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 X 2n n 2 2 S = f(y; z1;:::; zn) 2 R × C j y + jzij = 1g (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 +", 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, 8p 2 MG, 9 Up 3 p : G-inv. open subset of M, 9 Dp 3 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, 8p 2 MG, 9 Up 3 p : G-inv. open subset of M, 9 Dp 3 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 8 1 n p 2 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 2 M(S ) , 9 1 n 'p : Up ! Dp ⊂ TpM (S ) -eq. biholo. For g 2 (C∗)n, '(g) ◦ ' ◦ −1 ! p := g p g : gUp gDp is a biholomorphism. ..Claim ' '(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 g2(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. 8 (S1)n (C∗)n 9 p 2 M = M , Vp ⊂ M s.t. p 2 Vp ⊂ M C∗ n . Vp TpM ( ) -eq. biholo. .. We set 0 S ∗ ∗ n C n X := p2M(C ) Vp ( ) -inv. submfd. of M X : a connected component of X 0. 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. .. ..