Introduction Structure near a fixed point Toric structure ......

. . Complex torus manifolds . .. . .

Hiroaki Ishida

Osaka City University Advanced Mathematical Institute

November 28, 2011 Introduction Structure near a fixed point Toric structure ...... Torus manifold

Let us consider M : connected smooth manifold, (S1)n y M : effective, 1 n M(S ) , ∅. ⇒ ≤ 1 n 2 dim M (see the tangential rep. of (S1)n at a fixed point).

1 We consider the extreme case (n = 2 dim M). Introduction Structure near a fixed point Toric structure ...... Torus manifold

. Definition (Hattori-Masuda) . .. M : torus manifold of dim 2n ⇐⇒ def. M : closed connected oriented manifold of dim 2n, (S1)n y M : effective, 1 n . M(S ) , ∅. .. . . Introduction Structure near a fixed point Toric structure ...... Complex torus manifold

We consider the case when M is a complex mfd. . . ..Definition A complex torus manifold M of dimC n is

M : closed conn. complex manifold of dimC n, (S1)n y M : effective, as biholomorphisms, 1 n . M(S ) , ∅. .. . . Introduction Structure near a fixed point Toric structure ...... Simple examples of torus manifolds . Example . .. S2n : 2n-dimensional sphere ∑ 2n n 2 2 S = {(y, z1,..., zn) ∈ R × C | y + |zi| = 1}

(S1)n y S2n given by

,..., · , ,..., , ,..., . (g1 gn) (y z1 zn) := (y g1 gnzn) .. . . . Example (Complex) . .. (S1)n y CPn given by

,..., · ,..., , ,..., . (g1 gn) [z0 zn] := [z0 g1z1 gnzn] .. . . Introduction Structure near a fixed point Toric structure ...... In Khabarovsk Combining BIG RESULTS +ε, Donaldson Kodaira Masuda-Panov Orlik-Raymond . Theorem (I.-Masuda) . .. M : closed connected complex mfd of dimC n, (S1)n y M : effectively, as biholomorphisms, Hodd(M) = 0. =⇒ .T(M) = 1. .. . . Introduction Structure near a fixed point Toric structure ...... Purpose . Main Theorem (I.-Karshon) . .. A complex torus manifold is equivariantly biholomorphic. to a toric manifold. .. . . In other words, . Corollary . .. M : closed connected complex mfd of dimC n, (S1)n y M : effectively, as biholomorphisms.

1 n M(S ) , ∅. ⇐⇒ M is equivariantly biholomorphic to a toric manifold...... Introduction Structure near a fixed point Toric structure ...... Purpose . . ..Definition Y : toric variety of dimC n ⇐⇒ def. Y is a normal algebraic variety over C, there is an embedding of (C∗)n as a Zariski open subset such that..... Y is called a toric manifold if Y is complete and non-singular...... (C∗)n × Y −→ Y action ⊂ ⊂ (C∗)n × (C∗)n −→ (C∗)n binary operation Introduction Structure near a fixed point Toric structure ...... Purpose

. Main Theorem (I.-Karshon) . .. A complex torus manifold M is equivariantly biholomorphic. to a toric manifold. .. . . For our M, local structure near a fixed point how to find (C∗)n-action how to find the toric structure on M Introduction Structure near a fixed point Toric structure ...... Invariant neighborhood at a fixed point

. Proposition . .. M : smooth manifold G : compact Lie group, G y M smoothly Then, ∀p ∈ MG, ∃ Up ∋ p : G-inv. open subset of M, ∃ Dp ∋ 0 : G-inv. open subset of TpM

such that Up  Dp. . eq. diff .. . . Introduction Structure near a fixed point Toric structure ...... Invariant neighborhood at a fixed point

. Proposition . .. M: complex manifold G : compact Lie group, G y M as biholo. Then, ∀p ∈ MG, ∃ Up ∋ p : G-inv. open subset of M, ∃ Dp ∋ 0 : G-inv. open subset of TpM

such that Up  Dp. . eq. biholo .. . . Introduction Structure near a fixed point Toric structure ...... Extension of the action of the torus

From now, we assume

M : closed connected complex mfd of dimC n (S1)n y M : effectively and as biholo. 1 n M(S ) , ∅. We know that

∀ 1 n p ∈ M(S ) , (S1)n y M is standard near p. Introduction Structure near a fixed point Toric structure ...... Extension of the action of the torus

1 n ξ1, . . . , ξn : fundamental vector fields of (S ) J : complex structure on M

Then, −Jξ1,..., −Jξn, ξ1, . . . , ξn satisfy holomorphic, commutative, R-linearly independent, complete. Introduction Structure near a fixed point Toric structure ...... Extension of the action of the torus

−Jξ1,..., −Jξn, ξ1, . . . , ξn allow us to define

Cn y M holomorphic

whose global stabilizer is discrete subgroup of rank n. =⇒ Cn y M induces

∗ (C )n y M effective, holomorphic. Introduction Structure near a fixed point Toric structure ...... Structure near a fixed point

1 n For p ∈ M(S ) ,

∃ 1 n φp : Up → Dp ⊂ TpM (S ) -eq. biholo.

For g ∈ (C∗)n,

φ(g) ◦ φ ◦ −1 → p := g p g : gUp gDp

is a biholomorphism. . . ..Claim φ φ(g) ∩ . p = p on gUp Up. .. . . Introduction Structure near a fixed point Toric structure ...... Structure near a fixed point

Identity theorem tells us that ∪ Vp := gUp g∈(C∗)n

∗ n is (C ) -eq. biholo. to TpM. Introduction Structure near a fixed point Toric structure ......

M : closed connected complex mfd of dimC n (S1)n y M : effectively and as biholo. 1 n M(S ) , ∅. . . ..What we have (S1)n y M extends to (C∗)n y M. ∀ (S1)n (C∗)n ∃ p ∈ M = M , Vp ⊂ M s.t.

p ∈ Vp ⊂ M  C∗ n . Vp TpM ( ) -eq. biholo. .. . . We set ′ ∪ ∗ ∗ n C n X := p∈M(C ) Vp ( ) -inv. submfd. of M X : a connected component of X ′. Introduction Structure near a fixed point Toric structure ...... lemmas

X : (C∗)n-inv. connected open submfd of M, (C∗)n y X locally looks like a rep. of (C∗)n.

We can see the followings : . . ..Lemma .X is a non-singular toric variety...... Lemma .X is compact. Therefore, X = M. .. . . Introduction Structure near a fixed point Toric structure ...... Summary

. Theorem (I.-Karshon) . .. M : closed connected complex mfd of dimC n, (S1)n y M : effectively, as biholomorphisms. 1 n M(S ) , ∅. ⇐⇒ M is equivariantly biholomorphic to a toric manifold......