stability and hermitian-einstein metrics for vector bundles on …stemmler/tifr/... ·...
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![Page 1: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/1.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Stability and Hermitian-Einstein metricsfor vector bundles on framed manifolds
Matthias Stemmler
Department of Mathematics and Computer SciencePhilipps University Marburg
Lie Theory and Complex GeometryMarburg, November 4, 2010
![Page 2: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/2.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Contents
IntroductionStabilityHermitian-Einstein metricsKobayashi-Hitchin correspondence
Framed manifoldsDefinitionPoincare-type metric
Adapting the notionsProblemFramed stabilityFramed Hermitian-Einstein metricsRelationship
Outlook
![Page 3: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/3.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Contents
IntroductionStabilityHermitian-Einstein metricsKobayashi-Hitchin correspondence
Framed manifoldsDefinitionPoincare-type metric
Adapting the notionsProblemFramed stabilityFramed Hermitian-Einstein metricsRelationship
Outlook
![Page 4: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/4.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Stability
Let (X, g) be a compact Kahler manifold of complexdimension n and E a torsion-free coherent sheaf on X.
Definition
I The g-degree of E is defined as
degg(E) =∫Xc1(E) ∧ ωn−1
g .
I If rank(E) > 0, the g-slope of E is defined as
µg(E) =degg(E)rank(E)
.
![Page 5: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/5.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Stability
Let (X, g) be a compact Kahler manifold of complexdimension n and E a torsion-free coherent sheaf on X.
Definition
I The g-degree of E is defined as
degg(E) =∫Xc1(E) ∧ ωn−1
g .
I If rank(E) > 0, the g-slope of E is defined as
µg(E) =degg(E)rank(E)
.
![Page 6: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/6.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Stability
Let (X, g) be a compact Kahler manifold of complexdimension n and E a torsion-free coherent sheaf on X.
Definition
I The g-degree of E is defined as
degg(E) =∫Xc1(E) ∧ ωn−1
g .
I If rank(E) > 0, the g-slope of E is defined as
µg(E) =degg(E)rank(E)
.
![Page 7: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/7.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Stability
DefinitionE is called g-stable if
µg(F) < µg(E)
holds for every coherent subsheaf F of E with0 < rank(F) < rank(E).
RemarkThis is also defined for a holomorphic vector bundle E:Consider E = OX(E).
![Page 8: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/8.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Stability
DefinitionE is called g-stable if
µg(F) < µg(E)
holds for every coherent subsheaf F of E with0 < rank(F) < rank(E).
RemarkThis is also defined for a holomorphic vector bundle E:Consider E = OX(E).
![Page 9: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/9.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Hermitian-Einstein metrics
Let E be a holomorphic vector bundle on X.
DefinitionA Hermitian metric h in E is called ag-Hermitian-Einstein metric if
iΛgFh = λ idE with λ ∈ R,
where
I iΛg = contraction with ωg,
I Fh = Chern curvature form of (E, h).
![Page 10: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/10.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Kobayashi-Hitchin correspondence
Let E be a holomorphic vector bundle.
Theorem (S. Kobayashi ’82, Lubke ’83)
E is irreducible and admits a g-Hermitian-Einstein metric=⇒ E is g-stable.
Theorem (Donaldson ’83–’87, Uhlenbeck/Yau ’86)
E is g-stable =⇒ E admits a unique (up to a constantmultiple) g-Hermitian-Einstein metric.
![Page 11: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/11.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Kobayashi-Hitchin correspondence
Let E be a holomorphic vector bundle.
Theorem (S. Kobayashi ’82, Lubke ’83)
E is irreducible and admits a g-Hermitian-Einstein metric=⇒ E is g-stable.
Theorem (Donaldson ’83–’87, Uhlenbeck/Yau ’86)
E is g-stable =⇒ E admits a unique (up to a constantmultiple) g-Hermitian-Einstein metric.
![Page 12: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/12.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Kobayashi-Hitchin correspondence
Let E be a holomorphic vector bundle.
Theorem (S. Kobayashi ’82, Lubke ’83)
E is irreducible and admits a g-Hermitian-Einstein metric=⇒ E is g-stable.
Theorem (Donaldson ’83–’87, Uhlenbeck/Yau ’86)
E is g-stable =⇒ E admits a unique (up to a constantmultiple) g-Hermitian-Einstein metric.
![Page 13: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/13.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Kobayashi-Hitchin correspondence
Proof.
I Start with a fixed “background metric” h0.
I Solve an evolution equation for all finite values of atime parameter t.
I In case of convergence as t→∞:The limit is a Hermitian-Einstein metric.
I In case of divergence:Construct a “destabilizing subsheaf” contradictingthe stability hypothesis.
![Page 14: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/14.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Kobayashi-Hitchin correspondence
Proof.
I Start with a fixed “background metric” h0.
I Solve an evolution equation for all finite values of atime parameter t.
I In case of convergence as t→∞:The limit is a Hermitian-Einstein metric.
I In case of divergence:Construct a “destabilizing subsheaf” contradictingthe stability hypothesis.
![Page 15: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/15.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Kobayashi-Hitchin correspondence
Proof.
I Start with a fixed “background metric” h0.
I Solve an evolution equation for all finite values of atime parameter t.
I In case of convergence as t→∞:The limit is a Hermitian-Einstein metric.
I In case of divergence:Construct a “destabilizing subsheaf” contradictingthe stability hypothesis.
![Page 16: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/16.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Kobayashi-Hitchin correspondence
Proof.
I Start with a fixed “background metric” h0.
I Solve an evolution equation for all finite values of atime parameter t.
I In case of convergence as t→∞:The limit is a Hermitian-Einstein metric.
I In case of divergence:Construct a “destabilizing subsheaf” contradictingthe stability hypothesis.
![Page 17: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/17.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Contents
IntroductionStabilityHermitian-Einstein metricsKobayashi-Hitchin correspondence
Framed manifoldsDefinitionPoincare-type metric
Adapting the notionsProblemFramed stabilityFramed Hermitian-Einstein metricsRelationship
Outlook
![Page 18: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/18.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed manifolds
Definition
I A framed manifold or logarithmic pair is a pair(X,D) consisting of
I a compact complex manifold X andI a smooth divisor D in X.
I A framed manifold (X,D) is called canonicallypolarized if KX ⊗ [D] is ample.
Example
(Pn, V ) is canonically polarized if V ⊂ Pn is a smoothhypersurface of degree > n+ 2.
![Page 19: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/19.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed manifolds
Definition
I A framed manifold or logarithmic pair is a pair(X,D) consisting of
I a compact complex manifold X andI a smooth divisor D in X.
I A framed manifold (X,D) is called canonicallypolarized if KX ⊗ [D] is ample.
Example
(Pn, V ) is canonically polarized if V ⊂ Pn is a smoothhypersurface of degree > n+ 2.
![Page 20: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/20.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed manifolds
Definition
I A framed manifold or logarithmic pair is a pair(X,D) consisting of
I a compact complex manifold X andI a smooth divisor D in X.
I A framed manifold (X,D) is called canonicallypolarized if KX ⊗ [D] is ample.
Example
(Pn, V ) is canonically polarized if V ⊂ Pn is a smoothhypersurface of degree > n+ 2.
![Page 21: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/21.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Poincare-type metric
Theorem (R. Kobayashi ’84)
If (X,D) is canonically polarized, there exists a unique(up to a constant multiple) complete Kahler-Einsteinmetric on X ′ := X \D with negative Ricci curvature.
Remark
I We call this metric the Poincare-type metric on X ′.
I Choose local coordinates (σ, z2, . . . , zn) such that Dis given by σ = 0. Then in these coordinates, wehave
ωPoin ∼ 2i
(dσ ∧ dσ
|σ|2 log2(1/|σ|2)+
n∑k=2
dzk ∧ dzk).
![Page 22: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/22.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Poincare-type metric
Theorem (R. Kobayashi ’84)
If (X,D) is canonically polarized, there exists a unique(up to a constant multiple) complete Kahler-Einsteinmetric on X ′ := X \D with negative Ricci curvature.
Remark
I We call this metric the Poincare-type metric on X ′.
I Choose local coordinates (σ, z2, . . . , zn) such that Dis given by σ = 0. Then in these coordinates, wehave
ωPoin ∼ 2i
(dσ ∧ dσ
|σ|2 log2(1/|σ|2)+
n∑k=2
dzk ∧ dzk).
![Page 23: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/23.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Poincare-type metric
Theorem (R. Kobayashi ’84)
If (X,D) is canonically polarized, there exists a unique(up to a constant multiple) complete Kahler-Einsteinmetric on X ′ := X \D with negative Ricci curvature.
Remark
I We call this metric the Poincare-type metric on X ′.
I Choose local coordinates (σ, z2, . . . , zn) such that Dis given by σ = 0. Then in these coordinates, wehave
ωPoin ∼ 2i
(dσ ∧ dσ
|σ|2 log2(1/|σ|2)+
n∑k=2
dzk ∧ dzk).
![Page 24: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/24.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Poincare-type metric
Theorem (R. Kobayashi ’84)
If (X,D) is canonically polarized, there exists a unique(up to a constant multiple) complete Kahler-Einsteinmetric on X ′ := X \D with negative Ricci curvature.
Remark
I We call this metric the Poincare-type metric on X ′.
I Choose local coordinates (σ, z2, . . . , zn) such that Dis given by σ = 0. Then in these coordinates, wehave
ωPoin ∼ 2i
(dσ ∧ dσ
|σ|2 log2(1/|σ|2)+
n∑k=2
dzk ∧ dzk).
![Page 25: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/25.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Bounded geometry
(Cheng-Yau ’80, R. Kobayashi ’84, Tian-Yau ’87, ...)
DefinitionA local quasi-coordinate map is a holomorphic map
V −→ X ′, V ⊂ Cn open
which is of maximal rank everywhere. In this case, Vtogether with the Euclidean coordinates of Cn is called alocal quasi-coordinate system.
![Page 26: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/26.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Bounded geometry
TheoremX ′ together with the Poincare-type metric is of boundedgeometry, i. e. there is an (infinite) familyV = (V ; v1, . . . , vn) of local quasi-coordinate systemssuch that:
I X ′ is covered by the images of the V in V.
I There is an open neighbourhood U of D such thatX \ U is covered by the images of finitely many Vwhich are coordinate systems in the ordinary sense.
I Every V contains an open ball of radius 12 .
I The coefficients of the Poincare-type metric andtheir derivatives in quasi-coordinates are uniformlybounded.
![Page 27: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/27.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Bounded geometry
TheoremX ′ together with the Poincare-type metric is of boundedgeometry, i. e. there is an (infinite) familyV = (V ; v1, . . . , vn) of local quasi-coordinate systemssuch that:
I X ′ is covered by the images of the V in V.
I There is an open neighbourhood U of D such thatX \ U is covered by the images of finitely many Vwhich are coordinate systems in the ordinary sense.
I Every V contains an open ball of radius 12 .
I The coefficients of the Poincare-type metric andtheir derivatives in quasi-coordinates are uniformlybounded.
![Page 28: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/28.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Bounded geometry
TheoremX ′ together with the Poincare-type metric is of boundedgeometry, i. e. there is an (infinite) familyV = (V ; v1, . . . , vn) of local quasi-coordinate systemssuch that:
I X ′ is covered by the images of the V in V.
I There is an open neighbourhood U of D such thatX \ U is covered by the images of finitely many Vwhich are coordinate systems in the ordinary sense.
I Every V contains an open ball of radius 12 .
I The coefficients of the Poincare-type metric andtheir derivatives in quasi-coordinates are uniformlybounded.
![Page 29: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/29.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Bounded geometry
TheoremX ′ together with the Poincare-type metric is of boundedgeometry, i. e. there is an (infinite) familyV = (V ; v1, . . . , vn) of local quasi-coordinate systemssuch that:
I X ′ is covered by the images of the V in V.
I There is an open neighbourhood U of D such thatX \ U is covered by the images of finitely many Vwhich are coordinate systems in the ordinary sense.
I Every V contains an open ball of radius 12 .
I The coefficients of the Poincare-type metric andtheir derivatives in quasi-coordinates are uniformlybounded.
![Page 30: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/30.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Bounded geometry
TheoremX ′ together with the Poincare-type metric is of boundedgeometry, i. e. there is an (infinite) familyV = (V ; v1, . . . , vn) of local quasi-coordinate systemssuch that:
I X ′ is covered by the images of the V in V.
I There is an open neighbourhood U of D such thatX \ U is covered by the images of finitely many Vwhich are coordinate systems in the ordinary sense.
I Every V contains an open ball of radius 12 .
I The coefficients of the Poincare-type metric andtheir derivatives in quasi-coordinates are uniformlybounded.
![Page 31: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/31.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Construction of quasi-coordinates
Choose local coordinates
(∆n; z1, . . . , zn) on U ⊂ X
such thatD ∩ U = z1 = 0.
Then the quasi-coordinates are(BR(0)×∆n−1; v1, . . . , vn) with 1
2 < R < 1 such that
v1 =w1 − a1− aw1
, where z1 = exp(w1 + 1w1 − 1
),
andvi = wi = zi for 2 6 i 6 n,
where a varies over real numbers in ∆ close to 1.
![Page 32: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/32.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Construction of quasi-coordinates
Choose local coordinates
(∆n; z1, . . . , zn) on U ⊂ X
such thatD ∩ U = z1 = 0.
Then the quasi-coordinates are(BR(0)×∆n−1; v1, . . . , vn) with 1
2 < R < 1 such that
v1 =w1 − a1− aw1
, where z1 = exp(w1 + 1w1 − 1
),
andvi = wi = zi for 2 6 i 6 n,
where a varies over real numbers in ∆ close to 1.
![Page 33: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/33.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Asymptotics
Theorem (Schumacher ’98)
There exists 0 < α 6 1 such that for all k ∈ 0, 1, . . .and β ∈ (0, 1), the volume form of the Poincare-typemetric is of the form
2Ω||σ||2 log2(1/||σ||2)
(1 +
ν
logα(1/||σ||2)
),
where
I Ω is a smooth volume form on X,
I σ is a canonical section of [D], ||·|| is a norm in [D],I ν lies in the Holder space of Ck,β functions in
quasi-coordinates.
![Page 34: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/34.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Contents
IntroductionStabilityHermitian-Einstein metricsKobayashi-Hitchin correspondence
Framed manifoldsDefinitionPoincare-type metric
Adapting the notionsProblemFramed stabilityFramed Hermitian-Einstein metricsRelationship
Outlook
![Page 35: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/35.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Problem
Let
I (X,D) be a canonically polarized framed manifoldand
I E a holomorphic vector bundle on X.
Questions
1. How can we define a notion of “framed stability”of E?
2. What should a “framed Hermitian-Einstein metric”in E be?
3. Do we obtain existence and uniqueness of framedHermitian-Einstein metrics in the case of framedstability?
![Page 36: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/36.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Problem
Let
I (X,D) be a canonically polarized framed manifoldand
I E a holomorphic vector bundle on X.
Questions
1. How can we define a notion of “framed stability”of E?
2. What should a “framed Hermitian-Einstein metric”in E be?
3. Do we obtain existence and uniqueness of framedHermitian-Einstein metrics in the case of framedstability?
![Page 37: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/37.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Problem
Let
I (X,D) be a canonically polarized framed manifoldand
I E a holomorphic vector bundle on X.
Questions
1. How can we define a notion of “framed stability”of E?
2. What should a “framed Hermitian-Einstein metric”in E be?
3. Do we obtain existence and uniqueness of framedHermitian-Einstein metrics in the case of framedstability?
![Page 38: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/38.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Problem
Let
I (X,D) be a canonically polarized framed manifoldand
I E a holomorphic vector bundle on X.
Questions
1. How can we define a notion of “framed stability”of E?
2. What should a “framed Hermitian-Einstein metric”in E be?
3. Do we obtain existence and uniqueness of framedHermitian-Einstein metrics in the case of framedstability?
![Page 39: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/39.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Problem
Let
I (X,D) be a canonically polarized framed manifoldand
I E a holomorphic vector bundle on X.
Questions
1. How can we define a notion of “framed stability”of E?
2. What should a “framed Hermitian-Einstein metric”in E be?
3. Do we obtain existence and uniqueness of framedHermitian-Einstein metrics in the case of framedstability?
![Page 40: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/40.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Two notions of degree for a torsion-free coherent sheaf E :
I On X: The degree with respect to the polarizationKX ⊗ [D]:
degKX⊗[D](E) = (c1(E) ∪ c1(KX ⊗ [D])n−1) ∩ [X]
=∫Xc1(E) ∧ ωn−1,
where ω = i2π ·curvature form of a positive Hermitian
metric in KX ⊗ [D].I On X ′: The degree with respect to the
Poincare-type metric:
degX′(E) =∫X′c1(E) ∧ ωn−1
Poin .
![Page 41: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/41.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Two notions of degree for a torsion-free coherent sheaf E :
I On X: The degree with respect to the polarizationKX ⊗ [D]:
degKX⊗[D](E) = (c1(E) ∪ c1(KX ⊗ [D])n−1) ∩ [X]
=∫Xc1(E) ∧ ωn−1,
where ω = i2π ·curvature form of a positive Hermitian
metric in KX ⊗ [D].I On X ′: The degree with respect to the
Poincare-type metric:
degX′(E) =∫X′c1(E) ∧ ωn−1
Poin .
![Page 42: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/42.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Theorem (S. ’09)
For every torsion-free coherent sheaf E on X, the numberdegX′(E) is well-defined and satisfies
degKX⊗[D](E) = degX′(E).
![Page 43: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/43.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Proof.Use the results on the asymptotics of the Poincare-typemetric. More precisely:
I For convenience only n = 2.
I Exhaustion
X ′ =⋃ε>0
Xε with Xε = x ∈ X : ||σ(x)|| > ε.
I We have
degKX⊗[D](E) = limε→0
∫Xε
c1(E) ∧ ω,
degX′(E) = limε→0
∫Xε
c1(E) ∧ ωPoin.
![Page 44: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/44.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Proof.Use the results on the asymptotics of the Poincare-typemetric. More precisely:
I For convenience only n = 2.
I Exhaustion
X ′ =⋃ε>0
Xε with Xε = x ∈ X : ||σ(x)|| > ε.
I We have
degKX⊗[D](E) = limε→0
∫Xε
c1(E) ∧ ω,
degX′(E) = limε→0
∫Xε
c1(E) ∧ ωPoin.
![Page 45: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/45.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Proof.Use the results on the asymptotics of the Poincare-typemetric. More precisely:
I For convenience only n = 2.
I Exhaustion
X ′ =⋃ε>0
Xε with Xε = x ∈ X : ||σ(x)|| > ε.
I We have
degKX⊗[D](E) = limε→0
∫Xε
c1(E) ∧ ω,
degX′(E) = limε→0
∫Xε
c1(E) ∧ ωPoin.
![Page 46: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/46.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Proof.Use the results on the asymptotics of the Poincare-typemetric. More precisely:
I For convenience only n = 2.
I Exhaustion
X ′ =⋃ε>0
Xε with Xε = x ∈ X : ||σ(x)|| > ε.
I We have
degKX⊗[D](E) = limε→0
∫Xε
c1(E) ∧ ω,
degX′(E) = limε→0
∫Xε
c1(E) ∧ ωPoin.
![Page 47: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/47.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Proof.
I Use asymptotic results to compare ω and ωPoin.
I We obtain
degX′(E) = degKX⊗[D](E)
− 2i limε→0
∫Xε
c1(E) ∧ ∂∂ log log(1/||σ||2)
+ i limε→0
∫Xε
c1(E) ∧ ∂∂ log(
1 +ν
logα(1/||σ||2)
)
I Show the vanishing of the integrals for ε→ 0 usingI Stokes’ theorem andI that ν is Ck,β with k > 2 in quasi-coordinates.
![Page 48: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/48.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Proof.
I Use asymptotic results to compare ω and ωPoin.
I We obtain
degX′(E) = degKX⊗[D](E)
− 2i limε→0
∫Xε
c1(E) ∧ ∂∂ log log(1/||σ||2)
+ i limε→0
∫Xε
c1(E) ∧ ∂∂ log(
1 +ν
logα(1/||σ||2)
)
I Show the vanishing of the integrals for ε→ 0 usingI Stokes’ theorem andI that ν is Ck,β with k > 2 in quasi-coordinates.
![Page 49: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/49.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Proof.
I Use asymptotic results to compare ω and ωPoin.
I We obtain
degX′(E) = degKX⊗[D](E)
− 2i limε→0
∫Xε
c1(E) ∧ ∂∂ log log(1/||σ||2)
+ i limε→0
∫Xε
c1(E) ∧ ∂∂ log(
1 +ν
logα(1/||σ||2)
)
I Show the vanishing of the integrals for ε→ 0 usingI Stokes’ theorem andI that ν is Ck,β with k > 2 in quasi-coordinates.
![Page 50: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/50.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Let E be a torsion-free coherent sheaf on X.
DefinitionWe call
deg(X,D)(E) := degKX⊗[D](E)Th.= degX′(E)
the framed degree of E .
![Page 51: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/51.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Definition
I If rank(E) > 0, we define the framed slope of E as
µ(X,D)(E) =deg(X,D)(E)
rank(E).
I We say that E is stable in the framed sense if
µ(X,D)(F) < µ(X,D)(E)
holds for every coherent subsheaf F of E with0 < rank(F) < rank(E).
RemarkNote that we only consider subsheaves F over X ratherthan X ′.
![Page 52: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/52.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Definition
I If rank(E) > 0, we define the framed slope of E as
µ(X,D)(E) =deg(X,D)(E)
rank(E).
I We say that E is stable in the framed sense if
µ(X,D)(F) < µ(X,D)(E)
holds for every coherent subsheaf F of E with0 < rank(F) < rank(E).
RemarkNote that we only consider subsheaves F over X ratherthan X ′.
![Page 53: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/53.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Definition
I If rank(E) > 0, we define the framed slope of E as
µ(X,D)(E) =deg(X,D)(E)
rank(E).
I We say that E is stable in the framed sense if
µ(X,D)(F) < µ(X,D)(E)
holds for every coherent subsheaf F of E with0 < rank(F) < rank(E).
RemarkNote that we only consider subsheaves F over X ratherthan X ′.
![Page 54: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/54.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Corollary
Every holomorphic vector bundle E which is stable in theframed sense is simple, i. e.
End(E) = C · idE .
RemarkThe simplicity seems to hold only over X and notover X ′.
![Page 55: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/55.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed stability
Corollary
Every holomorphic vector bundle E which is stable in theframed sense is simple, i. e.
End(E) = C · idE .
RemarkThe simplicity seems to hold only over X and notover X ′.
![Page 56: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/56.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed Hermitian-Einstein metrics
Consider Hermitian-Einstein metrics in E over X ′ withrespect to the Poincare-type metric.
Difficulty
For a uniqueness statement we need the simplicity of Eover X ′, which is not given in the case of framed stability.
SolutionWe impose additional conditions.
![Page 57: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/57.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed Hermitian-Einstein metrics
Consider Hermitian-Einstein metrics in E over X ′ withrespect to the Poincare-type metric.
Difficulty
For a uniqueness statement we need the simplicity of Eover X ′, which is not given in the case of framed stability.
SolutionWe impose additional conditions.
![Page 58: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/58.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed Hermitian-Einstein metrics
Consider Hermitian-Einstein metrics in E over X ′ withrespect to the Poincare-type metric.
Difficulty
For a uniqueness statement we need the simplicity of Eover X ′, which is not given in the case of framed stability.
SolutionWe impose additional conditions.
![Page 59: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/59.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed Hermitian-Einstein metrics
Set
P =h Hermitian metric in E over X ′
with
∫X′|ΛPoinFh|h dVPoin <∞
.
We know (Simpson ’88): P decomposes into componentswhich are covered by charts
s 7−→ hes,
where s is a positive definite self-adjoint section ofEnd(E) over X ′ with respect to h satisfying
supX′|s|h + ||∇′′s||L2 + ||∆′s||L1 <∞.
![Page 60: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/60.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed Hermitian-Einstein metrics
Set
P =h Hermitian metric in E over X ′
with
∫X′|ΛPoinFh|h dVPoin <∞
.
We know (Simpson ’88): P decomposes into componentswhich are covered by charts
s 7−→ hes,
where s is a positive definite self-adjoint section ofEnd(E) over X ′ with respect to h satisfying
supX′|s|h + ||∇′′s||L2 + ||∆′s||L1 <∞.
![Page 61: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/61.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed Hermitian-Einstein metrics
One of these components contains all the metrics whichextend to X. We denote this component by P0.
DefinitionBy a framed Hermitian-Einstein metric in E we mean aHermitian metric h in E over X ′ which satisfies
I h ∈ P0 and
I iΛPoinFh = λ idE over X ′ with λ ∈ R.
![Page 62: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/62.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Framed Hermitian-Einstein metrics
One of these components contains all the metrics whichextend to X. We denote this component by P0.
DefinitionBy a framed Hermitian-Einstein metric in E we mean aHermitian metric h in E over X ′ which satisfies
I h ∈ P0 and
I iΛPoinFh = λ idE over X ′ with λ ∈ R.
![Page 63: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/63.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Uniqueness
Theorem (S. ‘09)
If E is simple and h and h are framed Hermitian-Einsteinmetrics in E, we have
h = c · h
with a positive constant c.
![Page 64: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/64.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Uniqueness
Proof.
I h, h ∈ P0 guarantees that the framed degree of Ecan be computed using h or h.
I Therefore, we have
iΛPoinFh = λ idE = iΛPoinFh
with λ =2πµ(X,D)(E)
(n− 1)! volPoin(X ′).
![Page 65: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/65.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Uniqueness
Proof.
I h, h ∈ P0 guarantees that the framed degree of Ecan be computed using h or h.
I Therefore, we have
iΛPoinFh = λ idE = iΛPoinFh
with λ =2πµ(X,D)(E)
(n− 1)! volPoin(X ′).
![Page 66: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/66.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Uniqueness
Proof.
I Write h = hes, join h and h by the path ht = hets
and use Donaldson’s functional
L(t) =∫X′
∫ t
0tr(s(iΛPoinFhu − λ idE)
)du
ωnPoin
n!.
I h, h ∈ P0 guaranteesI well-definedness of L(t),
I L′′(t) = ||∂s||2L2 ,
I simplicity of E over X is sufficient to conclude thats is a multiple of idE .
![Page 67: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/67.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Uniqueness
Proof.
I Write h = hes, join h and h by the path ht = hets
and use Donaldson’s functional
L(t) =∫X′
∫ t
0tr(s(iΛPoinFhu − λ idE)
)du
ωnPoin
n!.
I h, h ∈ P0 guaranteesI well-definedness of L(t),
I L′′(t) = ||∂s||2L2 ,
I simplicity of E over X is sufficient to conclude thats is a multiple of idE .
![Page 68: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/68.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Uniqueness
Proof.
I Write h = hes, join h and h by the path ht = hets
and use Donaldson’s functional
L(t) =∫X′
∫ t
0tr(s(iΛPoinFhu − λ idE)
)du
ωnPoin
n!.
I h, h ∈ P0 guaranteesI well-definedness of L(t),
I L′′(t) = ||∂s||2L2 ,
I simplicity of E over X is sufficient to conclude thats is a multiple of idE .
![Page 69: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/69.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Uniqueness
Proof.
I Write h = hes, join h and h by the path ht = hets
and use Donaldson’s functional
L(t) =∫X′
∫ t
0tr(s(iΛPoinFhu − λ idE)
)du
ωnPoin
n!.
I h, h ∈ P0 guaranteesI well-definedness of L(t),
I L′′(t) = ||∂s||2L2 ,
I simplicity of E over X is sufficient to conclude thats is a multiple of idE .
![Page 70: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/70.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Existence in the case of framed stability
Theorem (S. ’09)
If E is stable in the framed sense, there exists a unique(up to a constant multiple) framed Hermitian-Einsteinmetric in E.
Proof.
I Carry over the arguments from the classical case.
I Critical point: In the case of framed stability, oneonly considers subsheaves of E = OX(E) over Xand not over X ′.
I However: In the classical proof, the destabilizingsubsheaf is produced from an L2
1 section of End(E).
![Page 71: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/71.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Existence in the case of framed stability
Theorem (S. ’09)
If E is stable in the framed sense, there exists a unique(up to a constant multiple) framed Hermitian-Einsteinmetric in E.
Proof.
I Carry over the arguments from the classical case.
I Critical point: In the case of framed stability, oneonly considers subsheaves of E = OX(E) over Xand not over X ′.
I However: In the classical proof, the destabilizingsubsheaf is produced from an L2
1 section of End(E).
![Page 72: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/72.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Existence in the case of framed stability
Theorem (S. ’09)
If E is stable in the framed sense, there exists a unique(up to a constant multiple) framed Hermitian-Einsteinmetric in E.
Proof.
I Carry over the arguments from the classical case.
I Critical point: In the case of framed stability, oneonly considers subsheaves of E = OX(E) over Xand not over X ′.
I However: In the classical proof, the destabilizingsubsheaf is produced from an L2
1 section of End(E).
![Page 73: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/73.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Existence in the case of framed stability
Theorem (S. ’09)
If E is stable in the framed sense, there exists a unique(up to a constant multiple) framed Hermitian-Einsteinmetric in E.
Proof.
I Carry over the arguments from the classical case.
I Critical point: In the case of framed stability, oneonly considers subsheaves of E = OX(E) over Xand not over X ′.
I However: In the classical proof, the destabilizingsubsheaf is produced from an L2
1 section of End(E).
![Page 74: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/74.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Existence in the case of framed stability
Therefore, it suffices to prove the following lemma.
Lemma (S. ’09)
We have
L21(X,End(E),Poincare) ⊂ L2
1(X,End(E)).
Proof.using the results on the asymptotics of the Poincare-typemetric
![Page 75: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/75.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Existence in the case of framed stability
Therefore, it suffices to prove the following lemma.
Lemma (S. ’09)
We have
L21(X,End(E),Poincare) ⊂ L2
1(X,End(E)).
Proof.using the results on the asymptotics of the Poincare-typemetric
![Page 76: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/76.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Contents
IntroductionStabilityHermitian-Einstein metricsKobayashi-Hitchin correspondence
Framed manifoldsDefinitionPoincare-type metric
Adapting the notionsProblemFramed stabilityFramed Hermitian-Einstein metricsRelationship
Outlook
![Page 77: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/77.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Outlook
Let (X,D) be a canonically polarized framed manifold.
Observation
I For large m, (X,D) is m-framed in the sense thatthe Q-divisor
KX +m− 1m
D
is ample.
![Page 78: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/78.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Outlook
Observation
I (Tian-Yau ’87) For such m, there exist (incomplete)Kahler-Einstein metrics gm on X ′ constructed froman initial metric of the form
i∂∂ log(
2Ωm2||σ||2(1−1/m)(1− ||σ||2/m)2
),
I whereas the Poincare-type Kahler-Einstein metricgPoin on X ′ is constructed from
i∂∂ log(
2Ω||σ||2 log2(1/||σ||2)
).
![Page 79: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/79.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Outlook
Observation
I (Tian-Yau ’87) For such m, there exist (incomplete)Kahler-Einstein metrics gm on X ′ constructed froman initial metric of the form
i∂∂ log(
2Ωm2||σ||2(1−1/m)(1− ||σ||2/m)2
),
I whereas the Poincare-type Kahler-Einstein metricgPoin on X ′ is constructed from
i∂∂ log(
2Ω||σ||2 log2(1/||σ||2)
).
![Page 80: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/80.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Outlook
QuestionCan the framed situation be seen as a “limit” of them-framed situation as m→∞?
Problems
I Kobayashi-Hitchin correspondence in the m-framedcase
I Convergence of gm to gPoin
I Convergence of the corresponding Hermitian-Einsteinmetrics
![Page 81: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/81.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Outlook
QuestionCan the framed situation be seen as a “limit” of them-framed situation as m→∞?
Problems
I Kobayashi-Hitchin correspondence in the m-framedcase
I Convergence of gm to gPoin
I Convergence of the corresponding Hermitian-Einsteinmetrics
![Page 82: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/82.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Outlook
QuestionCan the framed situation be seen as a “limit” of them-framed situation as m→∞?
Problems
I Kobayashi-Hitchin correspondence in the m-framedcase
I Convergence of gm to gPoin
I Convergence of the corresponding Hermitian-Einsteinmetrics
![Page 83: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/83.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Outlook
QuestionCan the framed situation be seen as a “limit” of them-framed situation as m→∞?
Problems
I Kobayashi-Hitchin correspondence in the m-framedcase
I Convergence of gm to gPoin
I Convergence of the corresponding Hermitian-Einsteinmetrics
![Page 84: Stability and Hermitian-Einstein metrics for vector bundles on …stemmler/tifr/... · 2010-11-06 · Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein](https://reader033.vdocuments.us/reader033/viewer/2022042116/5e93faca8ed1b60b1800d241/html5/thumbnails/84.jpg)
Framed stability andHermitian-Einstein
metrics
Matthias Stemmler
Introduction
Stability
Hermitian-Einstein metrics
K-H correspondence
Framed manifolds
Definition
Poincare-type metric
Adapting the notions
Problem
Framed stability
Framed H-E metrics
Relationship
Outlook
Thank you.