Tangent cones of K¨ ahler-Einstein metrics Hans-Joachim Hein UMd College Park & Fordham University joint work with Song Sun Stony Brook University July 13, 2016
Calabi-Yau Theorem: Let X be a compact K¨ ahler manifold of complex dimension n with c 1 ( X ) = 0 . Then every K¨ ahler class k ∈ H 2 ( X ) contains a unique Ricci-flat K¨ ahler metric ω ∈ k . Example: Let f be a homogeneous complex polynomial of degree n + 2 in n + 2 complex variables. Let X = X f = { [ z 1 : . . . : z n +2 ] ∈ CP n +1 : f ( z 1 , . . . , z n +2 ) = 0 } . If f is generic, then X is smooth with c 1 ( X ) = 0. Can take k = 2 πc 1 ( O (1) | X ). Then ω FS | X ∈ k , so there exists a smooth function ϕ : X → R , unique up to constants, such that ahler form ω = ω FS | X + i∂ ¯ the K¨ ∂ϕ ∈ k is Ricci-flat. Today: Let f = f t move in a holomorphic family parametrized by t ∈ C . Assume X t = X f t is smooth as above for all t � = 0 but X 0 is singular. What happens to the Ricci-flat metric ω t ( t � = 0) representing k t = 2 πc 1 ( O (1) | X t ) as t → 0?
Calabi-Yau Theorem: Let X be a compact K¨ ahler manifold of complex dimension n with c 1 ( X ) = 0 . Then every K¨ ahler class k ∈ H 2 ( X ) contains a unique Ricci-flat K¨ ahler metric ω ∈ k . Example: Let f be a homogeneous complex polynomial of degree n + 2 in n + 2 complex variables. Let X = X f = { [ z 1 : . . . : z n +2 ] ∈ CP n +1 : f ( z 1 , . . . , z n +2 ) = 0 } . If f is generic, then X is smooth with c 1 ( X ) = 0. Can take k = 2 πc 1 ( O (1) | X ). Then ω FS | X ∈ k , so there exists a smooth function ϕ : X → R , unique up to constants, such that ahler form ω = ω FS | X + i∂ ¯ the K¨ ∂ϕ ∈ k is Ricci-flat. Today: Let f = f t move in a holomorphic family parametrized by t ∈ C . Assume X t = X f t is smooth as above for all t � = 0 but X 0 is singular. What happens to the Ricci-flat metric ω t ( t � = 0) representing k t = 2 πc 1 ( O (1) | X t ) as t → 0?
Calabi-Yau Theorem: Let X be a compact K¨ ahler manifold of complex dimension n with c 1 ( X ) = 0 . Then every K¨ ahler class k ∈ H 2 ( X ) contains a unique Ricci-flat K¨ ahler metric ω ∈ k . Example: Let f be a homogeneous complex polynomial of degree n + 2 in n + 2 complex variables. Let X = X f = { [ z 1 : . . . : z n +2 ] ∈ CP n +1 : f ( z 1 , . . . , z n +2 ) = 0 } . If f is generic, then X is smooth with c 1 ( X ) = 0. Can take k = 2 πc 1 ( O (1) | X ). Then ω FS | X ∈ k , so there exists a smooth function ϕ : X → R , unique up to constants, such that ahler form ω = ω FS | X + i∂ ¯ the K¨ ∂ϕ ∈ k is Ricci-flat. Today: Let f = f t move in a holomorphic family parametrized by t ∈ C . Assume X t = X f t is smooth as above for all t � = 0 but X 0 is singular. What happens to the Ricci-flat metric ω t ( t � = 0) representing k t = 2 πc 1 ( O (1) | X t ) as t → 0?
Calabi-Yau Theorem: Let X be a compact K¨ ahler manifold of complex dimension n with c 1 ( X ) = 0 . Then every K¨ ahler class k ∈ H 2 ( X ) contains a unique Ricci-flat K¨ ahler metric ω ∈ k . Example: Let f be a homogeneous complex polynomial of degree n + 2 in n + 2 complex variables. Let X = X f = { [ z 1 : . . . : z n +2 ] ∈ CP n +1 : f ( z 1 , . . . , z n +2 ) = 0 } . If f is generic, then X is smooth with c 1 ( X ) = 0. Can take k = 2 πc 1 ( O (1) | X ). Then ω FS | X ∈ k , so there exists a smooth function ϕ : X → R , unique up to constants, such that ahler form ω = ω FS | X + i∂ ¯ the K¨ ∂ϕ ∈ k is Ricci-flat. Today: Let f = f t move in a holomorphic family parametrized by t ∈ C . Assume X t = X f t is smooth as above for all t � = 0 but X 0 is singular. What happens to the Ricci-flat metric ω t ( t � = 0) representing k t = 2 πc 1 ( O (1) | X t ) as t → 0?
We are lightyears away from understanding this properly. Main enemy is collapsing. In the n = 1 cubic example, ( X t , ω t ) is a flat 2-torus for all t � = 0 that GH-converges to a line as t → 0. To avoid collapsing it is necessary to assume that the singularities of the complex variety X 0 are sufficiently mild (’canonical’). • n = 1: canonical ⇔ smooth, i.e. no singularity at all • n = 2: canonical ⇔ locally biholomorphic to C 2 / Γ for a finite group Γ ⊂ SU(2) acting freely on S 3 ⊂ C 2 In particular, canonical singularities are isolated for n = 2. • For n ≥ 3, a canonical singularity need not be isolated. Even if it is isolated, it is rarely (for us: never) of the form C n / Γ.
We are lightyears away from understanding this properly. Main enemy is collapsing. In the n = 1 cubic example, ( X t , ω t ) is a flat 2-torus for all t � = 0 that GH-converges to a line as t → 0. To avoid collapsing it is necessary to assume that the singularities of the complex variety X 0 are sufficiently mild (’canonical’). • n = 1: canonical ⇔ smooth, i.e. no singularity at all • n = 2: canonical ⇔ locally biholomorphic to C 2 / Γ for a finite group Γ ⊂ SU(2) acting freely on S 3 ⊂ C 2 In particular, canonical singularities are isolated for n = 2. • For n ≥ 3, a canonical singularity need not be isolated. Even if it is isolated, it is rarely (for us: never) of the form C n / Γ.
We are lightyears away from understanding this properly. Main enemy is collapsing. In the n = 1 cubic example, ( X t , ω t ) is a flat 2-torus for all t � = 0 that GH-converges to a line as t → 0. To avoid collapsing it is necessary to assume that the singularities of the complex variety X 0 are sufficiently mild (’canonical’). • n = 1: canonical ⇔ smooth, i.e. no singularity at all • n = 2: canonical ⇔ locally biholomorphic to C 2 / Γ for a finite group Γ ⊂ SU(2) acting freely on S 3 ⊂ C 2 In particular, canonical singularities are isolated for n = 2. • For n ≥ 3, a canonical singularity need not be isolated. Even if it is isolated, it is rarely (for us: never) of the form C n / Γ.
We are lightyears away from understanding this properly. Main enemy is collapsing. In the n = 1 cubic example, ( X t , ω t ) is a flat 2-torus for all t � = 0 that GH-converges to a line as t → 0. To avoid collapsing it is necessary to assume that the singularities of the complex variety X 0 are sufficiently mild (’canonical’). • n = 1: canonical ⇔ smooth, i.e. no singularity at all • n = 2: canonical ⇔ locally biholomorphic to C 2 / Γ for a finite group Γ ⊂ SU(2) acting freely on S 3 ⊂ C 2 In particular, canonical singularities are isolated for n = 2. • For n ≥ 3, a canonical singularity need not be isolated. Even if it is isolated, it is rarely (for us: never) of the form C n / Γ.
We are lightyears away from understanding this properly. Main enemy is collapsing. In the n = 1 cubic example, ( X t , ω t ) is a flat 2-torus for all t � = 0 that GH-converges to a line as t → 0. To avoid collapsing it is necessary to assume that the singularities of the complex variety X 0 are sufficiently mild (’canonical’). • n = 1: canonical ⇔ smooth, i.e. no singularity at all • n = 2: canonical ⇔ locally biholomorphic to C 2 / Γ for a finite group Γ ⊂ SU(2) acting freely on S 3 ⊂ C 2 In particular, canonical singularities are isolated for n = 2. • For n ≥ 3, a canonical singularity need not be isolated. Even if it is isolated, it is rarely (for us: never) of the form C n / Γ.
If n = 2 and if X 0 has only canonical singularities (i.e. isolated orbifold singularities of the form C 2 / Γ), then the behavior of the Ricci-flat metrics ω t on X t as t → 0 is completely understood. 1) orbifold version of the Calabi-Yau theorem (folklore) ⇒ there is a unique Ricci-flat K¨ ahler orbifold metric ω 0 ∈ k 0 on X 0 I.e. if π : C 2 → C 2 / Γ is the quotient map, then locally π ∗ ω 0 = ω C 2 + smooth errors . I.e. ( X 0 , ω 0 ) is locally asymptotic to a flat cone C 2 / Γ. Not even quasi-isometric to the singularities of ω FS | X 0 ! 2) Gluing construction ( ∃ many complete noncompact Ricci-flat ahler manifolds asymptotic to C 2 / Γ at infinity) ⇒ ω t converges K¨ smoothly to ω 0 away from X sing , and globally in the GH sense. 0 (Biquard-Rollin 2012, Spotti 2012)
If n = 2 and if X 0 has only canonical singularities (i.e. isolated orbifold singularities of the form C 2 / Γ), then the behavior of the Ricci-flat metrics ω t on X t as t → 0 is completely understood. 1) orbifold version of the Calabi-Yau theorem (folklore) ⇒ there is a unique Ricci-flat K¨ ahler orbifold metric ω 0 ∈ k 0 on X 0 I.e. if π : C 2 → C 2 / Γ is the quotient map, then locally π ∗ ω 0 = ω C 2 + smooth errors . I.e. ( X 0 , ω 0 ) is locally asymptotic to a flat cone C 2 / Γ. Not even quasi-isometric to the singularities of ω FS | X 0 ! 2) Gluing construction ( ∃ many complete noncompact Ricci-flat ahler manifolds asymptotic to C 2 / Γ at infinity) ⇒ ω t converges K¨ smoothly to ω 0 away from X sing , and globally in the GH sense. 0 (Biquard-Rollin 2012, Spotti 2012)
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