Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks K¨ahler-Einstein metrics and K-stability Chi Li Department of Mathematics, Princeton University May 3, 2012 Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Table Of Contents 1 Preliminaries 2 Continuity method 3 Review of Tian’s program 4 Special degeneration and K-stability 5 Thanks Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Basic K¨ahler geometry (X, J, g) ←→ (X, J, ωg) g(·, ·) = ωg(·,J·) ←→ ωg(·, ·) = g(J·, ·) LC ∇ J = 0 ⇐⇒ dωg = 0 Locally, √ X i j ωg = −1 gi¯jdz ∧ dz¯ , g = (gi¯j) > 0 i,j Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Curvature form Riemannian curvature 2 ∂ gk¯l rq¯∂gkq¯ ∂gr¯l Ri¯jk¯l = − + g ∂zi∂z¯j ∂zi ∂z¯j Ricci curvauture: √ √ 2 −1 n −1 X ∂ i j Ric(ω) = − log ω = − log det(g ¯)dz ∧dz¯ 2π 2π ∂zi∂z¯j kl i,j Chern-Weil theory implies: Ric(ω) ∈ c1(X) Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Scalar curvature Scalar curvature n−1 i¯j nRic(ω) ∧ ω S(ω) = g Ric(ω) ¯ = ij ωn Average of Scalar curvature is a topological constant: hnc (X)[ω]n−1, [X]i S = 1 h[ω]n, [X]i Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks K¨ahler-Einstein equation Ric(ωKE) = λωKE Generalization of uniformization theorem of Riemann surfaces. 1 λ = −1: −c1(X) > 0, canonically polarized. Existence: Aubin, Yau 2 λ = 0: c1(X) = 0, Calabi-Yau manifold. Existence: Yau 3 λ > 0: c1(X) > 0, Fano manifold. There are obstructions. Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks K¨ahler-Einstein metric on Fano manifold Obstructions: 1 Automorphism group is reductive (Matsushima’57) 2 Futaki invariant vanishes (Futaki’83) 3 K-stability (Tian’97, Donaldson’02) Existence result when Futaki invariant vanishes: 1 Homogeneous Fano manifold 2 Del Pezzo surface: Tian (1990) 3 toric Fano manifold: Wang-Zhu (2000) Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Complex Monge-Amp`ereequation PDE of K¨ahler-Einstein metric: √ (ω + −1∂∂φ¯ )n = ehω−φωn Ricci potential: Z Z hω n n Ric(ω) − ω = ∂∂h¯ ω, e ω = ω X X Space of K¨ahler metrics in [ω]: √ ∞ ¯ H(ω) = {φ ∈ C (X); ωφ := ω + −1∂∂φ > 0} Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Aubin-Yau Continuity Method and invariant R(X) For 0 ≤ t ≤ 1, √ n hω−tφ n (ω + −1∂∂φ¯ t) = e ω This equivalent to Ric(ωφt ) = tωφt + (1 − t)ω (∗)t Define R(X) = sup{t :(∗)t is solvable } Tian(’92) studied this invariant first. He estimated 2 15 R(BlpP ) ≤ 16 . Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Other previous results 1 Sz´ekelyhidi(’09) showed: R(X) is the same as sup{t :∃ a K¨ahler metric ω ∈c1(X) such that Ric(ω) > tω} In particular, R(X) is independent of reference metric ω. 2 6 2 21 R(BlpP ) = 7 . R(Blp,qP ) ≤ 25 2 Shi-Zhu(’10) studied the limit behavior of solutions of 2 continuity method on BlpP . Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Toric Fano manifolds { toric Fano manifold X4} ←→ { reflexive lattice polytope 4} 2 Example: BlpP . Pc: Center of mass @ @ @ @ O Pc@ @ @ @ qq @ Q @ q @ @ Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks R(X) on any toric Fano manifold Theorem (Li’09) If Pc 6= O, |OQ| R(X4) = |PcQ| Here |OQ|, |PcQ| are lengths of line segments OQ and PcQ. If Pc = O, then there is K¨ahler-Einstein metric on X4 and R(X4) = 1. Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks limit as t → R(X) Theorem (Li’11) There exist biholomorphic transformations σt : X4 → X4, such ∗ ¯ that σti ωti converge to a K¨ahler current ω∞ = ω + ∂∂ψ∞, which satisfy a complex Monge-Amp`ereequation of the form !−(1−R(X)) n −R(X)ψ∞ X 0 2 (ω + ∂∂ψ¯ ∞) = e bαksαk Ω α In particular, ¯ X 0 2 Ric(ωψ∞ ) = R(X)ωψ∞ + (1 − R(X))∂∂ log( bα|sα| ) α Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Corollaries 1 Conic type singularity for the limit metric (compatible with Cheeger-Colding’s theory) 0 2 Partial C -estimate along continuity method (∗)t for toric metrics. (Follows from convergence up to gauge transformation) 3 Calculate multiplier ideal sheaf for toric Fano manifold with large symmetries. Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Singularity from the polytope The nonzero terms in P0 can be determined by the geometry of the polytope! 2 21 Example: R(Blp,qP ) = 25 Q5 D3 Q1 @ 21 q qD@4 − 4 Pc D @ Q 2 r @ 2 Pc r qD5 Q4 Q3 D1 q q Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Idea of the proof Aubin-Yau Continuity method: √ n hω−tφ n (ω + −1∂∂φ¯ t) = e ω 0 Solvability for t ∈ [0, t0] ⇐⇒ uniform C -estimate on φt for t ∈ [0, t0]. Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks toric invariant K¨ahler metrics ∼ ∗ n n 1 n X4\D = (C ) = R × (S ) 2 zi zi = (log |zi| , ) |zi| (S1)n-invariant K¨ahler metric ←→ convex function u = u(x) on n R . Potential of reference ω = ωFS: ! X 2 X hpα,xi u˜ = log |sα| = log e . α α ¯ Assume potential of ωφ = ω + ∂∂φ is a convex function u. n u˜, u are proper convex functions on R satisfying n n n Du˜(R ) = Du(R ) = Dw(R ) = 4 Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Real Monge-Amp`ereby torus symmetry −(1−t)˜u−tu det(uij) = e (∗∗)t u =u ˜ + φ Combined potential: wt(x) = tu(x) + (1 − t)˜u Define n mt = inf{wt(x): x ∈ R } = wt(xt) Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Basic estimates Proposition (Wang-Zhu) 1 there exists a constant C, independent of t < R(X4), such that |mt| < C 2 There exists κ > 0 and a constant C, both independent of t < R(X4), such that wt ≥ κ|x − xt| − C (1) Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Equivalent criterion for C0 estimate Proposition (Wang-Zhu) 0 Uniform bound of |xt| for any 0 ≤ t ≤ t0 ⇐⇒ C -estimates for the solution φt in (∗)t for t ∈ [0, t0]. Corollary 1 If R(X4) < 1, then limti→R(X4) |xti | = +∞. 2 If R(X4) < 1, then there exists y∞ ∈ ∂4, such that lim Du˜0(xti ) = y∞ (2) ti→R(X4) Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks Key identity Z 1 −w t Du˜0e dx = − Pc (3) V ol(4) n 1 − t R Remark General formula: for any holomorphic vector field v, Z n t −1 − divΩ(v)ωt = F (KX , v) X 1 − t Chi Li Department of Mathematics, Princeton University K¨ahler-Einstein metrics and K-stability Table Of Contents Preliminaries Continuity method Review of Tian’s program Special degeneration and K-stability Thanks key observation Assume the reflexive polytope 4 is defined by hλr, xi ≥ −1, r = 1,...,N Proposition (Li) If Pc 6= O, R(X4) Q := − Pc ∈ ∂4 1 − R(X4) Precisely, λr(Q) ≥ −1 Equality holds if and only if λ(y∞) = −1.