with a Haar measure dh. Let f ∈ L1(G) and suppose that its Fourier transform fˆ is also L1-function on H⊥ where H⊥ is the group of topological characters χ : G → S1 which are trivial on H. Then Z Z f(h) dh = fˆ(χ) dχ, H H⊥ where dχ is the Plancherel Haar measure on H⊥.

It is expected that for the anticanonical bundle −KX , the main term of a Height zeta function Z(s, g) is given by

Z Z0(s, g) = Z(s, ug) du. U(F )\U(AF )

Note that Z0(s, ut) = Z0(s, t) where u ∈ U(AF ) and t ∈ T (AF ), and so we have

1 2 Z0(s, t) ∈ L (T (F )\T (AF ), dt) ∩ L (T (F )\T (AF ), dt)

54 which follows from Jensen’s inequality. We apply Poisson formula 40 and obtain

Z Z Z0(s, id) = Z(s, g)χ ¯(g) dg dχ ∗ (T (F )\T (AF )) G(F )\G(AF ) Z Z = H(s, g)−1 χ¯(g) dg dχ ∗ (T (F )\T (AF )) G(AF ) Z = Hb(s, χ) dχ, ∗ (T (F )\T (AF ))

∗ where (T (F )\T (AF )) is the set of automorphic characters and

Z Hb(s, χ) = H(s, g)−1 χ¯(g) dg. G(AF )

For any non-archimedean place v, let KT,v be the image of Kv by a homomorphism G → T . KT,v ⊂ T (ov) is a compact open subgroup, and KT,v = T (ov) for almost all non-archimedean places v. For simplicity, we may assume that the equality holds for all non-archimedean places. This is possible after replacing metrics by the average of metrics over compact subgroups T (ov). For any archimedean

r 1 r place v, let KT,v = {±1} if Fv = R and KT,v = (S ) if Fv = C. Again, we may Q assume that the height pairing is right KT,v-invariant. Let KT = v∈Val(F ) KT,v. Since the local height pairing H(s, g) is right K-invariant (Lemma 37), we have

Hb(s, χ) = 0,

∗ if χ∈ / (T (F )KT \T (AF )) . It follows that

Z Z0(s, id) = Hb(s, χ) dχ. (3.1) ∗ (T (F )KT \T (AF ))

∼ We now describe the set of characters on T (F )KT \T (AF ). Let M = Hom(T, Gm) =

55 Zr and N = M ∗ = Hom(M, Z). We have a natural surjective map:

T (AF ) → NR,

P mapping T (AF ) 3 (tv) 7→ (M 3 m 7→ v log |m(tv)|v ∈ R) ∈ NR. We denote its 1 kernel by T (AF ).

Proposition 41. We have the following properties:

1 ∼ 1. T (AF )\T (AF ) = NR;

1 2. T (F )\T (AF ) is compact;

1 3. T (F )KT \T (AF ) is isomorphic to a product of a ﬁnite abelian group and a compact abelian group.

4. w(T ) = KT ∩ T (F ) is a ﬁnite abelian group of torsion elements.

See [BT96a, Proposition 3.2]. We look at the third statement in more detail. First we have a natural surjective homomorphism to the class group of T :

1 T (F )KT \T (AF ) → Cl(T ) → 0,

Q 0 and its kernel is given by T (oF )KT \( T (ov) × T ) where v-∞

0 Y X T = {(tv) ∈ T (Fv) | log |m(tv)|v = 0, for any m ∈ M}. v|∞ v|∞

Q 0 Q 0 We have T (oF )KT \( T (ov) × T ) = (T (oF ) KT,v)\T and a natural iso- v-∞ v|∞ morphism ∼ ⊕v|∞T (Fv)/KT,v = ⊕v|∞NR =: NR,∞.

56 Let N 1 , E ⊂ N be the images of T 0 and T (o ) under this isomorphism R,∞ T R,∞ F respectively. Dirichlet’s unit theorem implies that N 1 /E is compact and E ∼= R,∞ T T

T (oF )/w(T ). We have the following isomorphism:

Y (T (o ) K )\T 0 ∼= N 1 /E . F T,v R,∞ T v|∞

Thus non-canonically T (F )K \T 1( ) is isomorphic to the product of N 1 /E T AF R,∞ T 1 ∗ and Cl(T ). Let UG = (T (F )KT \T (AF )) . UG is a ﬁnitely generated abelian group, and

∗ (T (F )KT \T (AF ))

∗ is non-canonically isomorphic to MR×UG. The natural map MR → (T (F )KT \T (AF )) is given by

Y i ∗ M 3 m 7→ ((tv)v 7→ |m(tv)|v) ∈ (T (F )KT \T (AF )) . v

Let M 1 be the quotient of the diagonal map: R,∞

MR 3 m 7→ ⊕v|∞m ∈ MR,∞ := ⊕v|∞MR, which is the dual of N 1 . We have a natural map R,∞

Φ:(T (F )K \T ( ))∗ → M 1 T AF R,∞ whose image is the kernel of M 1 → E ∗ and is a maximal rank lattice. The kernel R,∞ T ∗ of Φ is non-canonically isomorphic to the product of MR and Cl(T ) .

57 Next we discuss the local height integrals and its Euler product:

Z −1 Hbv(s, χv) = Hv(s, g) χ¯v(g) dgv G(Fv) Y Hb(s, χ) = Hbv(s, χv). v

Let X div(ω) = − κιDι, ι where ω is the top degree right invariant form. When χ = 1, then Hbv(s, χv) has the following analytic properties:

Proposition 42. For any v ∈ Val(F ),

Z −1 Hbv(s, 1) = Hv(s, g) dgv G(Fv)

is holomorphic when <(sι) > κι − 1, and

Y −1 ζFv (sι − κι + 1) Hbv(s, 1) ι∈I

−s −1 is holomorphic when <(sι) − κι + 1 > −1/2. Here ζFv (s) = (1 − qv ) is the

Dedekind zeta function of Fv.

See [CLT10, Lemma 4.1.1 and Proposition 4.1.2]. Moreover, for almost all non- archimedean places v, we have a formula of J. Denef ([Den87, Theorem 3.1]). Let S be a ﬁnite set of non-archimedean places such that for any v∈ / S, we have a smooth integral model X at v, the metric at v is induced from X , the reduction of boundary components are smooth geometrically irreducible divisors with normal

58 crossings, and µv(ov) = 1. For any I ⊂ I, we deﬁne

◦ 0 0 XI := ∩ι∈I Dι,XI := XI \ ∪I )I XI ,

which is the stratiﬁcation of the boundary; by convention X∅ = G. Then we have

Proposition 43. [CLT10, Proposition 4.1.6] For any ﬁnite place v∈ / S,

Z ◦ ! −1 −1 X #XI (kv) Y qv − 1 Hbv(s, 1) = Hv(s, gv) dgv = τv(G) , dim(X) qsι−κι+1 − 1 G(Fv) I⊂I qv ι∈I v

where τv(G) is the local Tamagawa number of G,

#G(k ) τ (G) = v . v qdim(G)

Convergence of Euler products follows from Proposition 43:

Proposition 44. Y Hb(s, 1) = Hbv(s, 1), v is holomorphic when <(sι) > κι, and

Y −1 ζF (sι − κι + 1) Hb(s, 1), ι

has a holomorphic continuation to the domain deﬁned by <(sι) > κι − 1/2.

See [CLT10, Proposition 4.3.4].

Let χ = χaχu where χa ∈ MR and χu ∈ UT . Let m(χa) be the coordinate of

59 ∗ G the image of a character χa by X (G)R → Pic (X)R = ⊕ιRDι. Then we have

Hb(s, χ) = Hb(s − im(χa), χu).

Let χu ∈ T (F )KT \T (AF ) be a character of ﬁnite order. Choose a basis χ1, ··· , χr

∗ for X (T ) = M. Letχ ˆ1, ··· , χˆr be the dual basis for X∗(T ) = Hom(Gm,T ). For any ι ∈ I, we deﬁne a ﬁnite order character from Gm as follows:

r Y mι(χi) χu,ι = (χu ◦ χˆi) . i=1

This deﬁnition does not depend on the choice of a basis. Analytic properties of

Hb(s, χ) also can be obtained by using techniques in [CLT10].

Proposition 45. Write χ = χaχuχ∞ where χa ∈ MR, χu : a ﬁnite order character, and χ∞ : a character at archimedean places. Then

Y −1 Φ(s, χ) := LF (sι − κι + 1 − imι(χa), χu,ι) Hb(s, χ), ι∈I

has a holomorphic continuation to the domain deﬁned by <(sι) > κι − 1/2 where

LF (s, χu,ι) is a Hecke L-function of χu,ι. Fix a compact set K in the domain deﬁned by <(sι) > κι − 1/2. Then we have

|Φ(s, χ)| = O(1),

∗ for any <(s) ∈ K and χ ∈ (T (F )KT \T (AF )) .

For a proof, one can apply techniques in [CLT10]. Also see [BT96a, Proposition 4.11] for example.

60 So far we have not justiﬁed the identity (3.1) obtained from the Poisson formula. To do this, one can combine integration by parts with respect to left invariant vector ﬁelds ti∂/∂ti at archimedean places, where (t1, ··· , tr) ∈ T , with techniques in [CLT02, Proposition 2.2 and Proposition 8.4]. Fix a norm k · k on MR,∞ =

⊕v|∞MR. We consider the natural map

∗ m∞ :(T (F )KT \T (AF )) → MR,∞.

The image of this homomorphism is non-canonically isomorphic to the direct sum of M and the lattice in M 1 . R R,∞

Proposition 46. For any N ∈ N and a compact set K in a domain deﬁned by

<(sι) − κι > −1, there exists a positive constant C > 0 such that

Y C Hbv(s, χ) ≤ N , (1 + km∞(χ)k) v|∞ for any <(s) ∈ K.

For a proof, see [CLT02, Proposition 2.2 and Proposition 8.4]. Upshot is:

• A function Y X F(s) = (sι − κι) Hb(s, χu),

ι∈I χu∈UG

is holomorphic when <(sι) − κι > −1/2.

• The main term of the Height zeta function

Z Y −1 Z0(s, id) = (sι − κι − imι(χa)) F(s − im(χa)) dχa, (3.2) χ ∈M a R ι∈I

61 ◦ is holomorphic in the interior of the shifted cone −KX + Λeﬀ (X) .

Assertions follow from Propositions 45 and 46. Note that for any χ ∈ X∗(G) and t ∈ R,

Z0(s + itm(χ), id) = Z0(s, id).

Proposition 29 implies that Z0(s, id) descends to a function on Pic(X)C, and it is ◦ holomorphic in −KX + Λeff (X) because the image of a simplicial cone is exactly the cone of effective divisors. [CLT01, Section 3] discussed this type of integration (3.2), and the main theorem is that the analytic properties of such integrals match those of the X - function of Λeff (X), which is introduced in Subsection 2.1.3. Analytic properties of X -functions are explained in [BT98a, Section 5], and for any big line bundle

◦ L ∈ Λeﬀ (X) , the pole and its order of the X -function

XΛeﬀ (X)(sL + KX ), coincides with geometric invariants a(L) and b(L) introduced in Subsection 2.1.1. See [BT98a, Proposition 5.3 and 5.4].

At last, we discuss the leading constant c(−KX ). We have not determined the normalization of the Plancherel measure dχ. Let dn be the Lebesgue measure on

∼ 1 1 1 NR = T (AF )/T (AF ) normalized by vol(NR/N) = 1. Let dt = ω dn where ω is a 1 measure on T (AF ). We deﬁne b(T ) by

Z b(T ) = ω1. 1 T (AF )/T (F )

62 Lemma 47. We have

! 1 Z X Z0(s, id) = r Hb(s − im(χa), χu) dχa, (2π) b(T ) χa∈M R χu∈UG

where r = rank M and dχa is the Lebesgue measure on MR normalized by

vol(MR/M) = 1.

0 See [BT98a] or [BT96a]. Let UG be the set of characters χu ∈ UG such that the restriction of χu to T (AF,ﬁn) is trivial. Then Proposition 45 implies that

Y X lim F(s) = lim (sι − κι) Hb(s, χu). s→κ s→κ ι∈I 0 χu∈UG

It follows from the Poisson formula ([BT96a, Lemma 5.7]) that

Z X −1 Hb(s, χu) = H(s, g) dg. 0 G(AF ) χu∈UG

Hence it follows from [CLT10, Proposition 4.3.4] that

Z Y −1 lim F(s) = lim (sι − κι) H(s, g) dg s→κ s→κ ι∈I G(AF ) Z ∗ r = (ζF (1)) dτX X(AF ) ∗ r = (ζF (1)) τ(−KX ),

∗ where ζF (1) is the residue of Dedekind zeta function at s = 1. Note that since T is split and U is isomorphic to an aﬃne space over F , the weak approximation holds

63 ∗ r for X. According to [BT98a, Deﬁnition 2.4 and Theorem 2.9], b(T ) = (ζF (1)) .

Now [BT98a, Theorem 5.5] concludes that the residue of Z0(−sKX , id) at s = 1 is given by

α(X)τ(−KX ).

Note that since X is rational over F , β(X) = 1. Our conclusion of this subsection is:

Theorem 48. Let G be an extension of a splitting algebraic torus T by a unipotent group U over a number ﬁeld F . Let X be a smooth projective equivariant compactiﬁcation of G over F and

X Z(s, g) = H(s, γg)−1, γ∈G(F ) the height zeta function with respect to an adelic height pairing. Let

Z Z0(s, g) = Zχ(s, g) dχ,

be the integral over automorphic characters. Then Z0(s, id) satisﬁes Manin’s conjecture.

3.3 Height zeta functions of equivariant compact-

iﬁcations of the ax + b-group

In this section, we study Height zeta functions of equivariant compactiﬁcations of the ax + b-group. First, we discuss harmonic analysis on semi-direct products of Ga and Gm.

64 3.3.1 Harmonic analysis

In this subsection we study the local and adelic representation theory of

G := Ga oϕ Gm,

an extension of T := Gm by N := Ga via a homomorphism ϕ : Gm → GL1. The group law given by (x, a) · (y, b) = (x + ϕ(a)y, ab).

This subsection is extracted from [TT12, Section 3]. We ﬁx the standard Haar measures which are introduced in Subsection 2.1.3 and 3.2.1:

Y × Y × dx = dxv and da = dav , v v on N(AF ) and T (AF ). Note that G(AF ) is not unimodular, unless ϕ is trivial. The

× product measure dg := dxda is a right invariant measure on G(AF ) and dg/ϕ(a) is a left invariant measure.

Let % be the right regular unitary representation of G(AF ) on the Hilbert space:

2 H := L (G(F )\G(AF ), dg).

We now discuss the decomposition of H into irreducible representations. Let

Y 1 ψ = ψv : AF → S , v be the standard automorphic character which is introduced in [Tat67, Subsection

65 2.2]. Let ψn be the character deﬁned by

x → ψ(nx), for n ∈ F ×. Let W := ker(ϕ : F × → F ×), which is a ﬁnite cyclic group, and

G(AF ) πn := Ind (ψn), N(AF )×W

× 2 for n ∈ F . More precisely, the underlying Hilbert space of πn is L (W \T (AF )), and the group action is given by

(x, a) · f(b) = ψn(ϕ(b)x)f(ab),

where f is a square-integrable function on T (AF ).

Proposition 49. Irreducible automorphic representations, i.e., irreducible unitary

2 representations occurring in H = L (G(F )\G(AF )), are parametrized as follows:

2 M H = L (T (F )\T ( F )) ⊕ d πn, A n∈(F ×/ϕ(F ×))

Remark 50. Up to unitary equivalence, the representation πn does not depend on the choice of a representative n ∈ F ×/ϕ(F ×).

Proof. Deﬁne

H0 := {φ ∈ H |φ((x, 1)g) = φ(g)} ,

66 and let H1 be the orthogonal complement of H0. It is straightforward to prove that ∼ 2 H0 = L (T (F )\T (AF )).

Lemma 52 concludes our assertion.

1 Lemma 51. For any φ ∈ L (G(F )\G(AF )) ∩ H, the projection of φ onto H0 is given by Z φ0(g) := φ((x, 1)g) dx N(F )\N(AF )

0 Proof. It is easy to check that φ0 ∈ H0. Also, for any φ ∈ H0, we have

Z 0 (φ − φ0)φ dg = 0. G(F )\G(AF )

Lemma 52. We have ∼ M H1 = d πn. n∈F ×/ϕ(F ×)

∞ Proof. For φ ∈ Cc (G(F )\G(AF )) ∩ H1, deﬁne

Z fn,φ(a) := φ(x, a)ψn(x) dx. N(F )\N(AF )

67 Then,

Z Z 2 2 × k φ kL2 = |φ(x, a))| dxda T (F )\T (AF ) N(F )\N(AF ) Z Z 2 X × = φ(x, a)ψ(αx)dx da

T (F )\T (AF ) α∈F N(F )\N(AF ) Z Z 2 X × = φ(x, a)ψ(αx)dx da

T (F )\T (AF ) α∈F × N(F )\N(AF ) Z Z 2 X X 1 × = φ(x, a)ψ(nϕ(α)x)dx da #W T (F )\T (AF ) α∈F × n∈F ×/ϕ(F ×) N(F )\N(AF ) Z Z 2 X X 1 × = φ(x, αa)ψ(nx)dx da #W T (F )\T (AF ) α∈F × n∈F ×/ϕ(F ×) N(F )\N(AF ) Z Z 2 X 1 × = φ(x, a)ψ (x)dx da #W n n∈F ×/ϕ(F ×) T (AF ) N(F )\N(AF )

X 2 = k fn,φ kL2 . n∈F ×/ϕ(F ×)

The second equality is the Plancherel theorem for N(F )\N(AF ). Third equality follows from Lemma 51. The ﬁfth equality follows from the left G(F )-invariance of φ. Thus, we obtain an unitary operator:

M I : H1 → d πn. n∈F ×/ϕ(F ×)

Compatibility with the group action is straightforward, so I is actually a morphism of unitary representations. We construct the inverse map of I explicitly. For

∞ f ∈ Cc (W \T (AF )), deﬁne

1 X φ (x, a) := ψ (ϕ(α)x)f(αa). n,f #W n α∈F ×

68 The orthogonality of characters implies that

R φn,f (x, a) · φn,f (x, a) dx N(F )\N(AF )

2 R P P = (#W ) ( × ψn(ϕ(α)x)f(αa)) · ( × ψn(ϕ(α)x)f(αa)) dx N(F )\N(AF ) α∈F α∈F

1 P 2 = #W α∈F × |f(αa)| .

Substituting, we obtain

Z Z 2 2 × k φn,f k = |φn,f (x, a)| dxda T (F )\T (AF ) N(F )\N(AF ) 1 Z X = |f(αa)|2da× =k f k2 . #W n T (F )\T (AF ) α∈F ×

Lemma 51 implies that φf ∈ H1 and we obtain a morphism

M Θ: d πn → H1. n∈F ×/ϕ(F ×)

Now we only need to check that ΘI = id and IΘ = id. The ﬁrst follows from the

∞ Poisson formula: For any φ ∈ Cc (G(F )\G(AF )) ∩ H1,

X 1 X Z ΘIφ = ψ (ϕ(α)x) φ(y, αa)ψ (y)dy #W n n n∈F ×/ϕ(F ×) α∈F × N(F )\N(AF ) X 1 X Z = φ(ϕ(α)y, αa)ψ (ϕ(α)(y − x))dy #W n n∈F ×/ϕ(F ×) α∈F × N(F )\N(AF ) X 1 X Z = φ((y + x, a))ψ (ϕ(α)y)dy #W n n∈F ×/ϕ(F ×) α∈F × N(F )\N(AF ) X Z = φ((y, 1)(x, a))ψ(αy)dy = φ(x, a), α∈F N(F )\N(AF )

69 where we apply Poisson formula for the last equality. The other identity, IΘ = id is checked by a similar computation.

To simplify notation, we now restrict to F = Q. For our applications in Sec- tions 3.3.2, we need to know an explicit orthonormal basis for the unique inﬁnite- dimensional representation π = L2( ×) of G = G . For any n ≥ 1, deﬁne compact AQ 1 subgroups of G(Zp)

n n n G(p Zp) := {(x, a) | x ∈ p Zp, a ∈ 1 + p Zp}.

Let vp : Qp → Z be the discrete valuation on Qp.

n Lemma 53. Let Kp = G(p Zp).

2 × Kp • When n = 0, an orthonormal basis for L (Qp ) is given by

{1 j × | j ≥ 0}. p Zp

2 × Kp • When n ≥ 1, an orthonormal basis for L (Qp ) is given by

j {λp(·/p )1 j × | j ≥ −n, λp ∈ Mp}, p Zp

× n where Mp is the set of characters on Zp /(1 + p Zp).

Q Moreover, let Kﬁn = p Kp, where the local compact subgroups are given by Kp =

np G(p Zp), with np = 0 for almost all p. Let S be the set of primes with np 6= 0 and

N = Q pnp . Then an orthonormal basis for L2( × )Kﬁn is given by p AQ,ﬁn

−vp(ap) 0 {⊗p∈Sλp(ap · p )1 m × (ap) ⊗p/∈S 1 × (ap) | m ∈ , λp ∈ Mp}. N Zp mZp N

70 2 × Kp Proof. For the ﬁrst assertion, let f ∈ L (Qp ) where Kp = G(Zp). Since it is

Kp-invariant, we have

f(bp · ap) = f(ap),

× for any b ∈ Zp . Hence f takes the form of

∞ X f = cj1 j × p Zp j=−∞

j P∞ 2 where cj = f(p ) and j=−∞ |cj| < +∞. On the other hand we have

ψp(ap · xp)f(ap) = f(ap)

j for any xp ∈ Zp. This implies that f(p ) = 0 for any j < 0. Thus the ﬁrst assertion follows. The second assertion is treated similarly. The last assertion follows from the ﬁrst and the second assertions.

Q We denote these vectors by vm,λ where m ∈ N and λ ∈ M := p∈S Mp. Note that M is a ﬁnite set. Also we deﬁne

it θm,λ,t(g) := Θ(vm,λ ⊗ | · |∞)(g)

X it = ψ(αx)vm,λ(αaﬁn) |αa∞|∞. × α∈Q

The following proposition is a combination of Lemma 53 and the standard Fourier analysis on the real line:

K Proposition 54. Let f ∈ H1 . Suppose that

71 1. I(f) is integrable, i.e.,

I(f) ∈ L2( ×)K ∩ L1( ×), AQ AQ

2. the Fourier transform of f is also integrable, i.e.,

Z +∞ |(f, θm,λ,t)| dt < +∞, −∞

for any m ∈ N and λ ∈ M.

Then we have ∞ X X 1 Z +∞ f(g) = (f, θ )θ (g) dt 4π m,λ,t m,λ,t λ∈M m=1 −∞ where Z (f, θm,λ,t) = f(g)θm,λ,t(g) dg. G(Q)\G(AQ)

Proof. For simplicity, we assume that np = 0 for all primes p. Let I(f) = h ∈ L2( ×)K ∩ L1( ×). It follows from the proof of Lemma 52 that AQ AQ

X f(g) = Θ(h)(g) = ψ(αx)h(αa). × α∈Q

Note that this inﬁnite sum exists in both L1 and L2 sense. It is easy to check that

Z −it × h(a)vm(aﬁn)|a∞|∞ da = (f, θm,t). × AQ

Write X h = vm ⊗ hm, m

72 2 × where hm ∈ L (R>0, da∞). The ﬁrst and the second assumptions imply that hm and the Fourier transform of bhm both are integrable. Hence the inverse formula of Fourier transformation on the real line implies that

X 1 Z +∞ h(a) = (f, θ )v (a )|a |it dt. 4π m,t m ﬁn ∞ ∞ m −∞

Apply Θ to both sides, and our assertion follows.

We recall some results regarding Igusa integrals with rapidly oscillating phase, studied in [CLT09]:

Proposition 55. Let p be a ﬁnite place of Q and d, e ∈ Z. Let

2 2 Φ: Qp × C → C,

2 be a function such that for each (x, y) ∈ Qp, Φ((x, y), s) is holomorphic in s =

2 (s1, s2) ∈ C . Assume that the function (x, y) 7→ Φ(x, y, s) belongs to a bounded subset of the space of smooth compactly supported functions when <(s) belongs to a

ﬁxed compact subset of R2. Let Λ be the interior of a closed convex cone generated by (1, 0), (0, 1), (d, e).

× Then, for any α ∈ Qp ,

Z s1 s2 d e × × ηα(s) = |x|p |y|p ψp(αx y )Φ(x, y, s)dxp dyp , 2 Qp

is holomorphic on TΛ. The same argument holds for the inﬁnite place when Φ is a smooth function with compact supports.

73 Proof. For the inﬁnite place, use integration by parts and apply the convexity principle. For ﬁnite places, assume that d, e are both negative. Let δ(x, y) = 1 if

|x|p = |y|p = 1 and 0 else. Then we have

Z X s1 s2 d e −n −m × × ηα(s) = |x|p |y|p ψp(αx y )Φ(x, y, s)δ(p x, p y) dxp dyp 2 n,m∈Z Qp

X −(ns1+ms2) = p · ηα,n,m(s), n,m∈Z where Z nd+me d e n m × × ηα,n,m(s) = ψp(αp x y )Φ(p x, p y, s) dxp dyp . |x|p=|y|p=1

Fix a compact subset of C2 and assume that <(s) is in that compact set. The assumptions in our proposition mean that the support of Φ(·, s) is contained in a

2 ﬁxed compact set in Qp, so there exists an integer N0 such that ηα,n,m(s) = 0 if n < N0 or m < N0. Moreover our assumptions imply that there exists a positive

2 real number δ such that Φ(·, s) is constant on any ball of radius δ in Qp. This

n × implies that if 1/p < δ, then for any u ∈ Zp ,

Z nd+me d e d n m × × ηα,n,m(s) = ψp(αp x y u )Φ(p xu, p y, s) dxp dyp Z nd+me d e d n m × × = ψp(αp x y u )Φ(p x, p y, s) dxp dyp ZZ nd+me d e d × n m × × = ψp(αp x y u ) du Φ(p x, p y, s) dxp dyp , × Zp and the last integral is zero if n is suﬃciently large because of [CLT09, Lemma

2.3.5]. Thus we conclude that there exists an integer N1 such that ηα,n,m(s) = 0 if

74 n > N1 or m > N1. Hence we obtained that

X −(ns1+ms2) ηα(s) = p · ηα,n,m(s),

N0≤n,m≤N1 and this is holomorphic everywhere. The case of d < 0 and e = 0 is treated similarly. Next assume that d < 0 and e > 0. Then again we have a constant c such that

n ηα,n,m(s) = 0 if 1/p < δ and n|d| − me > c. We may assume that c is suﬃciently large so that the ﬁrst condition is unnecessary. Then we have

n (n|d|−me) X X − (e<(s1)+|d|<(s2)) <(s2) |ηα(s)| ≤ p e · p e · |ηα,n,m(s)|

N0≤n m c <(s2) X − n (e<(s )+|d|<(s )) p e ≤ p e 1 2 · <(s2) − e N0≤n 1 − p

Thus ηα(s) is holomorphic on TΛ.

From the proof of Proposition 55, we can claim more for ﬁnite places:

Proposition 56. Let > 0 be any small positive real number. Fix a compact subset K of Λ, and assume that <(s) is in K. Deﬁne:

 n o  <(s1) <(s2) max 0, − |d| , − |e| if d < 0 and e < 0, κ(K) := n o  <(s1) max 0, − |d| if d < 0 and e ≥ 0,.

Then we have

κ(K)+ |ηα(s)| 1/|α|p

75 as |α|p → 0.

−k Proof. Let |α|p = p , and assume that both d, e are negative. By changing variables, if necessary, we may assume that N0 in the proof of Proposition 55 is zero. If k is suﬃciently large, then one can prove that there exists a constant c such that ηα,n,m(s) = 0 if n|d| + m|e| > k + c. Also it is easy to see that

|p−(ns1+ms2)| ≤ p(n|d|+m|e|)κ(K).

Hence we can conclude that

2 κ(K) κ(K)+ |ηα(s)| k 1/|α|p 1/|α|p .

The case of d < 0 and e = 0 is treated similarly.

Assume that d < 0 and e > 0. Then we have a constant c such that ηα,n,m(s) = 0 if n|d| − me > k + c. Thus we can conclude that

X X −(n<(s1)+m<(s2)) |ηα(s)| ≤ p |ηα,n,m(s)| m≥0 n≥0 X p−m<(s2)(me + k)p(me+k)κ(K,s2) m≥0

κ(K,s2) X −m(<(s2)−eκ(K,s2)) k1/|α|p (m + 1)p m≥0

κ(K,s2)+ 1/|α|p . where <(s ) κ(K, s ) = max 0, − 1 :(<(s ), <(s )) ∈ K . 2 |d| 1 2

76 Thus we can conclude that

κ(K,s2)+ κ(K)+ |ηα(s)| 1/|α|p 1/|α|p .

3.3.2 the ax + b-group

In this subsection, we study Height zeta functions of equivariant compactiﬁca- tions of the ax+b-group by using harmonic analysis developed in Subsection 3.3.1. This subsection is extracted from [TT12, Section 5]. Our main theorem is:

Theorem 57. Let X be a smooth projective equivariant compactiﬁcation of G = G1 over Q, under the right action. Assume that the boundary divisor has strict normal crossings. Let a, x ∈ Q(X) be rational functions, where (x, a) are the standard coordinates on G ⊂ X. Let E be the Zariski closure of {x = 0} ⊂ G. Assume that:

• the union of the boundary and E is a divisor with strict normal crossings,

• div(a) is a reduced divisor, and

• for any pole Dι of a, one has −ordDι (x) > 1.

Then Manin’s conjecture holds for X.

The remainder of this subsection is devoted to a proof of this fact. Blowing up the zero-dimensional subscheme

Supp(div0(a)) ∩ Supp(div∞(a)),

77 if necessary, we may assume that

Supp(div0(a)) ∩ Supp(div∞(a)) = ∅.

Here div0 and div∞ stand for the divisors of zeroes, respectively poles, of the rational functions a on X. The local height functions are invariant under the right action of some compact subgroup Kp ⊂ G(Zp). Moreover, we can assume that K = G(pnp ), for some n ∈ . Let S be the set of bad places for p Zp p Z>0 X; a priori, this set depends on a choice of an integral model for X and for the action of G. Specifically, we insist that for p∈ / S, the reduction of X at p is smooth, the reduction of the boundary is a union of smooth geometrically irreducible divisors with normal crossings, and the action of G lifts to the integral models. In particular, we insist that np = 0 for all p∈ / S. The proof works with X being any sufficiently large finite set. For simplicity, we assume that the height function at the infinite place is invariant under the action of K∞ = {(0, ±1)}.

Lemma 58. We have

2 K 1 Z(s, g) ∈ L (G(Q)\G(AQ)) ∩ L (G(Q)\G(AQ)).

See Proposition 39. By Proposition 49, the height zeta function decomposes as

Z(s, id) = Z0(s, id) + Z1(s, id).

Analytic properties of Z0(s, id) were established in Subsection 3.2.2. It remains to show that Z1(s, id) is holomorphic on a tube domain over an open neighborhood of the shifted eﬀective cone −KX + Λeﬀ (X). To conclude this, we use the spectral

78 decomposition of Z1:

Lemma 59. We have

∞ X X 1 Z +∞ Z (s, id) = (Z(s, g), θ )θ (id) dt, 1 4π m,λ,t m,λ,t λ∈M m=1 −∞ for suﬃciently large <(s).

Proof. To apply Proposition 54, we need to check that Z1 satisﬁes the assumptions of Proposition 54. The proof of Lemma 52 implies that

Z I(Z1) = Z(s, g)ψ(x) dx. N(Q)\N(A)

Thus we have

Z Z Z × × |I(Z1)| da = Z(s, g)ψ(x) dx da T (A) T (A) N(Q)\N(A) Z Z X −1 × ≤ H(s, g) ψ(αx) dx da × T ( ) N( ) α∈Q A A

Z Z X Y −1 × = Hp(s, gp) ψp(αxp) dxp dap × T ( p) N( p) α∈Q p Q Q Z Z −1 × × H∞(s, g∞) ψ∞(αx) dx∞ da∞. T (R) N(R)

Assume that p∈ / S. Since the height function is right Kp-invariant, we obtain that

79 for any yp ∈ Zp,

Z Z −1 −1 Hp(s, gp) ψp(αxp) dxp = Hp(s, (xp + apyp, ap)) ψp(αxp) dxp N(Qp) N(Qp) Z Z −1 = Hp(s, gp) ψp(αxp) ψp(αapyp) dyp dxp N(Qp) Zp

= 0 if |αap|p > 1.

Hence we can conclude that

Z Z Z H (s, g )−1ψ (αx ) dx da× ≤ H (<(s), g )−11 (αa ) dg . p p p p p p p p Zp p p T (Qp) N(Qp) G(Qp)

Similarly, for p ∈ S, we can conclude that

Z Z Z −1 × −1 Hp(s, gp) ψp(αxp) dxp dap ≤ Hp(<(s), gp) 1 1 (αap) dgp, N Zp T (Qp) N(Qp) G(Qp) where N is an integer introduced in Lemma 53. Then the convergence of the following sum

Z Z Z X Y −1 −1 × H 1 1 (αap) dgp · H ψ∞(αx) dx∞ da , p N Zp ∞ ∞ × G( p) T ( ) N( ) α∈Q p Q R R can be veriﬁed from the detailed study of the local integrals which we will conduct later. See proofs of Lemmas 62, 66, and 67. Next we need to check that

Z +∞ |(Z(s, g), θm,λ,t)| dt < +∞. −∞

80 Note that

Z (Z(s, g), θm,λ,t) = Z(s, g)θm,λ,t dg G(Q)\G(AQ) Z −1 = H(s, g) θm,λ,t dg G(AQ) Z X −1 −it = H(s, g) ψ(αx)vm,λ(αaﬁn)|αa∞|∞ dg × G( ) α∈Q AQ X Y 0 0 = Hp(s, m, λ, α) · H∞(s, t, α), × α∈Q p

0 where Hp(s, m, λ, α) is given by

Z −1 = Hp(s, gp) ψ (αxp)1 × (αap) dgp, p∈ / S p mZp G(Qp) Z −1 vp(αap) = Hp(s, gp) ψp(αxp)λp(αap/p )1 m × (αap) dgp, p ∈ S N Zp G(Qp) and Z 0 −1 −it H∞(s, t, α) = H∞(s, g∞) ψ∞(αx∞)|αa∞|∞ dg∞. G(R) The integrability follows from the proof of Lemma 66. Thus we can apply Proposi- tion 54, and the identity in our statement follows from the continuity of Z(s, g).

We obtained that

∞ X X 1 Z +∞ Z (s, id) = (Z(s, g), θ )θ (id) dt 1 4π m,λ,t m,λ,t λ∈M m=1 −∞ ∞ Z +∞ it X X 1 Y m −vp(m/N) m = (Z(s, g), θm,λ,t) λp · p dt. 2π N N ∞ λ∈M, λ(−1)=1 m=1 −∞ p∈S

81 We will use the following notation:

Y vp(αap) λS(αap) := λq(p ), p∈ / S q∈S αap Y vp(αap) λS,p(αap) := λp λq(p ), p ∈ S. pvp(αap) q∈S\p

Proposition 60. If <(s) is suﬃciently large, then

X X 1 Z +∞ Y Z1(s, id) = Hbp(s, λ, t, α) · Hb∞(s, t, α) dt, 2π × −∞ λ∈M, λ(−1)=1 α∈Q p

where Hbp(s, λ, t, α) is given by

Z −1 −it Hp(s, gp) ψp(αxp)λS(αap)1Zp (αap)|ap|p dgp, p∈ / S G(Qp) Z −1 −it Hp(s, gp) ψ (αxp)λS,p(αap)1 1 (αap)|ap| dgp, p ∈ S p N Zp p G(Qp) and Z −1 −it Hb∞(s, t, α) = H∞(s, g∞) ψ∞(αx∞)|a∞|∞ dg∞ G(R)

Proof. For simplicity, we assume that S = ∅. We have seen that

∞ X X 1 Z +∞ Y Z (s, id) = H0 (s, m, α) · H0 (s, t, α)|m|it dt. 1 2π p ∞ ∞ × −∞ m=1 α∈Q p

On the other hand, we have

∞ Z j −it X −1 p Hbp(s, t, α) = Hp(s, gp) ψ (αxp)1 j × (αap) dgp. p p Zp α j=0 G(Qp) p

82 Hence we have the formal identity:

∞ Y X Y 0 0 it Hbp(s, t, α) · Hb∞(s, t, α) = Hp(s, m, α) · H∞(s, t, α)|m|∞, p m=1 p and our assertion follows from this. To justify the above identity, we need to ad- dress convergence issues; this will be discussed below. (See the proof of Lemma 62.)

Thus we need to study the local integrals in Proposition 60. We introduce some notation:

I1 = {ι ∈ I | Dι ⊂ Supp(div0(a))}

I2 = {ι ∈ I | Dι ⊂ Supp(div∞(a))}

I3 = {ι ∈ I | Dι 6⊂ Supp(div(a))}.

Note that I = I1 t I2 t I3 and I1 6= ∅. Also Dι ⊂ Supp(div∞(x)) for any ι ∈ I3 because

D = ∪ι∈I Dι = Supp(div(a)) ∪ Supp(div∞(x)).

Let X −div(ω) = dιDι, ι∈I where ω = dxda/a is the top degree right invariant form on G. Note that ω deﬁnes a measure |ω| on an analytic manifold G(Qv), and for any ﬁnite place p,

1 |ω| = 1 − dg , p p

83 where dgp is the standard Haar measure deﬁned in Section 3.3.1.

G Lemma 61. Consider an open convex cone Ω in Pic (X)R, deﬁned by the following relations:   sι − dι + 1 > 0 if ι ∈ I1   sι − dι + 1 + eι > 0 if ι ∈ I2    sι − dι + 1 > 0 if ι ∈ I3

where eι = |ordDι (x)|. Then Hbp(s, λ, t, α) and Hb∞(s, t, α) are holomorphic on TΩ.

Proof. First we prove our assertion for Hb∞. We can assume that

Hbv(s, t) = Hbv(s − itm(a), 0), where m(a) ∈ X∗(G) ⊂ PicG(X) is the character associated to the rational function a (by choosing appropriate adelic metrizations). It suffices to discuss the case when t = 0. Choose a finite covering {Uη} of X(R) by open subsets and local coordinates yη, zη on Uη such that the union of the boundary divisor D and E is locally defined by yη = 0 or yη · zη = 0. Choose a partition of unity {θη}; the local integral takes the form Z X −1 Hb∞(s, α) = H∞(s, g∞) ψ∞(αx∞)θη dg∞. η G(R)

Each integral is an oscillatory integral in the variables yη, zη. For example, assume

0 that Uη meets Dι,Dι0 , where ι, ι ∈ I2. Then

Z −1 H∞(s, g∞) ψ∞(αx∞)θηdg∞ G(R) ! Z αf = |y |sι−dι |z |sι0 −dι0 ψ Φ(s, y , z )dy dz , η η ∞ e eι0 η η η η 2 ι R yη zη

84 where Φ is a smooth function with compact support and f is a nonvanishing analytic function. Shrinking Uη and changing variables, if necessary, we may assume that f is a constant. Proposition 55 implies that this integral is holomorphic everywhere. The other integrals can be studied similarly. Next we consider ﬁnite places. Let p be a prime of good reduction. Since

Supp(div0(a)) ∩ Supp(div∞(a)) = ∅,

the smooth function 1Zp (αap) extends to a smooth function h on X(Qp). Let

U = {h = 1}.

Then Z −1 Hbp(s, λ, α) = Hp(s, gp) ψp(αxp)λS(αap)dgp. U

Now the proof of [CLT10, Lemma 4.4.1] implies that this is holomorphic on TΩ because U ∩ (∪ι∈I2 Dι(Qp)) = ∅. Places of bad reduction are treated similarly.

k Lemma 62. Let |α|p = p > 1. Then, for any compact set in Ω and for any δ > 0, there exists a constant C > 0 such that

− minι∈I1 {<(sι)−dι+1−δ} |Hbp(s, λ, t, α)| < C|α|p , for <(s) in that compact set.

Proof. First assume that p is a good reduction place. Let ρ : X (Zp) → X (Fp) be the reduction map modulo p where X is a smooth integral model of X over

85 Spec(Zp). Note that

ρ({|a|p < 1}) ⊂ ∪ι∈I1 Dι(Fp), where Dι is the Zariski closure of Dι in X . Thus Hbp(s, λ, α) is given by

Z X −1 Hbp(s, λ, α) = Hp(s, gp) ψp(αxp)λS(αap)1Zp (αap)dgp. ρ−1(˜x) x˜∈∪ι∈I1 Dι(Fp)

0 Letx ˜ ∈ Dι(Fp) for some ι ∈ I1, butx ˜ ∈ / Dι0 (Fp) for any ι ∈ I \ {ι}. Since p is a good reduction place, we can ﬁnd analytic coordinates y, z such that

Z Z −1 ≤ Hp(<(s), gp) 1Zp (αap)dgp ρ−1(˜x) ρ−1(˜x) −1 Z 1 −1 = 1 − Hp(<(s) − d, gp) 1Zp (αap)dτX,p p ρ−1(˜x) −1 Z 1 <(sι)−dι = 1 − |y|p 1Zp (αy)dypdzp p 2 mp 1 p−k(<(sι)−dι+1) = · , p 1 − p−(<(sι)−dι+1)

where dτX,p is the local Tamagawa measure. For the construction of such local analytic coordinates, see [Wei82], [Den87], or [Sal98]. Ifx ˜ ∈ Dι(Fp) ∩ Dι0 (Fp) for

0 ι ∈ I1, ι ∈ I3, then we can ﬁnd local analytic coordinates y, z such that

Z 1 Z <(sι)−dι+1 <(sι0 )−dι0 +1 × × ≤ 1 − |y|p |z|p 1Zp (αy)dyp dzp −1 p 2 ρ (˜x) mp 1 p−k(<(sι)−dι+1) p−(<(sι0 )−dι0 +1) = 1 − . p 1 − p−(<(sι)−dι+1) 1 − p−(<(sι0 )−dι0 +1)

0 0 Ifx ˜ ∈ Dι(Fp) ∩ Dι0 (Fp) for ι, ι ∈ I1, ι 6= ι , then we can ﬁnd analytic coordinates

86 x, y such that

Z 1 Z <(sι)−dι+1 <(sι0 )−dι0 +1 × × ≤ 1 − |y|p |z|p 1Zp (αyz) dyp dzp −1 p 2 ρ (˜x) mp 1 Z min{<(sι)−dι+1, <(sι0 )−dι0 +1} × × ≤ 1 − |yz|p 1Zp (αyz) dyp dzp p 2 mp 1 p−kr p−(k+1)r = 1 − (k − 1) + , p 1 − p−r (1 − p−r)2 where

r = min{<(sι) − dι + 1, <(sι0 ) − dι0 + 1}.

It follows from these inequalities and Lemma 9.4 in [CLT02] that there exists a constant C > 0, independent of p, satisfying the inequality in the statement.

Next assume that p is a bad reduction place. Choose an open covering {Uη} of

∪ι∈I1 Dι(Qp) such that

(∪ηUη) ∩ (∪ι∈I2 Dι(Qp)) = ∅,

and each Uη has analytic coordinates yη, zη. Moreover, we can assume that the boundary divisor is deﬁned by yη = 0 or yη ·zη = 0 on Uη. Let V be the complement of ∪ι∈I1 Dι(Qp), and consider the partition of unity for {Uη,V } which we denote by {θη, θV }. If k is suﬃciently large, then

{1 1 (αa) = 1} ∩ Supp(θV ) = . N Zp ∅

87 Hence if k is suﬃciently large, then

Z X −1 |Hbp(s, λ, α)| ≤ Hp(<(s), gp) 1 1 (αap) · θη dgp. N Zp η Uη

When Uη meets only one component Dι(Qp) for ι ∈ I1, then

Z Z <(sι)−dι −k(<(sι)−dι+1) ≤ |yη|p 1cZp (αyη)Φ(s, yη, zη) dyη,pdzη,p p , 2 Uη Qp as k → ∞, where c is some rational number and Φ is a smooth function with compact support. Other integrals are treated similarly.

We record the following useful lemmas (see, e.g., [CLT09, Lemma 2.3.1]):

Lemma 63.   1 if |x|p ≤ 1,  Z  ¯ × ψp(bx)db = − 1 if |x| = p, × p−1 p Zp    0 otherwise.

Lemma 64. Let d be a positive integer and a ∈ Qp. If |a|p > p and p - d, then

Z d × ψp(ax ) dxp = 0. × Zp

Moreover, if |a|p = p and d = 2, then

 √ √ Z  p−1 or i p−1 if pa is a quadratic residue, d ×  p−1 p−1 ψp(ax )dxp = × √ √ Zp  − p−1 −i p−1  p−1 or p−1 if pa is a quadratic non-residue.

−k Lemma 65. Let |α|p = p < 1. Consider an open convex cone Ω in Pic(X)R,

88 deﬁned by the following relations:

  sι − dι + 1 > 0 if ι ∈ I1   sι − dι + 2 + > 0 if ι ∈ I2    sι − dι + 1 > 0 if ι ∈ I3

where 0 < < 1/3. Then, for any compact set in Ω, there exists a constant C > 0 such that

2 − 3 (1+2) |Hbp(s, λ, t, α)| < C|α|p , for <(s) in that compact set.

Proof. First assume that p is a good reduction place and that p - eι, for any ι ∈ I2. We have

Z X −1 Hbp(s, λ, α) = Hp(s, gp) ψp(αxp)λS(αap)1Zp (αap) dgp. ρ−1(˜x) x˜∈X (Fp)

A formula of J. Denef (see [Den87, Theorem 3.1] or [CLT10, Proposition 4.1.7]) and Lemma 9.4 in [CLT02] give us an uniform bound:

Z X X −1 | | ≤ Hp(<(s), gp) dgp. ρ−1(˜x) x/˜∈∪ι∈I2 Dι(Fp) x/˜∈∪ι∈I2 Dι(Fp)

Hence we need to study

Z X −1 Hp(s, gp) ψp(αxp)λS(αap)1Zp (αap) dgp. ρ−1(˜x) x˜∈∪ι∈I2 Dι(Fp)

0 Letx ˜ ∈ Dι(Fp) for some ι ∈ I2, butx ˜ ∈ / Dι0 (Fp) ∪ E(Fp) for any ι ∈ I \ {ι}, where

89 E is the Zariski closure of E in X . Then we can ﬁnd local analytic coordinates y, z such that

Z −1 Z 1 sι−dι eι −1 −1 = 1 − |y|p ψp(αf/y )λS(αy )1Zp (αy ) dypdzp, −1 p 2 ρ (˜x) mp

× where f ∈ Zp[[y, z]] such that f(0) ∈ Zp . Since p does not divide eι, there exists

eι g ∈ Zp[[y, z]] such that f = f(0)g . After a change of variables, we can assume

× that f = u ∈ Zp . Lemma 64 implies that

Z Z Z 1 sι−dι+1 −1 eι eι × −1 × = |y|p λS(αy ) ψp(αub /y )dbp 1Zp (αy )dyp −1 p × ρ (˜x) mp Zp Z Z 1 sι−dι+1 −1 eι eι × × = |y|p λS(αy ) ψp(αub /y ) dbp dyp p −(k+1) e × p ≤|y ι |p Zp

Thus it follows from the second assertion of Lemma 64 that

Z Z Z 1 <(sι)−dι+1 eι eι × × ≤ |y|p ψp(αub /y )dbp dyp −1 p −(k+1) e × ρ (˜x) p ≤|y ι | Zp   1 if eι > 2 1 k (1+) 1 k+1 (1+)  ≤ kp eι + p eι × p p  √ 1  p−1 if eι = 2

1 2 k(1+) kp 3 . p

90 Ifx ˜ ∈ Dι(Fp) ∩ E(Fp), for some ι ∈ I2, then we have

Z −1 Z 1 sι−dι eι −1 −1 = 1 − |y|p ψp(αz/y )λS(αy )1Zp (αy ) dypdzp −1 p 2 ρ (˜x) mp Z Z sι−dι+1 −1 −1 eι × = |y|p λS(αy )1Zp (αy ) ψp(αz/y )dzpdyp mp mp Z 1 sι−dι+1 −1 × = |y|p λS(αy ) dyp . −(k+1) eι p p ≤|y|p <1

Hence we obtain that

Z Z 1 k (1+) 2 <(sι)−dι+1 × e 3 k(1+) ≤ |y|p dyp ≤ kp ι < kp . −1 −(k+1) eι ρ (˜x) p p ≤|y|p <1

0 Ifx ˜ ∈ Dι(Fp) ∩ Dι0 (Fp) for some ι ∈ I2 and ι ∈ I3, then it follows from Lemma 64

Z −1 Z 1 sι−dι s 0 −d 0 αu −1 −1 = 1 − |y| |z| ι ι ψ λS(αy )1 (αy ) dypdzp p p p e e 0 Zp −1 p 2 y ι z ι ρ (˜x) mp −1 1 Z Z αubeι sι−dι sι0 −dι0 −1 × = 1 − |y|p |z|p λS(αy ) ψp e e dbp dypdzp, p × y ι z ι0 Zp where the last integral is over the domain

2 −(k+1) eι eι0 {(y, z) ∈ mp | p ≤ |y z |p}.

We conclude that

Z 1−1 Z <(sι)−dι <(sι0 )−dι0 ≤ 1 − |y|p |z|p dypdzp −1 −(k+1) e e 0 ρ (˜x) p p ≤|y ι z ι |p 1−1 Z Z <(sι)−dι <(sι0 )−dι0 ≤ 1 − |y|p dyp |z|p dzp −k e p p ≤|y ι |p<1 mp

−(<(sι0 )−dι0 +1) k (1+) p ≤ kp eι . 1 − p−(<(sι0 )−dι0 +1)

91 0 −1 Ifx ˜ ∈ Dι(Fp) ∩ Dι0 (Fp) for some ι, ι ∈ I2, then the local integral on ρ (˜x) is:

−1 Z 1 sι−dι s 0 −d 0 αu −1 −1 −1 −1 1 − |y| |z| ι ι ψ λS(αy z )1 (αy z ) dypdzp p p p e e 0 Zp p 2 y ι z ι mp Z Z eι 1 sι−dι s 0 −d 0 −1 −1 αub × × × = 1 − |y| |z| ι ι λS(αy z ) ψ db dy dz . p p p e e 0 p p p p 2 × y ι z ι mp Zp

We can assume that eι ≤ eι0 . Then we can conclude that

2 2 k(1+) k p 3 .

Thus our assertion follows from these estimates and Lemma 9.4 in [CLT02].

Next assume that p is a place of bad reduction or that p divides eι, for some

ι ∈ I2. Fix a compact subset of Ω and assume that <(s) is in that compact set.

Choose a ﬁnite open covering {Uη} of ∪ι∈I2 Dι(Qp) with analytic coordinates yη, zη such that the union of the boundary D(Qp) and E(Qp) is deﬁned by yη = 0 or yη · zη = 0. Let V be the complement of ∪ι∈I2 Dι(Qp), and consider a partition of unity {θη, θV } for {Uη,V }. Then it is clear that

Z −1 Hp(s, gp) ψ (αxp)λS,p(αap)1 1 (αap)θV dgp, p N Zp V

92 is bounded, so we need to study

Z −1 1 Hp(s, gp) ψ (αxp)λS,p(αap)1 (αap)θUη dgp. p N Zp Uη

Assume that Uη meets only one Dι(Qp) for some ι ∈ I2. Then, the above integral looks like

Z Z sι−dι eι 1 = |yη|p ψp(αf/yη ))λS,p(αg/yη)1 p (αg/yη)Φ(s, yη, zη)dyη,pdzη,p, 2 N Z Uη Qp where f and g are nonvanishing analytic functions, and Φ is a smooth function with compact support. By shrinking Uη and changing variables, if necessary, we can assume that f and g are constant. The proof of Proposition 56 implies our assertion for this integral. Other integrals are treated similarly.

Lemma 66. For any compact set in an open convex cone Ω0, deﬁned by

  sι − dι − 1 > 0 if ι ∈ I1   sι − dι + 3 > 0 if ι ∈ I2    sι − dι + 1 > 0 if ι ∈ I3 there exists a constant C > 0 such that

C |Hb∞(s, t, α)| < , |α|2(1 + t2) for <(s) in that compact set.

Proof. Consider the left invariant diﬀerential operators ∂a = a∂/∂a and ∂x =

93 a∂/∂x. Assume that <(s) 0. Integrating by parts, we have

Z 1 2 −1 −it Hb∞(s, t, α) = − ∂ H∞(s, g∞) ψ (αx∞)|a∞| dg∞ t2 a ∞ ∞ G(R) 1 Z ∂2 = (∂2H (s, g )−1)ψ (αx )|a |−it dg . (2π)2|α|2t2 ∂x2 a ∞ ∞ ∞ ∞ ∞ ∞ ∞ G(R)

According to Proposition 2.2. in [CLT02],

∂2 (∂2H (s, g )−1) = |a|−2∂2∂2H (s, g )−1 ∂x2 a ∞ ∞ x a ∞ ∞ −1 = H∞(s − 2m(a), g∞) × (a bounded smooth function).

Moreover, Lemma 4.4.1. of [CLT10] tells us that

Z −1 H∞(s − 2m(a), g∞) dg∞, G(R)

is holomorphic on TΩ0 . Thus we can conclude our lemma.

Lemma 67. The Euler product

Y Hbp(s, λ, t, α) · Hb∞(s, t, α) p

is holomorphic on TΩ0 .

Proof. First we prove that the Euler product is holomorphic on TΩ0 . To conclude this, we only need to discuss:

Y Hbp(s, λ, t, α),

p/∈S∪S3, |α|p=1,

94 where S3 = {p | p | eι for some ι ∈ I3}. Let p be a prime such that p∈ / S ∪ S3

0 and |α|p = 1. Fix a compact subset of Ω , and assume that <(s) is sitting in that compact set. From the deﬁnition of Ω0, there exists > 0 such that

  sι − dι + 1 > 2 + for any ι ∈ I1  sι − dι + 1 > for any ι ∈ I3.

Since we have

−1 {|a|p ≤ 1} = X(Qp) \ ρ (∪ι∈I2 Dι(Fp)), we can conclude that

Z X −1 Hbp(s, λ, α) = Hp(s, gp) ψp(αxp)λS(ap)dgp. ρ−1(˜x) x/˜∈∪ι∈I2 Dι(Fp)

Note that X Z Z = 1 dgp = 1. −1 ρ (˜x) G(Zp) x/˜∈∪ι∈I Dι(Fp) Also it follows from a formula of J. Denef (see [Den87, Theorem 3.1] or [CLT10, Proposition 4.1.7]) and Lemma 9.4 in [CLT02] that there exists an uniform bound

C > 0 such that for anyx ˜ ∈ ∪ι∈I1 Dι(Fp),

Z Z −1 C < Hp(<(s), gp) dgp < 2+ . ρ−1(˜x) ρ−1(˜x) p

R Hence we need to obtain uniform bounds of ρ−1(˜x) for

x˜ ∈ ∪ι∈I3 Dι(Fp) \ ∪ι∈I1∪I2 Dι(Fp).

95 Letx ˜ ∈ Dι(Fp) for some ι ∈ I3, butx ˜ ∈ / ∪ι∈I1∪I2 Dι(Fp) ∪ E(Fp). Then it follows from Lemmas 63 and 64 that

Z −1 Z 1 sι−dι eι = 1 − |y|p ψp(u/y ) dypdzp −1 p 2 ρ (˜x) mp Z Z 1 sι−dι eι eι × = |y|p ψp(ub /y ) dbp dyp p − 1 × mp Zp   0 if eι > 1 =  p−(sι−dι+2) − p−1 if eι = 1.

Ifx ˜ ∈ Dι(Fp) ∩ E(Fp) for some ι ∈ I3, then we have

Z −1 Z 1 sι−dι eι = 1 − |y|p ψp(z/y )dypdzp −1 p 2 ρ (˜x) mp −1 Z Z 1 sι−dι eι = 1 − |y|p ψp(z/y ) dzpdyp p mp mp   0 if eι > 1 =

 −(sι−dι+2) p if eι = 1.

0 Ifx ˜ ∈ Dι(Fp) ∩ Dι0 (Fp) for some ι, ι ∈ I3, then it follows from Lemma 64 that

Z −1 Z 1 sι−dι s 0 −d 0 u = 1 − |y| |z| ι ι ψ dypdzp p p p e e 0 −1 p 2 y ι z ι ρ (˜x) mp −1 Z Z eι 1 sι−dι s 0 −d 0 ub × = 1 − |y| |z| ι ι ψ db dypdzp p p p e e 0 p p 2 × y ι z ι mp Zp = 0.

Thus we can conclude from these estimates and Lemma 9.4 in [CLT02] that there

96 exists an uniform bound C0 > 0 such that

C0 Hbp(s, λ, t, α) − 1 < p1+

Our assertion follows from this.

0 Lemma 68. Let Ω be an open convex cone, deﬁned by

  sι − dι − 2 − > 0 if ι ∈ I1   sι − dι + 2 + 2 > 0 if ι ∈ I2    sι − dι + 1 > 0 if ι ∈ I3

0 where > 0 is suﬃciently small. Fix a compact subset of Ω and δ > 0. Then there exists a constant C > 0 such that

Y C | Hbp(s, λ, t, α) · Hb∞(s, α, t)| < 4 8 , 2 3 − 3 −δ 1+−δ p (1 + t )|β| |γ|

β for <(s) in that compact set, where α = γ with gcd(β, γ) = 1.

Proof. This lemma follows from Lemmas 62, 65, and 66, and from the proof of Lemma 67.

Theorem 69. The zeta function Z1(s, id) is holomorphic on the tube domain over an open neighborhood of the shifted eﬀective cone −KX + Λeﬀ(X).

Proof. Let 1 δ > 0. Lemma 68 implies that

X X 1 Z +∞ Y Z1(s, id) = Hbp(s, λ, t, α) · Hb∞(s, t, α) dt, 2π × −∞ λ∈M, λ(−1)=1 α∈Q p

97 0 0 is absolutely and uniformly convergent on Ω, so Z1(s, id) is holomorphic on TΩ .

0 G Now note that the image of Ω by Pic (X) → Pic(X) contains an open neighborhood of −KX + Λeﬀ (X). Thus our theorem is concluded.

Remark 70. Theorem 69 claims that only Z0(s, id) contributes to the main term of a Height zeta function for any big line bundle. However, this cannot be true in general. A reason why we achieved this result is that assumptions in Theorem 69

∼ 1 ensure that for any torus T = Gm, a G-rational map X 99K P mapping G → T \G never be a honest morphism. In terms of balanced line bundles which we will introduce in next part, we can phrase this phenomenon in the following way: Let Y be the Zariski closure of T . Then any big line bundle is balanced with respect to Y . See Part II and [BT98b].

98 Part II

Balanced line bundles

99 Chapter 4

Balanced line bundles

From now on, we discuss the notion of balanced line bundles. Balanced line bundles are introduced in the paper [HTT12], which is joint work with Brendan Hassett and Yuri Tschinkel, and the motivation of this research is coming from the study of ergodic method, mixing in [GTBT11]. We found that this concept can be explained in terms of birational geometry, e.g., the Minimal Model Program and Iitaka fibration. Such connections between Manin’s program and birational geometry were already observed by Victor Batyrev and Yuri Tschinkel in [BT98b] and [Tsc03], and we further develop the circle of these ideas in [HTT12]. In this chapter, first we discuss general properties of geometric invariants showing up in the context of Manin’s conjecture, and then introduce the notion of balanced line bundles. In this part, unless it is stated, we always assume that the ground field is an algebraically closed field of characteristic zero.

4.1 Generalities

This section is extracted from [HTT12, Section 2].

100 Let X be a Q-factorial projective variety, i.e., X is a normal projective variety such that for any Weil divisor D, some multiple of D is a Cartier divisor. A rigid eﬀective divisor is an eﬀective Weil divisor D ⊂ X such that

0 H (X, OX (nD)) = 1,

for suﬃciently divisible n ∈ N. If D is rigid with irreducible components D1,...,Dr then

0 H (OX (n1D1 + ... + nrDr)) = 1, for suﬃciently divisible n1, ··· , nr ∈ N and

◦ span(D1,...,Dr) ∩ Λeﬀ (X) = ∅.

4.1.1 The invariant a(L)

First we generalize the invariant a(L) in Conjecture 16 for a Q-factorial projective varieties with only canonical singularities. See [KM98, Deﬁnition 2.34] for deﬁnitions of various singularities.

Definition 71. Let X be a Q-factorial projective varieties with only canonical singularities. Assume that KX is not pseudo-effective and L is a big line bundle on X. The Fujita invariant is defined by

a(X,L) = inf{a ∈ R : aL + KX eﬀective }

= min{a ∈ R : a[L] + [KX ] ∈ Λeﬀ (X)}.

Remark 72. A smooth projective variety X is uniruled if and only if KX is not

101 pseudo-eﬀective [BDPP04], [Laz04b, Cor. 11.4.20].

For technical purposes, we need a slightly more general definition: Let (X, ∆) denote a Kawamata log terminal pair, i.e., a normal projective variety X and an effective Q-divisor ∆ on X, such that KX + ∆ is Q-Cartier and (X, ∆) has Kawamata log terminal singularities. The notions of pseudo-effective and big Q-

Cartier divisors make sense in this context [Laz04a, §2.2.B]. When KX + ∆ is not pseudo-eﬀective and L is a big line bundle, then the natural extension of Deﬁnition 71 may still be used:

a((X, ∆),L) = inf{a ∈ R : aL + KX + ∆ eﬀective }.

The following result was conjectured by Fujita and proved by Batyrev for threefolds and [BCHM10, Cor. 1.1.7] in general.

Theorem 73. Let (X, ∆) be a projective Kawamata log terminal pair and L an ample line bundle on X. Then, a((X, ∆),L) is rational.

However, this property can fail when L is big, but not ample:

Example 74. [Leh08, Example 4.9] Let Y be an abelian surface with Picard rank at least 3. The cone of nef divisors and the cone of pseudo-eﬀective divisors coincide, and its boundary is a quadratic cone. Let N be a line bundle on Y such that −N is ample. We consider X := P(O ⊕ O(N)), and we denote the projection morphism to Y by π. Let S ⊂ X be the section corresponding to the quotient map O ⊕ O(N) → O(N). Every line bundle on X is linearly equivalent to a divisor

∗ tS +π D where D is a divisor on Y , and in particular the canonical line bundle KX

∗ is linearly equivalent to −2S + π N. The cone of pseudo-eﬀective divisors Λeﬀ (X)

∗ is generated by S and π Λeﬀ (Y ).

102 Consider a big Q-divisor L = tS + π∗D where t > 0 and D is a big Q-divisor on Y . If t is sufficiently large, then a(X,L) is defined by a[D] + [N] ∈ ∂Λeff (Y ).

However, the boundary of Λeﬀ (Y ) is a quadratic cone, and this is not solvable in

Q in general.

From a point of view of Manin’s conjecture, global geometric invariants involving Manin’s conjecture should be birationally invariant, and this observation is true for Fujita invariant.

Proposition 75. Let β : X˜ → X be a birational morphism of projective varieties, ˜ where X is smooth and X has only canonical singularities. Assume KX is not pseudo-eﬀective and L is big. Setting L˜ = β∗L, we have

a(X,L) = a(X,˜ L˜).

Proof. Since X has canonical singularities, we have

∗ X KX˜ = β KX + diEi, i where the Ei are the irreducible exceptional divisors and the di are nonnegative rational numbers. It follows that for integers m, n > 0 we have

X ˜ Γ(OX (mKX + nL)) = Γ(OX˜ (mKX˜ − bmdicEi) + nL) i ˜ = Γ(OX˜ (mKX˜ + nL)), where the second equality reﬂects the fact that allowing poles in the exceptional locus does not increase the number of global sections. In particular, eﬀective

103 divisors supported in the exceptional locus of β are rigid. Note that it follows from the assumption that any multiple of KX˜ is not eﬀective, and at least this implies that a(X,˜ L˜) ≥ 0. So deﬁnition 71 gives a(X,˜ L˜) = a(X,L) > 0.

Remark 76. This proposition enables us to deﬁne Fujita invariants for singular varieties via passage to a smooth model.

4.1.2 The invariant of b(L)

Next, we discuss the second invariant showing up in the context of Manin’s conjecture. To ﬁx our terminology, we recall some basic deﬁnitions:

Definition 77. Let N be a finite dimensional vector space over R. A closed convex cone Λ ⊂ N is a closed subset which is closed under linear combinations with nonnegative real coefficients. An extremal face F ⊂ Λ is a closed convex subcone of Λ such that u, v ∈ Λ and u + v ∈ F imply that u, v ∈ F . It is well-known that every extremal face F is supported by a supporting function σ, i.e., there exists a linear functional σ : N → R such that σ ≥ 0 on Λ and

F = {σ = 0} ∩ Λ.

Definition 78. Let X be a projective variety with only Q-factorial terminal singularities such that KX is not pseudo-effective. Let L be a big line bundle on X. Then we define

b(X,L) = the codimension of the minimal extremal face of

Λeﬀ (X) containing a R-divisor a(X,L)L + KX .

The birational invariance of b(X,L) is also true in general:

104 Proposition 79. Let X be a Q-factorial terminal projective variety such that KX is not pseudo-eﬀective and L a big line bundle on X. Let β : X˜ → X be a smooth resolution. Setting L˜ = β∗L we have

b(X,L) = b(X,˜ L˜).

Proof. Let F be the minimal extremal face of Λeﬀ (X) containing a R-divisor ˜ ˜ ˜ ˜ ˜ a(X,L)L+KX and F be the minimal extremal face of Λeﬀ (X) containing a(X, L)L+ ˜ KX˜ . We denote vector spaces generated by F and F by MF and MF˜ respectively. ˜ There exists a nef cycle ξ ∈ NM1(X) such that

˜ ˜ F = {ξ = 0} ∩ Λeﬀ (X).

˜ Here NM1(X) is the cone of nef curves which is the dual of the cone of pseudo- eﬀective divisors. Let E1, ··· ,En be irreducible components of the exceptional locus of β. Note that the negativity lemma ([BCHM10, Lemma 3.6.2]) implies

˜ ∗ that NS(X) is a direct sum of β NS(X) and [Ei]’s. Since X has only terminal singularities, it follows from Proposition 75 that

˜ ˜ ˜ ∗ X a(X, L)L + KX˜ = β (a(X,L)L + KX ) + diEi, i

˜ ∗ where di’s are positive rational numbers. This implies that F contains β (a(X,L)L+

KX ) and Ei’s. Thus a(X,L)L + KX is contained in an extremal face supported by a supporting function β∗ξ, i.e.,

{β∗ξ = 0} ∩ Λeﬀ (X),

105 so this extremal face also contains F . Thus we can deﬁne a well-deﬁned injective map X Φ: MF ,→ MF˜/( REi). i

On the other hand, let η ∈ NM1(X) be a nef cycle supporting F . Consider a linear functionalη ˜ : NS(X˜) → R deﬁned by

∗ η˜ ≡ η on β NS(X) andη ˜ · Ei = 0 for any i.

The projection from NS(X˜) to β∗NS(X) maps pseudo-eﬀective divisors to pseudo- ˜ eﬀective divisors so thatη ˜ ∈ NM1(X). Moreover,

˜ ˜ ˜ ∗ X η˜ · (a(X, L)L + KX˜ ) =η ˜ · (β (a(X,L)L + KX ) + diEi) = 0, i

˜ ˜ so {η˜ = 0} ∩ Λeﬀ (X) contains F . This implies that Φ is actually bijective. Our assertion follows from this.

Remark 80. This proposition enables us to deﬁne b(X,L) for singular projective varieties via passage to a smooth model.

Example 81. Let X be a Q-factorial terminal projective variety with the big ˜ anticanonical line bundle. Let β : X → X be a smooth resolution, and E1, ··· ,En be irreducible components of the exceptional locus. Since X has only terminal singularities, we have

∗ X KX˜ = β KX + diEi, i where di’s are positive rational numbers. Hence the minimal extremal face con-

∗ taining −β KX + KX˜ contains a simplicial cone F := ⊕iR≥0[Ei]. We claim that

106 F is an extremal face. ˜ ˜ Take u, v ∈ Λeff (X) such that u + v ∈ F . The projection from NS(X) to β∗NS(X) maps pseudo-effective divisors to pseudo-effective divisors so that u, v P are linear combinations of [Ei]’s. Thus we need to prove that if E = i eiEi = E+ − E−, where E+ ≥ 0 and E− > 0 are effective R-divisors with no common component, then E is not pseudo-effective. By cutting by hyperplanes, we reduce to the case when X is a surface. Suppose that E is pseudo-effective. By per- turbing by Ei’s, we may assume that E is a Q-divisor. We consider the Zariski decomposition of E = P + N where P is a nef Q-divisor and N is an effective Q-divisor. Then, P and N are numerically equivalent to linear combinations of

+ − Ei’s. However (Ei · Ej) is negative deﬁnite so that P ≡ 0. We have E ≡ E + N. We may assume that E+ and N have no common component. Then negativity implies that 0 ≥ (E+)2 = E+ · (E− + N) ≥ 0, so we conclude that E+ = 0. This is a contradiction since −E− cannot be pseudo-eﬀective. Thus F is an extremal face, and we conclude that

˜ ∗ ˜ b(X, −β KX ) = rk NS(X, R) − n = rk NS(X, R) = b(X, −KX ).

However, the geometric meaning of the invariant b(X,L) is unclear in general, and we introduce some situations where it is more clear:

Deﬁnition 82. Let X be a Q-factorial terminal and projective variety such that

KX is not pseudo-eﬀective. Let L be a big line bundle on X. We say a(X,L)L+KX is locally rational polyhedral if there exist ﬁnitely many linear functionals

λi : NS(X, Q) → Q

107 such that λi(a(X,L)L + KX ) > 0 and

Λeff (X) ∩ {v : λi(v) ≥ 0 for any i}, is finite rational polyhedral and generated by effective Q-divisors.

There are a number of situations where Λeff (X) is generated by a finite collection of effective divisors:

• A projective variety X is log Fano if there exists an eﬀective Q-divisor ∆ on X

such that (X, ∆) is divisorially log terminal and −(KX +∆) is ample. If X is log Fano then the Cox ring of X is ﬁnitely generated [BCHM10, Cor. 1.3.2], so

in particular, Λeff (X) is finite rational polyhedral and generated by effective divisors.

• Let X be a smooth projective variety that is toric or an equivariant compact-

n iﬁcation of the additive group Ga . Then Λeﬀ (X) is generated by boundary divisors, i.e., irreducible components of the complement of the open orbit, [HT99, Thm. 2.5], [BT95, Prop.1.2.11].

Let X be a smooth projective variety with Λeff (X) generated by a finite number of effective divisors and Pic(X)Q = NS(X, Q). Since each irreducible rigid effective divisor on X is a generator of Λeff (X), we have

Z := ∪rigid eﬀectiveD is a Zariski closed proper subset of X.

Theorem 83. Let X be a Q-factorial terminal projective variety such that KX is not pseudo-eﬀective. Let L be a big line bundle on X. Suppose that a(X,L)L+KX

108 has the form of c(A+KX +∆) where A is an ample R-divisor, (X, ∆) a kawamata log terminal pair, and c > 0 a positive number. Then a(X,L)L + KX is locally rational polyhedral so that a(X,L) is rational.

Proof. The local finiteness of the pseudo-effective boundary is proved in [Leh08, Proposition 3.3] by using the finiteness of the ample models [BCHM10, Corollary 1.1.5], and moreover Lehmann proved that the pseudo-effective boundary is locally defined by movable curves. The generation by effective Q-divisors follows from the non-vanishing theorem [BCHM10, Theorem D].

In particular, a(X,L)L + KX is locally rational polyhedral when L is ample. As we observed in Example 74, the local ﬁniteness is no longer true if we only assume that L is big. However, there are certain cases where the local ﬁniteness of a(X,L)L + KX still holds for any big line bundle L:

Example 84 (Surfaces). Let X be a smooth projective surface such that KX is not pseudo-eﬀective. Let L be a big line bundle on X. Suppose that D = a(X,L)L +

KX is a non-zero pseudo-eﬀective divisor. Then D is locally rational polyhedral.

We consider the Zariski decomposition of D = P + N where P is a nef R-divisor and N is a negative part of D. [Bou04] proved that the boundary of Λeﬀ(X) is locally rational polyhedral away from the nef cone. (See [Bou04, Theorem 3.19] and [Bou04, Theorem 4.1].) Thus if N is non-zero, then our assertion follows.

Suppose that N is zero. Since a(X,L)L · P + KX · P = D · P = 0 and L · P > 0, we have KX · P < 0. Thus our assertion follows from Mori’s cone theorem. In particular, a(X,L) is a rational number.

Example 85 (Equivariant compactiﬁcations of the additive groups). Let X be

n a smooth projective equivariant compactiﬁcation of the additive group Ga . Then

109 Λeﬀ(X) is a simplicial cone generated by boundary components. (See [HT99, Theo- rem 2.5].) However this phenomena cannot be explained from Theorem 83. Indeed,

3 3 consider the standard embedding of Ga into P :

3 3 Ga 3 (x, y, z) 7→ (x : y : z : 1) ∈ P .

This is an equivariant compactiﬁcation, and the group action ﬁxes any points on the boundary divisor D which is a hyperplane section. Let X be an equivariant blow up of 12 generic points on a smooth cubic curve in the hyperplane D. Write

H for the pullback of the hyperplane class and E1, ··· ,E12 for the exceptional divisors. Consider

L = 4H − E1 − · · · − E12.

Then L is a big and nef divisor, but not semi-ample. (See [Laz04a, Section 2.3.A] for more details.) In particular, the section ring of L is not ﬁnitely generated. On the other hand, we consider

Λadj(X) = {Γ ∈ Λeff(X) | Γ = c(A + KX + ∆)} where A is an ample R-divisor, (X, ∆) a Kawamata log terminal pair, and c a positive number. Then Λadj(X) forms a covex cone. The existence of non finitely generated divisors and [BCHM10, Corollary 1.1.9] imply that Λadj(X) $ Λeff(X).

It is natural to expect that the invariant b(X,L) is related to the canonical

ﬁbration associated to a(X,L)L + KX . A sample result in this direction is:

Proposition 86. Let X be a smooth projective variety such that KX is not pseudo- eﬀective. Let L be a big line bundle and assume that D = a(X,L)L + KX is locally

110 rational polyhedral and semi-ample. Let π : X → Y be the semi-ample ﬁbration of D. Then

b(X,L) = rk NS(X) − rk NSπ(X),

where NSπ(X) is the lattice generated by π-vertical divisors, i.e., divisors M ⊂ X such that π(M) ( Y

Proof. Let F be the minimal extremal face of Λeﬀ(X) containing D = a(X,L)L +

KX and MF the vector space generated by F . We claim that MF = NSπ(X). Let

H be an ample Q-divisor on Y such that π∗H = D. Let M be a π-vertical divisor on X. Then for sufficiently large m, there exists an effective Cartier divisor H0 such that mH ∼ H0 and the support of H0 contains π(M). Thus mD = mπ∗H ∼ π∗H0 ∈ F and the support of π∗H0 contains M. We conclude that M ∈ F , and this proves that NSπ(X) ⊂ MF . Next, let Xy be a general fiber of π and C ⊂ Xy be a movable curve on X such that [C] is in the interior of NM1(Xy). Then the following set

FC = {[C] = 0} ∩ Λeﬀ(X),

is an extremal face containing D. The minimality implies F ⊂ FC . On the other hand, the local rational ﬁniteness of D implies that there exist eﬀective

Q-divisors D1, ··· ,Dn ∈ F such that D1, ··· ,Dn form a basis of MF . Since

D1 · C = ··· = Dn · C = 0, the supports of Di’s are π-vertical. Hence it follows that MF ⊂ NSπ(X).

Remark 87. When L is ample, it follows from [KMM87, Lemma 3.2.5] that the codimension of the minimal extremal face of Nef cone containing D is equal to the relative Picard rank ρ(X/Y ).

We explored this idea more in the case of toric varieties. See Section 6.3.

111 4.2 Balanced line bundles

This section is extracted from [HTT12, Section 3].

Deﬁnition 88. A big line bundle L on X is weakly balanced with respect to an irreducible subvariety Y ⊂ X if

• L|Y is big;

• a(Y,L|Y ) 6 a(X,L);

• if a(Y,L|Y ) = a(X,L) then b(Y,L|Y ) 6 b(X,L).

A big line bundle is called balanced with respect to Y if it is weakly balanced and one of the two inequalities is strict. It is weakly balanced on X if there exists a Zariski closed subset Z ⊂ X such that L is weakly balanced with respect to every irreducible subvariety Y not contained in Z. The subset Z will be called exceptional. A line bundle is called balanced on X if it is weakly balanced on X and if it is balanced with respect to every irreducible subvariety not contained in Z.

Example 89. Every ample line bundle on P2 is balanced. Consider X = P1 × P1. Direct arguments, or Proposition 96, show that an ample L is balanced on X if and only if L is proportional to −KX .

One hopes to use the invariant a(X,L) to identify the exceptional locus X◦, however, this is highly non-trivial even in the following simple situations:

Conjecture 90 (Debarre-de Jong conjecture). Let X ⊂ Pn be a Fano hypersurface of degree d ≤ n. Then the dimension of the variety of lines is 2n − d − 3.

112 In particular, when d = n, for any line C, we have

a(C, −KX |C ) = 2 > 1 = a(X, −KX ).

The conjecture predicts that the dimension of the variety of lines is n − 3 so that lines will not sweep out X.

Proposition 91. Let X be a smooth Fano variety of Picard rank one and Y ⊂ X an irreducible smooth eﬀective divisor. Then −KX is balanced with respect to Y .

Proof. For smooth divisors Y ⊂ X the claim follows from adjunction formula,

◦ −KX |Y + KY = Y |Y ∈ Λeﬀ (Y ) , because Y is an ample divisor on X. Thus we obtain

a(Y, −KX |Y ) < a(X, −KX ) = 1.

However this statement is no longer true when Y is singular:

Example 92 (Mukai-Umemura 3-folds, [MU83]). Consider the standard action of

12 SL2 on V = Cx⊕Cy. Let R12 = Sym (V ) be a space of homogeneous polynomials of degree 12 in two variables and f ∈ R a general form. Let X be a Zariski closure of SL2-orbit SL2 · [f] ⊂ P(R12). Then X is a smooth Fano 3-fold of index 1 with

Pic(X) = Z for general f. The complement of the open orbit SL2 · [f] is an

113 irreducible divisor

11 11 12 D = SL2 · [x y] = SL2 · [x y] ∪ SL2 · [x ].

D is a hyperplane section on P(R12) and generates Pic(X). Furthermore D is the image of P1 × P1 by a linear series of bidegree (11, 1), which is injective, open immersion outside of the diagonal, but not along the diagonal. In particular, D is singular along the diagonal.

Let β : D˜ → D be the normalization of D which is isomorphic to P1 ×P1. Then

∗ −β KX |D˜ is a line bundle of bidegree (11, 1) so that

˜ ∗ a(D, −KX |D) = a(D, −β KX |D˜ ) = 2 > 1 = a(X, −KX ).

Thus Proposition 91 does not hold for D.

Remark 93. We still do not know whether Proposition 91 holds for singular surfaces Y in P3. This question is related to an arithmetic question whether linear growth conjecture holds for big line bundles on very singular rational surfaces of general type in P3. See [Tsc09, Conjecture 4.10.1 and Conjecture 4.10.3].

Example 94. Let X ⊂ P4 be a smooth cubic threefold. The Picard group of X is spanned by the hyperplane class. By Proposition 91, −KX is balanced with respect to every smooth divisor on X. Let Y ⊂ X be a line. Note that O(2) on P4 restricts to the anticanonical line bundle on X and on Y , and b(Y,L|Y ) = b(X,L) = 1. Thus

−KX is weakly balanced, but not balanced, with respect to Y . Since the family of lines dominates X, −KX is not balanced on X.

114 Chapter 5

Fano varieties

In this chapter, we discuss balanced line bundles on del Pezzo surfaces and smooth Fano 3-folds of Picard rank 1.

5.1 Del Pezzo surfaces

This section is extracted from [HTT12, Section 4].

Let X be a smooth projective surface with ample −KX , i.e., a Del Pezzo surface. These are classiﬁed by the degree of the canonical class d := (KX ,KX ).

Basic examples are P2 and P1 × P1, more examples are obtained by blowing up 9 − d general points on P2. We have

• rk NS(X) = 10 − d;

• for 2 6 d 6 8 the cone Λeﬀ (X) is generated by classes of exceptional curves, i.e., smooth rational curves of self-intersection −1.

Let L be a big line bundle on X. When is it balanced? The only subvarieties of X on which we need to test the values of a and b are rational curves C ⊂ X,

115 and b(C,L|C ) = 1. It is easy to characterize curves breaking the balanced condition for the Fujita invariant:

Lemma 95. Let X be a del Pezzo surface of degree d. Let C be an irreducible rational curve with (C,C) 6= −1 and L a big line bundle on X. Then

a(C,L|C ) 6 a(X,L), (5.1) i.e., L is weakly balanced on X, where the exceptional set Z is the union of exceptional curves.

Proof. First observe that if C is an irreducible rational curve with (−KX ,C) = 1 then C is exceptional. Indeed, if (C,C) < 0, then (C,C) = −1, by adjunction, and C is exceptional. On the other hand, the Hodge index theorem implies that d(C,C) − 1 < 0, i.e., (C,C) = −1 or 0. The second case is not possible since

(KX ,C) + (C,C) must be even. Let C ⊂ X be a rational curve which is not exceptional. After rescaling, we may assume that a(X,L) = 1, in particular, we do not assume that L is an integral divisor. Writing L + KX = D, where D is an eﬀective Q-divisor, and computing the intersection with C we obtain

(L, C) = (−KX ,C) + (D,C) > (−KX ,C)

Since C is not exceptional, (−KX ,C) > 2, i.e., (L, C) > 2. It follows that

(L, C) + deg(KC˜) = (L, C) − 2 > 0,

116 ˜ where C is the normalization of C, i.e., a(C,L|C ) 6 1, as claimed.

We proceed with a characterization of b(X,L). Consider the Zariski decomposition

a(X,L)L + KX = P + E,

Pn where P is a nef Q-divisor and E = i=1 eiEi, ei ∈ Q>0,(Ei,Ej) < 0. We have (P,E) = 0. By basepoint free theorem (see [KM98, Theorem 3.3]), the divisor P is semi-ample and deﬁnes a semi-ample ﬁbration

π : X → B.

We have two cases:

Case 1. B is a point. Then

n X a(X,L)L + KX = eiEi i=1

is rigid, which implies that the classes Ei are linearly independent in NS(X). In particular, ⊕ E is an extremal face of Λ (X), and in fact the minimal extremal iR>0 i eﬀ face containing a(X,L)L + KX . It follows that

b(X,L) = rk NS(X) − n.

Case 2. B is a smooth rational curve. Then the minimal extremal face con-

117 taining n X a(X,L)L + KX = P + eiEi i=1 is given by

NSπ(X) ∩ Λeﬀ (X) = {P = 0} ∩ Λeﬀ (X),

where NSπ(X) ⊂ NS(X) is the subspace generated by vertical divisors, i.e., divisors D ⊂ X not dominating B. It follows that

b(X,L) = rk NS(X) − rk NSπ(X) = 1.

Proposition 96. Let X be a del Pezzo surface and L a big line bundle on X.

Then L is balanced if and only if a(X,L)L + KX = D where D is a rigid eﬀective divisor.

Proof. Assume that a(X,L) = 1. In Case 1, we must have

n X L + KX = D = eiEi, ei > 0, i=1 with Ei disjoint exceptional curves. Assume that L is not balanced so that

2 b(X,L) = 1. Let π : X → P be the blowdown of E1,...,En and h a hyperplane class on P2. Then

n ∗ X L = −KX + D = 3π h + (ei − 1)Ei. i=1

Let C be an irreducible rational curve which is not exceptional. If C does not meet any of the Ei then

∗ (L, C) = (3π h, C) > 3 > 2.

118 If C meets at least one of the Ei then

(L, C) = (−KX ,C) + (D,C) > 2

since the ﬁrst summand is > 2. It follows that a(C,LC ) < 1, i.e., L is balanced, contradicting our assumption. In Case 2, we have

n X L + KX = D = P + eiEi, ei > 0, i=1

1 where P is nef and Ei are disjoint exceptional divisors. Let π : X → P be the ﬁbration induced by the semi-ample line bundle P . The general ﬁber F of π is a conic and

rk NS(X) − rk NSπ(X) = 1.

We have (F,F ) = 0, (−KX ,F ) = 2, and the class of F is proportional to P . Hence, for any such F ,

a(F,L|F ) = a(X,L), b(F,L|F ) = b(X,L) = 1.

Thus L is not balanced.

2 2 Example 97. Let X = Blp1,p2 (P ) be the blowup of P in two points and ` the ˜ line passing through p1, p2. Let ` ⊂ X be the strict transform of ` and E1,E2 ⊂ X the exceptional divisors. Then

Λ (X) = `˜⊕ E ⊕ E . eﬀ R>0 R>0 1 R>0 2

119 We have ˜ ˜ ˜ −KX = 3` + 2E1 + 2E2, (`, `) = (Ei,Ei) = −1.

Every L such that L + KX is on the face spanned by E1 and E2 is balanced. On the other hand, the Zariski decomposition of

L + KX = λ` + Ei, λ > 0, > 0, is given by   ˜ λ(` + Ei) + ( − λ)Ei, when λ 6 ,  ˜ ˜ (` + Ei) + (λ − )` otherwise.

In particular, such L are balanced if and only if either λ or vanish.

5.2 Del Pezzo surface ﬁbrations

This section is extracted from [HTT12, Section 5]. As we observed in Example 19, the balance property can fail in general:

Example 98. [BT96b] Let f, g be general cubic forms on P3 and

1 3 X := {sf + tg = 0} ⊂ P × P the Fano threefold obtained by blowing up the base locus of the pencil. The projection onto the ﬁrst factor exhibits a cubic surface ﬁbration

1 π : X → P ,

120 so that −KX restricts to −KY , for every smooth ﬁber Y of π. Thus

a(Y, −KY ) = a(X, −KX ) = 1.

Furthermore, the N´eron-Severi rank of a smooth ﬁber of π is 7. On the other hand, by Lefschetz theorem, we have rk NS(X) = 2. It follows that

7 = b(Y, −KY ) > b(X, −KX ) = 2,

i.e., −KX is not balanced on X.

Let Z be the union of singular ﬁbers of π, −KY -lines in general smooth ﬁbers

3 Y , and the exceptional locus of the blow up to P . −KX is balanced with respect to every rational curve on X which is not contained in Z.

Failure of the balance property for Fano ﬁbrations is related to the monodromy action on the N´eron-Severi space. A sample result in this direction is:

Proposition 99. Let π : X → Y be a Fano ﬁbration from a smooth Fano variety to a smooth projective curve. Take a general smooth ﬁber Xy, and assume that the monodromy action on NS(Xy) is trivial. Then

rk NS(Xy) < rk NS(X).

Proof. See [dFH11, Proposition 6.5]. Also see [KM92, Section 12].

The following proposition follows from the rigidity of Fano varieties:

Proposition 100. Let π : X → B be a ﬂat family of Fano varieties with terminal

Q-factorial singularities over a connected smooth curve B and L a π-big line bundle

121 on X . Then

a(Xb,Lb) = a(Xb0 ,Lb0 ), b(Xb,Lb) = b(Xb0 ,Lb0 ), for all b, b0 ∈ B.

Proof. By [dFH09, Corollary 5.1], the cone of pseudo-eﬀective divisors for Fano varieties is locally constant in families.

5.3 Fano threefolds

In this section we investigate the geometry of smooth Fano threefolds of Picard rank one. There are ﬁnitely many families of smooth Fano threefolds; a list of these can be found in [MM82], [Sha99, Chapter 12], and also in [MM04], which includes one case that was missing previously. Here we follow the classiﬁcation scheme in [Man93]. The basic invariants of Fano threefolds are:

• the rank of the Picard group, i.e., b(X, −KX );

• the index r = r(X), which is the maximal integer such that KX is divisible by r in Pic(X);

3 • the degree d(X) := (−KX ) ;

• the Mori invariant m(X) which is the smallest integer m such that through

every point of X passes a rational curve C with (−KX ,C) 6 m.

The following proposition from [Man93, Section 2] describes smooth Fano threefolds:

122 Proposition 101. Every smooth Fano threefold over an algebraically closed ﬁeld of characteristic zero is isomorphic to one of the following:

(1) a generalized ﬂag variety P \G;

(2) a variety X with m(X) = 2;

(3) a blowup of varieties of type (1) or (2);

(4) a direct product of P1 and a del Pezzo surface.

We recall the ﬁner classiﬁcation from [Man93]. When rk Pic(X) = 1, we have the following possibilities for X:

• P3, or a quadric, or

• r(X) = 2 and d(X) ∈ {8, 16, 24, 32, 40}, or

• r(X) = 1 and d(X) ∈ {2, 4, 6, 8, 10, 12, 14, 16, 18, 22}.

The ﬁrst two cases are covered by Proposition 103. By Proposition 91, −KX is balanced with respect to smooth divisors on X. Thus we need to focus on curves.

Varieties in the second class are dominated by −KX -conics and there are no curves of smaller degree. Varieties in the third class are also dominated by −KX -conics but are not dominated by −KX -lines. Thus, −KX is weakly balanced but not balanced on X.

Remark 102. Let X be a Fano variety of the second or third class, deﬁned over a number ﬁeld. The families of −KX -conics dominating X are surfaces of general type, which embedded into their Albanese varieties. By Falting’s theorem, conics

123 defined over a fixed number field lie on a proper subvariety and cannot dominate X.

124 Chapter 6

Equivariant geometry

In this chapter, we discuss the balance property for projective varieties with group actions, e.g., ﬂag varieties, toric varieties, and equivariant compactiﬁcations of homogeneous spaces. Manin’s conjecture has been proved for many classes of these varieties, but, nevertheless, proving balanceness for these varieties is quite nontrivial. Results in this chapter have arithmetic applications in [CLT02] and [GTBT11].

6.1 Generalized ﬂag varieties

Let G be a connected semi-simple group, and P ⊂ G a parabolic subgroup. Let X = P \G be a generalized ﬂag variety. Manin’s conjecture for X was proved in [FMT89]. The following proposition is extracted from [HTT12, Section 3].

Proposition 103. Let G be a connected semi-simple algebraic group, P ⊂ G a parabolic subgroup and X = P \G the associated generalized ﬂag variety. Let L be a big line bundle on X. Then

125 • if L is not proportional to −KX , then L is not balanced;

• if L is proportional to −KX , then L is balanced with respect to any smooth subvariety Y ⊂ X.

Proof. Recall that for generalized flag varieties, the nef cone and the pseudo- effective cone coincide so that L is ample. Moreover, it is well-known that the nef cone of a flag variety is finitely generated by semi-ample line bundles. Also note that since every rationally connected smooth proper variety is simply connected, all parabolic subgroups are connected. Assume that L is not proportional to the anticanonical bundle, i.e., D = a(X,L)L + KX is a non-zero effective Q-divisor. Let π : X → Y be the semi- ample fibration of D. Then Y is also a G-variety so that there exists a parabolic subgroup P 0 ⊃ P such that Y = P 0\G and π is the natural projection map. We have the following exact sequence:

0 0 0 → Pic(P \G)Q → Pic(P \G)Q → Pic(P \P )Q → 0.

Indeed, the surjectivity follows from [KKV89, Proposition 3.2(i)]. Then the exact- ness of other parts follows from [KMM87, Lemma 3.2.5]. Let W be a ﬁber of π.

Then we have a(X,L) = a(W, L|W ) since KX |W = KW . The exact sequence and Remark 87 imply that

b(X,L) = ρ(X/Y ) = rk Pic(W ) = b(W, L|W ).

Thus L is not balanced with respect to any ﬁber of π.

Let L = −KX and Y ⊂ X a smooth subvariety. Let g be the Lie algebra of G.

126 For any ∂ ∈ g, we can construct a global vector ﬁeld ∂X on X such that for any open set U ⊂ X and any f ∈ OX (U),

X ∂ (f)(x) = ∂gf(x · g)|g=1.

It follows that NY/X is globally generated where NY/X is the normal bundle of Y in X. Thus we conclude that

−KX |Y + KY = det(NY/X ) ∈ Λeﬀ (Y ),

so that a(Y, −KX |Y ) ≤ a(X, −KX ) = 1.

Suppose that a(Y, −KX |Y ) = a(X, −KX ) = 1. Our goal is to prove that

b(Y, −KX |Y ) < b(X, −KX ) = rk NS(X).

First we assume that det(NY/X ) is trivial so that NY/X is the trivial vector bundle of rank r = codim(Y,X). The above construction of vector ﬁelds deﬁnes a surjective map:

0 ϕ : g → H (Y, NY/X ).

We may assume that e = P ∈ Y so that the Lie algebra p of P is contained in the

0 kernel of ϕ. We consider the Hilbert scheme Hilb(X). Note that H (Y, NY/X ) is naturally isomorphic to the Zariski tangent space of Hilb(X) at [Y ]. Consider the following morphism: π : G 3 g 7→ [Y · g] ∈ Hilb(X).

1 Since Y is Fano-type, H (Y, NY/X ) = 0. This implies that Y is unobstructed, and

127 it follows that Hilb(X) is smooth at [Y ] and

dim[Y ] Hilb(X) = r.

Moreover, since ϕ is surjective, we conclude that π is a smooth morphism and π(G) is a smooth open subscheme in Hilb(X). Let H be the connected component of Hilb(X) containing [Y ]. Let P 0 = Stab(Y ). Since the kernel of ϕ contains p, we have P ⊂ P 0. This implies that π(G) = P 0\G is open and closed so that

H = π(G) = P 0\G.

In particular, dim G − dim P 0 = r, so the kernel of ϕ is exactly equal to the Lie algebra p0 of P 0. Consider the universal family U ⊂ X × H on H. It follows that G acts on U transitively, and we conclude that U = P \G and Y = P \P 0. Our assertion follows from the exact sequence which we discussed before.

In general case, we consider the semi-ample fibration associated to det(NY/X ). Its general smooth fibers are smooth subvarieties with trivial normal bundles so that they can be realized as fibers of a fibration ρ : X → B. Then Y is a pullback of a subvariety H ⊂ B. Theorem 83 and Proposition 86 imply that

b(Y, −KX |Y ) = rk NS(Y ) − rk NSρ(Y ).

The restriction map:

Φ : NS(Y )/NSρ(Y ) → NS(Yh),

is injective where Yh is a general ﬁber of ρ. (For example, this follows from

128 [KMM87, Lemma 3.2.5].) Hence we have

b(Y, −KX |Y ) ≤ b(Yh, −KX |Yh ) < b(X, −KX ).

6.2 Equivariant compactiﬁcations of homogeneous

spaces

This section is extracted from [HTT12, Section 7]. Let G be a connected linear algebraic group, H ⊂ G a closed subgroup, and X a projective equivariant compactiﬁcation of X◦ := H\G. Using equivariant resolution of singularities we may assume that X is smooth and that the boundary

◦ ∪α∈ADα = X \ X is a divisor with normal crossings. If H is a parabolic subgroup of a semi-simple group G, then there is no boundary, i.e., A is empty, and H\G is a generalized ﬂag variety which was discussed in Section 6.1. Throughout, we will assume that A is not empty. Let X(G)∗ be the group of algebraic characters of G and

∗ X(G, H) = { χ : G → Gm | χ(hg) = χ(g), ∀h ∈ H } the subgroup of characters whose restrictions to H are trivial. Let PicG(X) be the group of equivalence classes of G-linearized line bundles on X and Pic(X) the

129 Picard group of X. For L ∈ PicG(X), the subgroup H ⊂ G acts linearly on the

ﬁber Lx at x = H ∈ H\G. This deﬁnes a homomorphism

PicG(X) → X(H)∗ to characters of H. Let Pic(G,H)(X) be the kernel of this map. We will identify line bundles and divisors with their classes in Pic(X).

Proposition 104. Let G be a connected linear algebraic group and H a closed subgroup of G. Let X be a smooth projective equivariant compactiﬁcation of X◦ :=

H\G with a boundary divisor ∪α∈ADα. Then

1. we have an exact sequence

∗ ◦ 0 → X(G, H) → ⊕α∈AZDα → Pic(X) → Pic(X ) → 0;

2. we have an exact sequence

0 → X(G, H)∗ → Pic(G,H)(X) → Pic(X) ; Q Q Q

and the last homomorphism is surjective when

C(G, H) := Coker(X(G)∗ → X(H)∗)

is ﬁnite. (Equivalently, Pic(X◦) is ﬁnite.)

3. and we have a canonical injective homomorphism

(G,H) Ψ: ⊕α∈AQDα ,→ Pic (X)Q;

130 and Ψ is an isomorphism when C(G, H) is ﬁnite.

Proof. The ﬁrst exact sequence easily follows. The second assertion follows from [MFK94, Corollary 1.6] and [KKV89, Proposition 3.2(i)]. For the last assertion: Corollary 1.6 of [MFK94] implies that some multiple of

Dα is G-linearizable. We may assume that G acts on the ﬁnite-dimensional vector

0 space H (X, OX (Dα)), via this G-linearization. Let sα be the section corresponding

0 to Dα. Then sα ∈ H (X, OX (Dα)) is an eigenvector of the action by G. After multiplying by a character of G, if necessary, we may assume that sα is ﬁxed by the action of G. We let Ψ(Dα) be this G-linearization. P Suppose that Φ( α dαDα) = OX , with trivial G-linearization, where dα ∈ Z. Then there exists a rational function f such that

X div(f) = dαDα. α

We may assume that f is a character of G whose restriction to H is trivial. By the definition of Ψ, the function f must be fixed by the G-linearization. This implies that f ≡ 1. When C(G, H) is finite, the surjectivity of Ψ follows from (1) and (2).

Proposition 105. Let G be a connected linear algebraic group and H a closed subgroup of G. Let X be a smooth projective equivariant compactiﬁcation of X◦ :=

H\G with a boundary divisor ∪α∈ADα. If C(G, H) is ﬁnite, then the anticanonical divisor −KX is big.

Proof. Let g, resp. h, be the Lie algebra of G, resp. H. For any ∂ ∈ g, there is a unique global vector ﬁeld ∂X on X such that for any open set U ⊂ X and any

131 f ∈ OX (U),

X ∂ (f)(x) = ∂gf(x · g)|g=1.

Let ∂1, . . . , ∂n ∈ g be a lift of a basis for g/h. Consider the following global section of the anticanonical bundle det(TX ):

X X δ = ∂1 ∧ · · · ∧ ∂n .

Note that this section is nonzero at x = H ∈ H\G = X◦. The proof of [CLT02, Lemma 2.4] implies that this section vanishes along the boundary. Hence we can conclude that

X ◦ div(δ) = dαDα + (an eﬀective divisor in X ), α

where nα > 0. When C(G, H) is ﬁnite, then Pic(X)Q is generated by boundary components so that every ample divisor can be expressed as a linear combination P of Dα’s. This implies that α dαDα is big so div(δ) is also big.

From now on we consider the following situation: Let H ⊂ M ⊂ G be connected linear algebraic groups. Typical examples arise when G is a unipotent group or a product of absolutely simple groups. Let X be a smooth projective equivariant compactiﬁcation of H\G, and Y the induced compactiﬁcation of H\M.

Lemma 106. Let π : X → X0 be an equivariant morphism onto a projective equivariant compactiﬁcation of M\G. Assume that the projection G → M\G admits a rational section. Then

0 • π(Dα) = X if and only if Dα ∩ Y 6= ∅;

132 • if Dα ∩ Y 6= ∅ then Dα ∩ Y is irreducible;

0 • if Dα ∩ Y 6= ∅ and Dα0 ∩ Y 6= ∅, for α 6= α then Dα ∩ Y 6= Dα0 ∩ Y .

Proof. We have the diagram

H\G ⊂ X ⊃ Y

π π M\G ⊂ X0 ⊃ M · e = point

The ﬁrst claim is evident. To prove the second assertion, choose a rational section

σ : M\G 99K G of the projection G → M\G. We may assume that a rational section is well-deﬁned at a point M ∈ M\G. Consider the diagram

◦ Dα ⊃ Dα

π π X0 ⊃ M\G

◦ −1 where Dα = Dα ∩ π (M\G). We deﬁne a rational map

◦ Ψ: Dα 99K Dα ∩ Y x 7→ x · (σ ◦ π(x))−1.

This map is dominant which implies that Dα ∩ Y is irreducible.

◦ Since the G-orbit of Dα ∩ Y is Dα, the third claim follows.

Remark 107. When M is a connected solvable group, then G is birationally isomorphic to M ×(M\G) so that the projection G → M\G has a rational section. See [Bor91, Corollary 15.8].

133 Theorem 108. Let H ⊂ M ⊂ G be connected linear algebraic groups. Let X be a smooth projective equivariant compactiﬁcation of H\G and Y ⊂ X the induced compactiﬁcation of H\M. Let L be a big line bundle on X. Assume that

• the projection G → M\G admits a rational section;

• and D := a(X,L)L + KX is a rigid eﬀective Q-divisor.

Furthermore, we assume that either

1. Λeff (X) is finitely generated by effective divisors;

2. or there exists a birational contraction map f : X 99K Z contracting D where Z is a normal projective variety.

Then L is balanced with respect to Y .

Proof. Let X0 be any smooth projective equivariant compactiﬁcation of M\G. We

0 consider a G-rational map π : X 99K X mapping

π : G 3 g 7→ Mg ∈ M\G.

After applying a G-equivariant resolution of the indeterminacy of the projection π if necessary, we may assume that π is a surjective morphism and X is a smooth equivariant compactiﬁcation of H\G with a boundary divisor ∪αDα. Note that

Y is a general ﬁber of π so that Y is smooth. Write a rigid eﬀective Q-divisor

134 D = a(X,L)L + KX by

n X D = a(X,L)L + KX = eiEi, i=1 where Ei’s are irreducible components of a(X,L)L + KX and ei ∈ Q>0. Our goal is to show that

(a(Y,L|Y ), b(Y,L|Y )) < (a(X,L), b(X,L)).

Since Ei’s are rigid eﬀective divisors, they are parts of boundary components. This implies that

n X a(X,L)L|Y + KY = (a(X,L)L + KX )|Y = eiEi|Y ∈ Λeﬀ (Y ). i=1

It follows that

a(Y,L|Y ) ≤ a(X,L).

Assume that a(Y,L|Y ) = a(X,L) =: a. Let F be the minimal extremal face of P Λeﬀ (X) containing D = aL + KX = eiEi and VF a vector subspace generated by F . Either condition (1) or (2) guarantees that F is generated by Ei’s so that

b(X,L) = rk NS(X) − n.

(See Proposition 79 and Example 81.) Let F 0 be the minimal extremal face of

Λeﬀ (Y ) containing X D|Y = aL|Y + KY = eiEi|Y .

0 0 Let V be a vector subspace generated by all components of Ei ∩ Y . Since F

0 contains all components of Ei ∩ Y , we have b(Y,L|Y ) ≤ codim V . Consider the

135 restriction map:

0 Φ : NS(X)/VF → NS(Y )/V .

It follows from [KKV89, Proposition 3.2(i)], Lemma 106, and the exact sequence (1) in Proposition 104 that Φ is surjective. On the other hand, π∗NS(X0) is contained in the kernel of Φ, so Φ has the nontrivial kernel. We conclude that

0 b(Y,L|Y ) ≤ codim V < b(X,L).

Corollary 109. Let H ⊂ M ⊂ G be connected linear algebraic groups and X a smooth projective equivariant compactiﬁcation of H\G with the big anticanonical bundle. Let Y ⊂ X be the induced compactiﬁcation of H\M. Assume that the projection G → M\G admits a rational section. Then −KX is balanced with respect to Y .

Example 110. Let G = PGL2, M = B, a Borel subgroup of G and H = 1. Let

X = P3 be the standard equivariant compactiﬁcation of G given by

  a b   3 PGL2 3   7→ [a : b : c : d] ∈ P , c d with boundary D := {ad − bc = 0} = P1 × P1. Then Y = P2; with boundary

0 1 DY = Y \ B = `1 ∪ `2, a union of two intersecting lines. We have X = P . The projection

0 π : X 99K X

136 has indeterminacy along one of lines, say `1. Blowing up `1, we obtain a ﬁbration

1 π˜ : X˜ → P .

We have ˜ ˜ ˜ a(X, −KX˜ ) = a(Y, −KX˜ |Y˜ ) = a(Y, −KY˜ ) = 1.

Every boundary component of X˜ dominates the base P1, since the G-action is transitive on the base. Lemma 106 shows that the number of boundary components of Y˜ is equal to the number of boundary components of X˜, which equals the rank of NS(X˜) = 2. However, X(B)∗ = Z, and in particular, the rank of the Picard group of Y˜ is one less than the number of boundary components, i.e.,

˜ ˜ b(Y, −KX˜ |Y˜ ) = 1 < 2 = b(X, −KX˜ ).

6.3 Toric varieties

In this section, we illustrate how to use the Minimal Model Program to determine balanced line bundles. This section is extracted from [HTT12, Section 8]. We refer to [FS04] for details concerning toric Mori theory.

P Proposition 111. Let X be a Q-factorial projective toric variety. Let D = i Di be the complement of the big torus regarded as a reduced divisor and its irreducible decomposition. Then

1. KX + D ∼ 0;

P 2. (X, i aiDi) is klt (resp. log-canonical) where 0 ≤ ai < 1 (resp, 0 ≤ ai ≤ 1);

137 3. the cone of curves and the cone of pseudo-effective divisors are convex rational polyhedral cones and generated by finitely many effective cycles;

4. X is a log Fano variety.

Proof. See Remark 2.6, Proposition 2.10, and Theorem 4.1 in [FS04]. Finite generation of the cone of pseudo-eﬀective divisors was addressed in [BT95, Proposition 1.2.11]. The last statement follows from (1), (2), and (3).

Next the following two propositions are shared with all Mori dream spaces (see [HK00, Section 1]):

Proposition 112 (D-Minimal Model Program). Let X be a Q-factorial projective toric variety and D be a Q-divisor. Then the minimal model program with respect to D runs i.e.

1. for any extremal ray R of NE1(X), there exists the contraction morphism

ϕR;

2. for any small contraction ϕR of a D-negative extremal ray R, the D-ﬂip

+ ψ : X 99K X exists;

3. any sequence of D-ﬂips terminates in ﬁnite steps;

4. and every nef line bundle is semi-ample.

Proof. See Theorem 4.5, Theorem 4.8, Theorem 4.9, and Proposition 4.6 in [FS04].

Proposition 113 (Zariski decomposition). Let X be a Q-factorial projective toric variety and D a Q-eﬀective divisor. We apply D-MMP and obtain a birational

138 0 0 contraction map f : X 99K X and the proper transform D of D, which is nef. Consider a common resolution:

X˜ µ }} }} ν }} ~}} ___ / 0 X f X

Then we have

1. µ∗D = ν∗D0 + E where E is a ν-exceptional eﬀective Q-divisor;

2. the support of E contains all divisors contracted by f;

3. let g : X˜ → Y be the semi-ample ﬁbration associated to ν∗D0, then for any ν-exceptional eﬀective Cartier divisor E0, the natural map

0 OY → g∗O(E ),

is an isomorphism.

Proof. The assertions (1) and (2) follow from the Negativity lemma (see [FS04, Lemma 4.10]). Also see [FS04, Theorem 5.4].

The invariant b(X,L) can be characterized in terms of Zariski decomposition of a(X,L)L + KX :

Proposition 114. Let X be a Q-factorial projective toric variety and D an eﬀec- tive Q-divisor on X. Suppose that

1. D = P + N where P is a nef and N ≥ 0;

139 2. let g : X → Y be the semi-ample ﬁbration associated to P . For any eﬀective Cartier divisor E which is supported by Supp(N), the natural map

OY → g∗O(E),

is an isomorphism.

Then the minimal extremal face of Λeﬀ (X) containing D is generated by vertical divisors of g and components of N.

Proof. When D is big, then the assertion is trivial. We may assume that dim Y < dim X. Let F be the minimal extremal face of Λeﬀ (X) containing D. Since F is extremal, it follows that F contains all vertical divisors of g and components of N.

On the other hand, our assumption implies that for general ﬁber Xy, N|Xy is a rigid divisor on Xy, and its irreducible components generate an extremal face

0 0 F of Λeﬀ (Xy). Let α ∈ NM1(Xy) be a nef cycle supporting F and we consider

Fα = {α = 0} ∩ Λeﬀ (X). Since Xy is a general ﬁber, α ∈ NM1(X) so that Fα is an

0 extremal face. D · α = 0 and the minimality of F imply that F ⊂ Fα. Let D ∈ F be an eﬀective Q-divisor. D0 · α = 0 implies that D0 is a sum of vertical divisors of g and components of N. Thus our assertion follows.

Proposition 115. Let X be a projective toric variety and Y an equivariant com- pactiﬁcation of a subtorus of codimension one (possibly singular). Let L be a big line bundle on X. Then L is weakly balanced with respect to Y .

Proof. Let M be the class of OX (Y ). By applying an equivariant embedded resolution of singularities if necessary, we may assume that X and Y are smooth or at

140 least Q-factorial terminal. Due to a group action of a torus, Y is not rigid so that

a(X,L)L|Y + KY = (a(X,L)L + KX )|Y + M|Y ∈ Λeﬀ (Y ).

Note that a(X,L)L + KX is an eﬀective Q-divisor on X. Thus we have

a(Y,L|Y ) ≤ a(X,L).

Suppose that a(Y,L|Y ) = a(X,L) =: a. Let D = aL + KX + Y and we consider the Zariski decomposition of D:

X˜ ⊃ Y˜ µ ~~ ~~ ν ~~ ~ ~ ___ / 0 0 D ⊂X f X ⊃ D where D0 is the strict transformation of D, which is nef, and Y˜ is the strict transformation of Y . We may assume that X˜ and Y˜ are both smooth. Let F be the ˜ minimal extremal face of Λeﬀ (X) containing

∗ ˜ aµ L + KX˜ + Y.

∗ Since aµ L + KX˜ ∈ F , it follows that codim F ≤ b(X,L). Note that since X has only terminal singularities, we have

∗ ˜ ∗ ∗ X ˜ aµ L + KX˜ + Y = aµ L + µ KX + diEi + Y i where di’s are positive integers and Ei’s are µ-exceptional divisors. Thus it follows that F is the minimal extremal face containing µ∗D and all µ-exceptional divisors.

141 Let g : X˜ → B be the semi-ample ﬁbration associated to ν∗D0. Note that dim B < dim X˜ since D is not big. Proposition 114 implies that F is generated by all vertical divisors of g and all ν-exceptional divisors. We denote the vector space, generated by F , by VF . 0 ˜ ∗ Let F be the minimal extremal face of Λeﬀ (Y ) containing aµ L|Y˜ + KY˜ = ∗ ˜ ˜ ˜ 0 (aµ L + KX˜ + M)|Y˜ where M be the class of OX˜ (Y ). Then F is also the minimal ∗ ˜ 0 extremal face containing µ D|Y˜ and all components of (Ei ∩ Y )’s so that F is the ∗ 0 ˜ minimal extremal face containing ν D |Y˜ and all components of (Gj ∩ Y )’s where

0 Gj’s are all ν-exceptional divisors. In particular F contains all vertical divisors ˜ ˜ ∗ 0 ˜ of g|Y˜ : Y → H = g(Y ). Since ν D admits a section vanishing along Y , H is a Weil divisor of B, which is a subtoric variety. Let V 0 ⊂ NS(Y˜ ) be a vector ˜ space generated by vertical divisors of g|Y˜ and components of (Gj ∩ Y )’s. Then b(Y,L) ≤ codim V 0. Consider the following restriction map:

˜ ˜ 0 Φ : NS(X)/VF → NS(Y )/V .

We claim that Φ is surjective. Let N be an irreducible component of the boundary divisor of Y˜ which dominates H. There exists an irreducible component N 0 of the boundary divisor of X˜ such that N 0 contains N. Then N 0 also dominates B. As in the proof of Lemma 106 one can prove that

0 ˜ N ∩ Y = mN + (vertical divisors of g|Y˜ ).

Our claim follows from this. Hence we conclude that

0 b(Y,L) ≤ codim V ≤ codim VF ≤ b(X,L).

142 Proposition 116. Let X be a Q-factorial terminal projective toric variety and L a big line bundle on X. Suppose that the positive part of Zariski decomposition of

D := a(X,L)L + KX is nontrivial. Then L is not balanced.

Proof. After applying blowing up if necessary, we may assume that D itself admits the Zariski decomposition i.e., D = P + N where P is a nef Q-divisor and N ≥ 0 is the negative part. Let g : X → Y be the semi-ample ﬁbration associated to P . We consider a general ﬁber Xy of

g. Since a(X,L)L|Xy + KXy = N|Xy is a rigid eﬀective divisor, we conclude that a(X,L) = a(Xy,L|Xy ). Let V ⊂ NS(X) be a vector space generated by vertical

0 divisors of g and components of N and V ⊂ NS(Xy) a vector space generated by

components of N|Xy . We consider the following restriction map:

0 Φ : NS(X)/V → NS(Xy)/V .

Lemma 106 implies that Φ is surjective. On the other hand, let T be the big torus

−1 of Y . Then the preimage g (T ) of T is a product of T and a general ﬁber Xy. It follows that Φ is injective. Thus we have

b(X,L) = b(Xy,L|Xy ).

Hence L is not balanced on X.

Alternative proof of Theorem 108 for toric varieties is provided below:

143 Proposition 117. Let X be a Q-factorial terminal projective toric variety, L a big line bundle on X, and Y an equivariant compactiﬁcation of a subtorus of codimension one (possibly singular). Suppose that the positive part of Zariski decomposition of a(X,L)L + KX is trivial. Then L is balanced with respect to Y .

Proof. We follow the notations in the proof of Proposition 115. Only part where we need to explain is why b(Y,L|Y ) < b(X,L) holds when a(Y,L|Y ) = a(X,L).

∗ Since aµ L + KX˜ is rigid, it follows that

codim VF < b(X,L).

Thus our assertion follows.

Corollary 118. Let X be a Q-factorial terminal projective toric variety. A big line bundle L is balanced with respect to all subtoric varieties if and only if a(X,L)L +

KX is rigid.

Remark 119. Propositions 115 and 117 hold when X is Q-factorial terminal and Y a general smooth divisor. The proofs work with small modiﬁcations. However we still do not know whether they hold for singular divisors.

3 3 Example 120. Consider the standard action of Gm = {(t0, t1, t2)} on P by

(t0, t1, t2) · (x0 : x1 : x2 : x3) 7→ (t0x0 : t1x1 : t2x2 : x3).

Consider the subtorus

−1 3 M = {(t0, t1, (t0t1) )} ⊂ Gm,

144 and let S be the equivariant compactiﬁcation of M deﬁned by

3 x0x1x2 = x3.

This is a singular cubic surface with three isolated singularities of type A2. We

3 3 3 denote them by p1, p2, p3 ∈ P . Since they are ﬁxed under the action of Gm on P ,

3 3 the blowup B := Blp1,p2,p3 (P ) is an equivariant compactiﬁcation of Gm. Moreover, the closure S˜ of M in B is the minimal desingularization of S and the class of S˜ in Pic(B) is ample. Put X := B × P1 and Y := S˜ × P1. We have a diagram

X o ? _Y

π B o ? _S,˜

Then Y is a nef divisor, and we have

rk NS(Y ) = 8 > rk NS(X) = 5.

However the anticanonical class −KX is still balanced with respect to Y since

a(Y, −KX |Y ) = a(X, −KX ) = 1

b(Y, −KX |Y ) = 1 < b(X, −KX ) = 5.

This shows that, in general, we cannot expect to control the subgroup of NS(X) generated by vertical divisors. In the proof of Proposition 115, we were able to control the quotient by this subgroup.

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