Kähler-Einstein metrics and K-stability

Transcription

1 May 3, 2012

2 Table Of Contents 1 Preliminaries 2 Continuity method 3 Review of Tian s program 4 Special degeneration and K-stability 5 Thanks

3 Basic Kähler geometry (X, J, g) (X, J, ω g ) g(, ) = ω g (, J ) ω g (, ) = g(j, ) LC J = 0 dω g = 0 Locally, ω g = 1 i,j g i jdz i d z j, g = (g i j) > 0

4 Curvature form Riemannian curvature Ricci curvauture: 1 Ric(ω) = 2π Chern-Weil theory implies: R i jk l = 2 g k l z i z j + g r q g k q z i g r l z j 1 log ωn = 2π i,j Ric(ω) c 1 (X) 2 z i z j log det(g k l )dzi d z j

5 Scalar curvature Scalar curvature S(ω) = g i j Ric(ω) i j = nric(ω) ωn 1 ω n Average of Scalar curvature is a topological constant: S = nc 1(X)[ω] n 1, [X] [ω] n, [X]

6 Kähler-Einstein equation Ric(ω KE ) = λω KE Generalization of uniformization theorem of Riemann surfaces. 1 λ = 1: c 1 (X) > 0, canonically polarized. Existence: Aubin, Yau 2 λ = 0: c 1 (X) = 0, Calabi-Yau manifold. Existence: Yau 3 λ > 0: c 1 (X) > 0, Fano manifold. There are obstructions.

7 Kähler-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)

8 Complex Monge-Ampère equation PDE of Kähler-Einstein metric: (ω + 1 φ) n = e hω φ ω n Ricci potential: Ric(ω) ω = h ω, e hω ω n = X ω n X Space of Kähler metrics in [ω]: H(ω) = {φ C (X); ω φ := ω + 1 φ > 0}

9 Aubin-Yau Continuity Method and invariant R(X) For 0 t 1, This equivalent to (ω + 1 φ t ) n = e hω tφ ω n Ric(ω φt ) = tω φt + (1 t)ω ( ) t Define R(X) = sup{t : ( ) t is solvable } Tian( 92) studied this invariant first. He estimated R(Bl p P 2 )

10 Other previous results 1 Székelyhidi( 09) showed: R(X) is the same as sup{t : a Kähler metric ω c 1 (X) such that Ric(ω) > tω} In particular, R(X) is independent of reference metric ω. R(Bl p P 2 ) = 6 7. R(Bl p,qp 2 ) Shi-Zhu( 10) studied the limit behavior of solutions of continuity method on Bl p P 2.

11 Toric Fano manifolds { toric Fano manifold X } { reflexive lattice polytope } Example: Bl p P 2. P c : Center of mass O P c Q

12 R(X) on any toric Fano manifold Theorem (Li 09) If P c O, R(X ) = OQ P c Q Here OQ, P c Q are lengths of line segments OQ and P c Q. If P c = O, then there is Kähler-Einstein metric on X and R(X ) = 1.

13 limit as t R(X) Theorem (Li 11) There exist biholomorphic transformations σ t : X X, such that σt i ω ti converge to a Kähler current ω = ω + ψ, which satisfy a complex Monge-Ampère equation of the form (ω + ψ ) n = e R(X)ψ ( α b α s α 2 ) (1 R(X)) Ω In particular, Ric(ω ψ ) = R(X)ω ψ + (1 R(X)) log( α b α s α 2 )

14 Corollaries 1 Conic type singularity for the limit metric (compatible with Cheeger-Colding s theory) 2 Partial C 0 -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.

15 Singularity from the polytope The nonzero terms in can be determined by the geometry of the polytope! Example: R(Bl p,q P 2 ) = Q 5 D 3 Q 1 D P c D 2 Q 2 P c D 5 Q 4 D Q 3 1

16 Idea of the proof Aubin-Yau Continuity method: (ω + 1 φ t ) n = e hω tφ ω n Solvability for t [0, t 0 ] uniform C 0 -estimate on φ t for t [0, t 0 ].

17 toric invariant Kähler metrics X \D = (C ) n = R n (S 1 ) n z i = (log z i 2, z i z i ) (S 1 ) n -invariant Kähler metric convex function u = u(x) on R n. Potential of reference ω = ω F S : ũ = log ( ) s α 2 = log e pα,x. α α Assume potential of ω φ = ω + φ is a convex function u. ũ, u are proper convex functions on R n satisfying Dũ(R n ) = Du(R n ) = Dw(R n ) =

18 Real Monge-Ampère by torus symmetry det(u ij ) = e (1 t)ũ tu ( ) t u = ũ + φ Combined potential: w t (x) = tu(x) + (1 t)ũ Define m t = inf{w t (x) : x R n } = w t (x t )

19 Basic estimates Proposition (Wang-Zhu) 1 there exists a constant C, independent of t < R(X ), such that m t < C 2 There exists κ > 0 and a constant C, both independent of t < R(X ), such that w t κ x x t C (1)

20 Equivalent criterion for C 0 estimate Proposition (Wang-Zhu) Uniform bound of x t for any 0 t t 0 C 0 -estimates for the solution φ t in ( ) t for t [0, t 0 ]. Corollary 1 If R(X ) < 1, then lim ti R(X ) x ti = +. 2 If R(X ) < 1, then there exists y, such that lim Dũ 0(x ti ) = y (2) t i R(X )

21 Key identity Remark 1 Dũ 0 e w dx = t V ol( ) R n 1 t P c (3) General formula: for any holomorphic vector field v, div Ω (v)ωt n = t 1 t F (K 1 X, v) X

22 key observation Assume the reflexive polytope is defined by λ r, x 1, r = 1,..., N Proposition (Li) If P c O, Precisely, Q := R(X ) 1 R(X ) P c λ r (Q) 1 Equality holds if and only if λ(y ) = 1. So Q and y lie on the same faces.

23 Transformation σ t : X X in toric coordinate: In complex coordinate: σ t : x x + x t σ t : z i e xi t /2 z i Transformation on the potential: U(x) = σt u(x) u(x t ) = u(x + x t ) u(x t ) Ũ t (x) = σt ũ 0 (x) ũ 0 (x t ) = ũ 0 (x + x t ) ũ 0 (x t )

24 Regularization of singular Monge-Ampère det(u ij ) = e tu (1 t)ũ w(xt) (ω + ψ) n = e tψ ( α b(p α, t) s α 2 ) (1 t) e hω w(xt) ω n ( ) t det(u ij ) = e tu (1 t)ũ w(xt) (ω + ψ) n = e R(X)ψ ( α b α s α 2 ) (1 R(X)) e hω c ω n ( )

25 Uniform a priori estimate 1 C 0 -estimate. The point is that: the gauge transformation σ t offset the blow up of φ so that the transformed potential ψ has uniformly bounded ψ. Proof: Prove Harnack estimate: sup( ψ) n sup ψ + C(n)t 1. X X 2 Partial C 2 -estimate: Uniform C 2 -estimate on compact set away from the blow up set. Proof: Adapt Yau s proof of C 2 -estimate. 3 C 2,α -estimate: Evans-Krylov estimate. This estimate is purely local.

26 My work on other continuity methods I (Conic Kähler-Einstein metric) Fix a smooth divisor D K X, construct Kähler-Einstein metric with conic singularity along the divisor D of cone angle 2πβ. This corresponds to solving the following equation: Ric(ω KE,β ) = βω KE,β + (1 β){d} My work in this direction: 1 Construction of Kähler-Einstein metric with cone angle in (π, 2π). (Joint with Yanir Rubinstein). 2 Formulate log-k-stability (obstruction to the existence).

27 My work on other continuity methods II (Kähler-Ricci flow) My work: Construction of rotationally symmetric Kähler-Ricci solitons. (Complex structure continuity method) My work: still need to be done.

28 Obstruction to existence: Calabi-Futaki invariant For any holomorphic vector field v, define θ v = L v v on K 1 X = n T X, s.t. 1 θ v = i v ω. F (K 1 X, v) = ω n (S(ω) S)θ v n! X Theorem (Futaki) F (K 1 X, v) does not depend on the choice of ω in c 1(X).

29 Mabuchi energy and Ding-energy Definition 1 (Mabuchi-energy) ν ω (ω φ ) = 1 2 (Ding-energy) ( 1 F ω (φ) = Fω(φ) 0 V log V 0 dt (S(ω φt ) S) φ t ωφ n /n! X X ) e hω φ ω n /n!

30 Tian s analytic criterion Theorem (Tian 97) If Aut(X, J) is discrete. There exists a Kähler-Einstein metric on X if and only if either F ω (ω φ ) or ν ω (ω φ ) is proper. Definition [Tian 97] A functional F : H(ω) R proper: F (ω φ ) f (J ω (ω φ )), for any ω φ H where f(t) : R + R is some monotone increasing function satisfying lim t + f(t) = +. J ω (ω φ ) is a positive energy norm defined on the space H ω.

31 Finite dimensional approximation I 1 H k = { inner products on the vector space H 0 (X, L k )} = GL(N k, C)/U(N k, C) 2 B k := { Nk 1 k log α=1 s (k) α 2 h ; {s (k) k α } is a basis of H 0 (X, L k ) } Note that B k H ω = {φ C (X); ω + 1 φ > 0}.

32 Finite dimensional approximation II Define two maps between H k and H. Hilb k : H H k h s 1, s 2 Hilbk (h) = FS k : H k B k H H k s 2 FS k (H k ) = s 2 ( Nk X (s 1, s 2 ) h k ω n h n!, α=1 s (k) α 2), s L. 1/k In the above definition, {s (k) α ; 1 α N k } is an orthonormal basis of the Hermitian complex vector space (H 0 (X, L k ), H k ).

33 Finite Approximation III Theorem (Tian) H = k B k More precisely, for any h H, In terms of Kähler form: F S k (Hilb k (h)) h 1 N k 1 log k α=1 s (k) α 2 ω h

34 One application: Quantization of Mabuchi-energy Theorem (Donaldson 05) 1 The k-th Chow-norm functionals approximate Mabuchi-energy as k +. 2 If Aut(X, L) is discrete, then constant scalar curvature Kähler metric in c 1 (L) obtains the minimum of Mabuchi-energy. Theorem (Li10) Without assuming Aut(X, L) is discrete, CSCK metric in c 1 (L) always obtains the minimum of Mabuchi-energy. Remark The more general case is proved by Chen-Tian( 08).

35 Tian s conjecture and partial C 0 -estimate Conjecture (Tian) There is a Kähler-Einstein metric on X if and only if for sufficiently large k, ν ω is proper on B k. This will follow from the following conjecture by Tian. Conjecture (Partial C 0 -estimate) If ω t = ω ht are solutions in a continuity method to KE problem, then when k 1, C k > 0 independent of parameter t, such that N k ρ k (ω t ) = α=1 s (k) α 2 h t C > 0. here {s (k) α } is orthonormal basis of (H 0 (X, K k X ), Hilb k(h t )).

36 Properness on B k : K-stability Assume these conjectures. Then to see whether there is a Kähler-Einstein metric, we need to test whether the Mabuchi energy is proper on B k. Tian ( 97) introduced algebraic condition: K-stability for testing such properness. Donaldson ( 02) reformulated it for the more general setting. Combine the above discussion, we have the following conjecture due to Tian, which is also a specialization of Yau-Tian-Donaldson conjecture to the Fano case: Conjecture (Tian) There exists a Kähler-Einstein metric on Fano manifold (X, J) if and only if (X, K X ) is K-polystable.

37 Test configuration A test configuration is a direction in B k. Definition Let X be a Fano manifold. A Q-test configuration (X, L) of (X, K k X ) consists of a variety X with a C -action, a C -equivariant ample line bundle L X, a flat C -equivariant map π : (X, L) C, where C acts on C by multiplication in the standard way (t, a) ta, such that for any t 0, (X t = π 1 (t), L Xt ) is isomorphic to (X, K k X ).

38 Algebraic definition of Futaki invariant d k = dim H 0 (X, O X ( rkk X )) = a 0 k n + a 1 k n 1 + O(k n 2 ) C -weight of H 0 (X, O X ( rkk X )): w k = b 0 k n+1 + b 1 k n + O(k n 1 ). Definition DF(X, L) = F 1 a 0 = a 1b 0 a 0 b 1 a 2 0 (4) This is asymptotic slope in the direction determined by (X, L).

39 K-stability Definition (X, L) is K-semistable along (X, L) if F (X, L) 0. Otherwise, it s unstable. (X, L) is K-polystable for any test configuration (X, L) if F (X, L) > 0, or F (X, L) = 0 and the normalization (X ν, L ν ) is a product test configuration. (X, L) is K-semistable (resp. K-polystable) if, for any integer k > 0, (X, L k ) is K-semistable (K-polystable) along any test configuration of (X, L k ).

40 Tian s original definition using special degeneration Definition (Special Degeneration) Special degeneration is a test configuration (X, L) of a Fano manifold (X, K k X ) such that X 0 is a Q-Fano variety and L = K k X /C. Definition (Tian 97) (X, K X ) is K-semistable (resp. K-polystable) if (X, K k X ) is K-semistable (resp. K-polystable) along any special degeneration.

41 Apply MMP to the problem of K-stability The following result is from joint work with Dr. Chenyang Xu. The result roughly says: Theorem (Li-Xu) For any test configuration, we can modify it using Minimal Model Program to get a special test configuration with smaller Donaldson-Futaki invariant.

42 K-stability and special test configuration Theorem (Li-Xu) For any test configuration (X, L) C 1, we can construct a special test configuration (X s, rk X s) and a positive integer m, such that mdf(x, L) DF(X s, rk X s). Furthermore, if we assume X is normal, then the equality holds only when (X, X 0 ) itself is a special test configuration.

43 Tian s conjecture on K-stability of Fano manifolds As a corollary of the above construction, we prove Theorem (Li-Xu) If X is destablized by a test configuration, then X is indeed destablized by a special test configuration. More precisely, the following statements true. 1 (unstable) If (X, kk X ) is not K-semi-stable, then there exists a special test configuration (X s, kk X s) with a negative Futaki invariant DF(X s, kk X s) < 0. 2 (semistable\stable) Let X be a K-semistable variety. If (X, kk X ) is not K-polystable, then there exists a special test configuration (X st, kk X s) with Donaldson-Futaki invariant 0 such that X s is not isomorphic to X C.

44 Intersection formula of Futaki invariant Proposition (Tian, Paul-Tian, Zhang, Donaldson, Wang, Odaka, Li-Xu) Assume X is normal, then ( ) 1 2a1 DF(X, L) = (n + 1)!2a 0 a Ln+1 + (n + 1)K X /P 1 L n. (5) 0

45 MMP: Variation of polarization in the direction of K X Assume L Xt C K X. We vary the polarization in the direction of K X : M s = L + sk X 1 s, Ms Xt K X d ds M 1 s = (1 s) 2 ( L + K X ) s=1/(1 s) == d d s M s = K X + L ( normlized Kähler-Ricci flow: ω ) s = Ric(ω s) + ω s

46 MMP with scaling Assume X has mild singularities (log canonical), we can run (K X + L)-MMP with scaling L: M λ = (K X + L) + λ L, M = λ L. Note that K X + M λ = λ + 1 (K X + L) λ d dλ M λ = 1 (K λ2 X + L) As λ decreases from + to 0, we get a sequence of critical points of λ and a sequence of models:

47 End product of MMP with scaling + λ 1... λ k > λ k+1 = 0 X 0 X 1... X k C The X k in the above sequence has very good properties: 1 L k C K X k is semiample. 2 Assume X 0 = N i=1 a ix 0,i = E 0, then X X 0 contracts precisely Supp(E). By property 1 above, we can define: X an = P roj(x k, K X k /C) so that K X an is ample.

48 Observation 1: Decreasing of Futaki invariant on a fixed model Derivative of Donaldson-Futaki invariant along M λ : d dλ DF(X, M λ) = C(n, λ) by the Hodge index theorem, because M n 1 λ ( L + K X /P 1) 2 0 L + K X /P 1 = i a i X 0,i only supports on X 0. This means that DF invariant decreases as λ decreases.

49 Observation 2: Invariance of Futaki invariant at transition point Two kinds of transitions: 1 Divisorial contraction 2 Flipping Invariance of Donaldson-Futaki invariant comes from projection formula for intersection product.

50 A weaker statement: unstable case Theorem Given any test configuration (X, L) C 1, for any ɛ 1, we can construct a special test configuration (X s, rk X s) and a positive integer m, such that m(ɛ + DF(X, L)) DF(X s, rk X s) Theorem Tian s conjecture in the un-stable case holds.

51 unstable case I Step 1: Equivariant semistable reduction: Y φ Y/X X X such that C z zk C Y is smooth Y 0 = N 1 i=1 Y 0,i is simple normal crossing.

52 unstable case II Step 2: perturb the pull back polarization L Y = ɛa + φ Y/X (L) by an ample divisor A (ɛ 1) such that L Y is still ample For some a Q L Y + K Y + ay 0 = with a i > 0 for any i 2. N a i Y 0,i i=2

53 unstable case III Step 3: Run (K Y + L Y )-MMP over C. The end product is a special test configuration with Q-Fano variety which is the strict transform of Y 0,1. Assume the DF(X, L) < 0 then 1 negativity is preserved under small perturbation 2 along MMP, Futaki invariant is always decreasing on a fixed model and does not change at the point of transition to a new model.

54 Stable case I How to eliminate the perturbation? More steps: Step 1: Equivariant semistable reduction. Y X X C C Y is smooth Y 0 = N 1 i=1 Y 0,i is simple normal crossing.

55 Step 2: log canonical modification: Run (K Y + Y 0 )-MMP over X to get X lc such that. X lc is not too singular(log canonical) so that we can run MMP K X lc is relatively ample so that we can perturb φ L along X lc the direction of K X lc to get L lc. X lc is a kind of canonical partial resolution of X. Step 3: Run (K X lc + L lc )-MMP over C to get X an such that L an K X an.

56 Stable case II Step 4: Take equivariant semistable reduction of X an : Z φ Z/X an X an X an C z zl C Z is smooth Z 0 = N 2 i=1 Z 0,i is simple normal crossing.

57 Then choose ample divisor A to perturb such that There exists a Q, and a i > 0 for i = 2,..., N 2 a(e 1, X an ) = 0. N 2 φ Z/X (Lan ) + ɛa + K Z + az 0 = a i Z 0,i Now one extract E 1 on X an to get φ X / X an : X X an such that φ X / X ank X an = K X. i=2

58 Stable Case III Step 5: Run (K X + L )-MMP over C to get X s. Note that N 3 K X + L = a i X 0,i with a i > 0 for i = 2,..., N 3. So X s 0 is the strict transform of Z 0,1 (or X 0,1 ). i=2

59 Steps of modification 1 (step 0): normalization 2 (step 1): equivariant semistable reduction 3 (step 2): log-canonical modification 4 (step 3): MMP with scaling 5 (step 4): base change and crepant blow up 6 (step 5): contracting extra components

60 Final remark: log (pair) case The above results can be generalized to the pair case (X, D). At least when D is smooth, we have the following connection: conic Kähler-Einstein metric on (X, D) log-k-stability of the pair (X, D).

61 Thank you! Thank you!