Remarks on hypersurface K-stability. Complex Geometry: A Conference Honoring Simon Donaldson
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1 Remarks on hypersurface K-stability Zhiqin Lu, UC Irvine Complex Geometry: A Conference Honoring Simon Donaldson October 26, 2009 Zhiqin Lu, UC. Irvine Hypersurface K-stability 1/42
2 The Result Theorem (L-Phong) If M is smooth hypersurface of CP n of degree 3 d n, then M is K-stable. Zhiqin Lu, UC. Irvine Hypersurface K-stability 2/42
3 The Result Theorem (L-Phong) If M is smooth hypersurface of CP n of degree 3 d n, then M is K-stable. Our K-stability is related to the K-stability in the sense of Donaldson, but they are not the same. Note that we don t assume the existence of csck metrics. Zhiqin Lu, UC. Irvine Hypersurface K-stability 2/42
4 Background Futaki (1983) defined an invariant (Futaki invariant): Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42
5 Background Futaki (1983) defined an invariant (Futaki invariant): Let M be a Fano manifold, and let ω c 1 (M) be a Kähler metric. Then there is a real function f such that Ric(ω) ω = 1 f Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42
6 Background Futaki (1983) defined an invariant (Futaki invariant): Let M be a Fano manifold, and let ω c 1 (M) be a Kähler metric. Then there is a real function f such that Ric(ω) ω = 1 f Let v be a holomorphic vector field of M. Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42
7 Background Futaki (1983) defined an invariant (Futaki invariant): Let M be a Fano manifold, and let ω c 1 (M) be a Kähler metric. Then there is a real function f such that Ric(ω) ω = 1 f Let v be a holomorphic vector field of M. Then the Futaki invariant can be defined as Fut(v) = v(f )ω n. M Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42
8 Background Futaki (1983) defined an invariant (Futaki invariant): Let M be a Fano manifold, and let ω c 1 (M) be a Kähler metric. Then there is a real function f such that Ric(ω) ω = 1 f Let v be a holomorphic vector field of M. Then the Futaki invariant can be defined as Fut(v) = v(f )ω n. If Fut(v) 0, then the Kähler-Einstein metric doesn t exist. M Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42
9 Donaldson s point of view Let (M, L) be a polarized Kähler manifold. Zhiqin Lu, UC. Irvine Hypersurface K-stability 4/42
10 Donaldson s point of view Let (M, L) be a polarized Kähler manifold. Let v be a holomorphic vector field of M which induces an action to the line bundle L (We say v linearizes on L). Zhiqin Lu, UC. Irvine Hypersurface K-stability 4/42
11 Donaldson s point of view Let (M, L) be a polarized Kähler manifold. Let v be a holomorphic vector field of M which induces an action to the line bundle L (We say v linearizes on L). Consider the space H 0 (M, L k ). Zhiqin Lu, UC. Irvine Hypersurface K-stability 4/42
12 Donaldson s point of view Let Tr v (H 0 (M, L k )) dim H 0 (M, L k ) = a 0k + a 1 + O( 1 k ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 5/42
13 Donaldson s point of view Let Tr v (H 0 (M, L k )) dim H 0 (M, L k ) = a 0k + a 1 + O( 1 k ) The Donaldson s Futaki invariant is defined to be Don Fut(v) = a 1 Zhiqin Lu, UC. Irvine Hypersurface K-stability 5/42
14 Donaldson s point of view Let Tr v (H 0 (M, L k )) dim H 0 (M, L k ) = a 0k + a 1 + O( 1 k ) The Donaldson s Futaki invariant is defined to be We can prove Don Fut(v) = a 1 Theorem Let L = K 1 M, then Don Fut(v) = 1 2 Fut(v). Zhiqin Lu, UC. Irvine Hypersurface K-stability 5/42
15 The Futaki invariant depends only on the cohomology class c 1 (M), not on the particular Kähler metric in the class. Zhiqin Lu, UC. Irvine Hypersurface K-stability 6/42
16 The Futaki invariant depends only on the cohomology class c 1 (M), not on the particular Kähler metric in the class. In Donaldson s definition, the metric doesn t even appear. Zhiqin Lu, UC. Irvine Hypersurface K-stability 6/42
17 Test Configuration Definition (Donaldson) Let L M be an ample line bundle over a compact Kähler manifold. A test configuration consists of the following data Zhiqin Lu, UC. Irvine Hypersurface K-stability 7/42
18 Test Configuration Definition (Donaldson) Let L M be an ample line bundle over a compact Kähler manifold. A test configuration consists of the following data 1. A scheme X with a C action ρ Zhiqin Lu, UC. Irvine Hypersurface K-stability 7/42
19 Test Configuration Definition (Donaldson) Let L M be an ample line bundle over a compact Kähler manifold. A test configuration consists of the following data 1. A scheme X with a C action ρ 2. A C equivariant line bundle L X, where C is ample on all fibers Zhiqin Lu, UC. Irvine Hypersurface K-stability 7/42
20 Test Configuration Definition (Donaldson) Let L M be an ample line bundle over a compact Kähler manifold. A test configuration consists of the following data 1. A scheme X with a C action ρ 2. A C equivariant line bundle L X, where C is ample on all fibers 3. A flat C equivariant map π : X C where C acts on C by multiplication. Zhiqin Lu, UC. Irvine Hypersurface K-stability 7/42
21 1. Let H be a finite dimensional space. Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42
22 1. Let H be a finite dimensional space. 2. Let A be an endomorphism of H. Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42
23 1. Let H be a finite dimensional space. 2. Let A be an endomorphism of H. 3. Let V be a subspace of H which is A invariant. Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42
24 1. Let H be a finite dimensional space. 2. Let A be an endomorphism of H. 3. Let V be a subspace of H which is A invariant. 4. Tr A (V ) is well-defined. Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42
25 1. Let H be a finite dimensional space. 2. Let A be an endomorphism of H. 3. Let V be a subspace of H which is A invariant. 4. Tr A (V ) is well-defined. 5. How to define Tr A (V ), if V is not A invariant? Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42
26 Nonlinear Trace We assume that, under some basis, that we can write A as where λ j are integers. A = λ 0... λ n Zhiqin Lu, UC. Irvine Hypersurface K-stability 9/42
27 Nonlinear Trace We assume that, under some basis, that we can write A as A = λ 0... where λ j are integers. Then there is a C action ρ(t), acts on H. Let λ n V 0 = lim t 0 ρ(t)v Zhiqin Lu, UC. Irvine Hypersurface K-stability 9/42
28 Nonlinear Trace We assume that, under some basis, that we can write A as A = λ 0... where λ j are integers. Then there is a C action ρ(t), acts on H. Let λ n V 0 = lim t 0 ρ(t)v Then V 0 is A invariant. Thus Tr A (V ) = Tr A (V 0 ) is well defined. Zhiqin Lu, UC. Irvine Hypersurface K-stability 9/42
29 Example Let M CP N. For any k, define H = H k = H 0 (CP N, H k ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 10/42
30 Example Let M CP N. For any k, define H = H k = H 0 (CP N, H k ) Define W = {S H S M = 0}, V = H/W Zhiqin Lu, UC. Irvine Hypersurface K-stability 10/42
31 Example Let M CP N. For any k, define H = H k = H 0 (CP N, H k ) Define W = {S H S M = 0}, V = H/W Using Tr A (V )/ dim V to define the Donaldson s Futaki invariant. Zhiqin Lu, UC. Irvine Hypersurface K-stability 10/42
32 Example (Stoppa) Let v be a holomorphic vector field on a (X, L). Let p X. Let W = {S H 0 (M, L k ) S(p) = 0}, V = H 0 (M, L k )/W Zhiqin Lu, UC. Irvine Hypersurface K-stability 11/42
33 Mabuchi s K-energy Nonlinearizing the Futaki s invariant, Mabuchi was able to define the following K-energy functional: Zhiqin Lu, UC. Irvine Hypersurface K-stability 12/42
34 Mabuchi s K-energy Nonlinearizing the Futaki s invariant, Mabuchi was able to define the following K-energy functional: Definition Let ω 0, ω 1 be two Kähler metrics of M such that Then we define K(ω 0, ω 1 ) = where ω s = ω 0 + s φ. ω 1 = ω φ. 1 0 M φ(ric(ω s ) ω s ) ωs n 1 ds, Zhiqin Lu, UC. Irvine Hypersurface K-stability 12/42
35 Mabuchi s K-energy satisfies the co-cycle condition K(ω 0, ω 1 ) + K(ω 1, ω 2 ) = K(ω 0, ω 2 ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 13/42
36 Mabuchi s K-energy satisfies the co-cycle condition K(ω 0, ω 1 ) + K(ω 1, ω 2 ) = K(ω 0, ω 2 ) If ω 1 = ω φ, and when φ is the Hamiltonian function of a holomorphic vector field, the Mabuchi s functional is the same as the Futaki invariant. Zhiqin Lu, UC. Irvine Hypersurface K-stability 13/42
37 Mabuchi s K-energy satisfies the co-cycle condition K(ω 0, ω 1 ) + K(ω 1, ω 2 ) = K(ω 0, ω 2 ) If ω 1 = ω φ, and when φ is the Hamiltonian function of a holomorphic vector field, the Mabuchi s functional is the same as the Futaki invariant. It is a very important tool in Kähler-Einstein geometry, in particular in the uniquesness of Kähler-Einstein metrics and the csck metrics. Zhiqin Lu, UC. Irvine Hypersurface K-stability 13/42
38 Let M CP N for some N. Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42
39 Let M CP N for some N. Let ω be the Kähler metric of M obtained by the restriction of the Fubini-Study metric to M. Define ρ(t) ω be the pull-back of the FS metric by the action ρ(t), where t C. Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42
40 Let M CP N for some N. Let ω be the Kähler metric of M obtained by the restriction of the Fubini-Study metric to M. Define ρ(t) ω be the pull-back of the FS metric by the action ρ(t), where t C. Define Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42
41 Let M CP N for some N. Let ω be the Kähler metric of M obtained by the restriction of the Fubini-Study metric to M. Define ρ(t) ω be the pull-back of the FS metric by the action ρ(t), where t C. Define Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Is this non-linear Futaki invariant depends only on the cohomology class of ω? Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42
42 Let M CP N for some N. Let ω be the Kähler metric of M obtained by the restriction of the Fubini-Study metric to M. Define ρ(t) ω be the pull-back of the FS metric by the action ρ(t), where t C. Define Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Is this non-linear Futaki invariant depends only on the cohomology class of ω? What are we interested in this question? Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42
43 Zhiqin Lu, UC. Irvine Hypersurface K-stability 15/42
44 Analysis/ Geometry Side Algebraic Sides Zhiqin Lu, UC. Irvine Hypersurface K-stability 15/42
45 Analysis/ Geometry Side Making nonlinear Futaki more algebraic, Algebraic Sides Zhiqin Lu, UC. Irvine Hypersurface K-stability 15/42
46 Analysis/ Geometry Side Algebraic Sides Making nonlinear Futaki more algebraic, which was done by Sean Paul. Zhiqin Lu, UC. Irvine Hypersurface K-stability 15/42
47 Sean Paul s result Nonlinear Fut(ω) = A + B where both A, B are nonnegative. A is related to the Chow weight, and B is related to the X-hyperdiscriminant and the Chow form. Both of which are algebraic. Zhiqin Lu, UC. Irvine Hypersurface K-stability 16/42
48 Zhiqin Lu K energy and K stability on hypersurfaces CAG, 2004, 12(3), Zhiqin Lu, UC. Irvine Hypersurface K-stability 17/42
49 Zhiqin Lu K energy and K stability on hypersurfaces CAG, 2004, 12(3), Zhiqin Lu, UC. Irvine Hypersurface K-stability 17/42
50 For given λ, set Λ = max λ, α. α I Zhiqin Lu, UC. Irvine Hypersurface K-stability 18/42
51 For given λ, set Λ = max λ, α. α I Chow weight? Zhiqin Lu, UC. Irvine Hypersurface K-stability 18/42
52 For given λ, set Λ = max λ, α. α I Chow weight? and, for u = (u 0,, u n ) an (n + 1)-vector, φ(u) = max u + λ, α. α I Finally, we define φ j (t) = φ(0,, t j,, 0) for 0 j n. Then the following theorem was proved: Zhiqin Lu, UC. Irvine Hypersurface K-stability 18/42
53 Theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) a generic rational vector with n i=0 λ i = 0, then the non-linear Futaki invariant is equal to lim t d t 0 dt K(ω FS, σ(t) (ω FS )) Λ(d 1)(n + 1) = + n n j=0 0 φ j(t)(φ j(t) 1)dt. Zhiqin Lu, UC. Irvine Hypersurface K-stability 19/42
54 Theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) a generic rational vector with n i=0 λ i = 0, then the non-linear Futaki invariant is equal to lim t d t 0 dt K(ω FS, σ(t) (ω FS )) Λ(d 1)(n + 1) = + n n j=0 0 φ j(t)(φ j(t) 1)dt. Here generic means that the set of λ such that the above equality holds is dense. Zhiqin Lu, UC. Irvine Hypersurface K-stability 19/42
55 Nonlinear Futaki = ( 2 n ( ) XFt d F t 2 M t k=0 k ) (F t ) k ω n 1 + (n d + 1) θω n 1. M t Zhiqin Lu, UC. Irvine Hypersurface K-stability 20/42
56 Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Zhiqin Lu, UC. Irvine Hypersurface K-stability 21/42
57 Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Theorem (Ding-Tian) If the center fiber of the test configuration is a normal variety, then Nonlinear Fut(ω) = Fut(v), where v is the vector field induced by ρ on the center fiber, and Fut(v) is the Futaki invariant of the center fiber. Zhiqin Lu, UC. Irvine Hypersurface K-stability 21/42
58 If the center fiber is normal, then obviously the nonlinear Futaki depends only on the cohomology class. Zhiqin Lu, UC. Irvine Hypersurface K-stability 22/42
59 If the center fiber is normal, then obviously the nonlinear Futaki depends only on the cohomology class. We can manage to generalize the above result when the center fiber is reduced. However, for generic test configuration, the center fiber is non-reduced. Zhiqin Lu, UC. Irvine Hypersurface K-stability 22/42
60 An example of test configuration Let M be defined by the zeroes of the homogeneous polynomial F (Z) = α I a α Z α (1) of degree d, where a α 0 for α in a set of indices I, and we have used the multi-index notation α = (α 0,, α n ), Z α = Z α0 0 Zn αn. Zhiqin Lu, UC. Irvine Hypersurface K-stability 23/42
61 An example of test configuration Let M be defined by the zeroes of the homogeneous polynomial F (Z) = α I a α Z α (1) of degree d, where a α 0 for α in a set of indices I, and we have used the multi-index notation α = (α 0,, α n ), Z α = Z α0 0 Zn αn. Let λ = (λ 0,, λ n ) be such that λ j = 0 and all λ j are integers. Zhiqin Lu, UC. Irvine Hypersurface K-stability 23/42
62 An example of test configuration Let M be defined by the zeroes of the homogeneous polynomial F (Z) = α I a α Z α (1) of degree d, where a α 0 for α in a set of indices I, and we have used the multi-index notation α = (α 0,, α n ), Z α = Z α0 0 Zn αn. Let λ = (λ 0,, λ n ) be such that λ j = 0 and all λ j are integers. Then λ induced an action ρ(t) = (t λ0,, t λn ). The scheme X is defined by a sub-scheme of CP n C defined by the equation F (t λ0 Z 0,, t λn Z n ) = 0 Zhiqin Lu, UC. Irvine Hypersurface K-stability 23/42
63 Generically, the center fiber is defined by Z β = 0, where β I is the unique element in I such that λ, β = max λ, α. Zhiqin Lu, UC. Irvine Hypersurface K-stability 24/42
64 Generically, the center fiber is defined by Z β = 0, where β I is the unique element in I such that λ, β = max λ, α. Thus unless β j = 0, 1 for all j, the center fiber will be non-reduced (with multiplicity). Zhiqin Lu, UC. Irvine Hypersurface K-stability 24/42
65 Theorem (S. Paul) The Mabuchi energy is bounded from below along all degenerations if and only if the Hyperdiscriminant polytope contains the Chow polytope. In particular, the asymptotic behavior of the Mabuchi energy along any degeneration is logarithmic. Zhiqin Lu, UC. Irvine Hypersurface K-stability 25/42
66 Arezzo-della Vedova-la Nave In order to check K-stability, one only needs to check those test configurations where the center fibers are of normal crossings. Zhiqin Lu, UC. Irvine Hypersurface K-stability 26/42
67 A theorem of Kodaira and Spencer Theorem Let M be a smooth hypersurface of CP n of degree at least 3, then M has no holomorphic vector field. Zhiqin Lu, UC. Irvine Hypersurface K-stability 27/42
68 By the above result, we can prove the following Theorem (Donaldson, Stoppa) If M admits a csck metric, then M is K-stable (in the sense of Donaldson). Zhiqin Lu, UC. Irvine Hypersurface K-stability 28/42
69 Definition of K-stability Definition Let M be the hypersurface defined by a homogeneous polynomial F. The hypersurface M is said to be K-stable if for any vector λ = (λ 0,, λ n ) with λ integers and n i=0 λ i = 0, we have lim t 0 t d dt K(ω FS, σ(t) (ω FS )) 0 for the K-energy on M. Moreover, the equality holds if and only if after a permutation of (0,, n), 1. λ = a(n, 1,, 1) for a positive rational number a > 0, and 2. F (1, 0,, 0) 0. Zhiqin Lu, UC. Irvine Hypersurface K-stability 29/42
70 The semi-continuity theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) an arbitrary rational vector with n i=0 λ i = 0. Define ξ(λ) = lim t 0 t d dt K(ω FS, σ(t) (ω FS )) Zhiqin Lu, UC. Irvine Hypersurface K-stability 30/42
71 The semi-continuity theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) an arbitrary rational vector with n i=0 λ i = 0. Define Theorem (L-Phong) Using the above result, we have ξ(λ) = lim t 0 t d dt K(ω FS, σ(t) (ω FS )) ξ(λ) lim λ λ,λ U ξ(λ ). Zhiqin Lu, UC. Irvine Hypersurface K-stability 30/42
72 The semi-continuity theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) an arbitrary rational vector with n i=0 λ i = 0. Define Theorem (L-Phong) Using the above result, we have As a corollary, we have ξ(λ) = lim t 0 t d dt K(ω FS, σ(t) (ω FS )) ξ(λ) Λ(d 1)(n + 1) ξ(λ) + n lim λ λ,λ U ξ(λ ). n j=0 0 φ j(t)(φ j(t) 1)dt. Zhiqin Lu, UC. Irvine Hypersurface K-stability 30/42
73 Chow-Mumford stability Definition Let F (Z) = α I a α Z α, where a α 0 for any α I. We say M is Chow-Mumford stable, if Zhiqin Lu, UC. Irvine Hypersurface K-stability 31/42
74 Chow-Mumford stability Definition Let F (Z) = α I a α Z α, where a α 0 for any α I. We say M is Chow-Mumford stable, if 1. for any (λ 0,, λ n ) with λ i all integers and λ i = 0, there is an element α = (α 0,, α n ) I such that λ, α > 0; Zhiqin Lu, UC. Irvine Hypersurface K-stability 31/42
75 Chow-Mumford stability Definition Let F (Z) = α I a α Z α, where a α 0 for any α I. We say M is Chow-Mumford stable, if 1. for any (λ 0,, λ n ) with λ i all integers and λ i = 0, there is an element α = (α 0,, α n ) I such that λ, α > 0; 2. there is no holomorphic vector field on M. Zhiqin Lu, UC. Irvine Hypersurface K-stability 31/42
76 By a theorem of Mumford, smooth hypersurface is Chow-Mumford stable. Zhiqin Lu, UC. Irvine Hypersurface K-stability 32/42
77 The technical heart of the paper is the following result: we define the function Λ(d 1)(n + 1) g(λ) = + n n j=0 0 φ j(t)(φ j(t) 1)dt. Zhiqin Lu, UC. Irvine Hypersurface K-stability 33/42
78 The technical heart of the paper is the following result: we define the function Λ(d 1)(n + 1) g(λ) = + n n j=0 0 φ j(t)(φ j(t) 1)dt. Theorem If M is Chow-Mumford stable, then g(λ) < 0, unless λ a(n, 1,, 1) for any positive number a. Zhiqin Lu, UC. Irvine Hypersurface K-stability 33/42
79 We first prove the following elementary lemma: Lemma Let be a finite set of liner functions. I = {ax + b a 0} Zhiqin Lu, UC. Irvine Hypersurface K-stability 34/42
80 We first prove the following elementary lemma: Lemma Let I = {ax + b a 0} be a finite set of liner functions. Assume that I contains at least one constant function. Define φ I = min f f I Zhiqin Lu, UC. Irvine Hypersurface K-stability 34/42
81 We first prove the following elementary lemma: Lemma Let I = {ax + b a 0} be a finite set of liner functions. Assume that I contains at least one constant function. Define φ I = min f f I Let J I, then we have 0 φ I (t)(φ I (t) 1)dt 0 φ J(t)(φ J(t) 1)dt Zhiqin Lu, UC. Irvine Hypersurface K-stability 34/42
82 Less linear function to minimize, larger value 0 φ (t)(φ (t) 1)dt. Zhiqin Lu, UC. Irvine Hypersurface K-stability 35/42
83 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42
84 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. By the Chow-Mumford stability, λ, β = max λ, α > 0 Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42
85 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. By the Chow-Mumford stability, λ, β = max λ, α > 0 Define κ = (n, 1,, 1) Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42
86 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. By the Chow-Mumford stability, λ, β = max λ, α > 0 Define κ = (n, 1,, 1) Define for a suitable constant c, we have λ = λ cκ Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42
87 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. By the Chow-Mumford stability, λ, β = max λ, α > 0 Define κ = (n, 1,, 1) Define λ = λ cκ for a suitable constant c, we have λ, β = max λ, α Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42
88 In our proof, we pick up finite sequences λ k = λ c k κ, β k Let the set of β k be J. Zhiqin Lu, UC. Irvine Hypersurface K-stability 37/42
89 In our proof, we pick up finite sequences λ k = λ c k κ, β k Let the set of β k be J. By the lemma above, φ (t)(φ (t) 1)dt 0 0 φ J(t)(φ J(t) 1)dt Zhiqin Lu, UC. Irvine Hypersurface K-stability 37/42
90 Define Λ(d 1)(n + 1) g(λ) = n + φ J(t)(φ J(t) 1)dt. 0 Zhiqin Lu, UC. Irvine Hypersurface K-stability 38/42
91 Define Λ(d 1)(n + 1) g(λ) = n + φ J(t)(φ J(t) 1)dt. 0 Then g(λ) g(λ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 38/42
92 Define Λ(d 1)(n + 1) g(λ) = n + φ J(t)(φ J(t) 1)dt. 0 Then g(λ) g(λ) The key observation is that g(λ) g(λ 1 ) g(λ s ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 38/42
93 Using the above inequality, we can prove the semi-k-stability: at least the function g is not positive, Zhiqin Lu, UC. Irvine Hypersurface K-stability 39/42
94 Using the above inequality, we can prove the semi-k-stability: at least the function g is not positive, by reducing an arbitrary text configuration to a trivial one. Zhiqin Lu, UC. Irvine Hypersurface K-stability 39/42
95 Using the above inequality, we can prove the semi-k-stability: at least the function g is not positive, by reducing an arbitrary text configuration to a trivial one. The only exception is that is λ is proportional to κ. By by continuity, we can always perturb it to prove the semi-k-stability. Zhiqin Lu, UC. Irvine Hypersurface K-stability 39/42
96 We can characterize κ in the following Lemma (L-Phong) Let J(t) be the J-functional restricted to the one-parameter group. Then J(t) = O(1) if and only if the test configuration is induced by κ. Here J(t) is Aubin s J-functional. Zhiqin Lu, UC. Irvine Hypersurface K-stability 40/42
97 Example The is a KE metric on the Fermat surface Z n Z n n = 0. However, along the line defined by the test configuration (n, 1,, 1), the Mabuchi functional is of O(1). Zhiqin Lu, UC. Irvine Hypersurface K-stability 41/42
98 Thank you! Zhiqin Lu, UC. Irvine Hypersurface K-stability 42/42
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