Framed stability and Hermitian-Einstein metrics Matthias Stemmler Introduction Stability Hermitian-Einstein metrics K-H correspondence Framed manifolds Definition Poincar´ e-type metric Adapting the notions Problem Framed stability Framed H-E metrics Relationship Outlook Stability and Hermitian-Einstein metrics for vector bundles on framed manifolds Matthias Stemmler School of Mathematics Tata Institute of Fundamental Research, Mumbai Colloquium at the Institute of Mathematical Sciences Chennai, March 24, 2011
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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
School of MathematicsTata Institute of Fundamental Research, Mumbai
Colloquium at the Institute of Mathematical SciencesChennai, March 24, 2011
Adapting the notionsProblemFramed stabilityFramed Hermitian-Einstein metricsRelationship
Outlook
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.
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.
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.
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
Classical result:
Theorem (Yau ’78)
If KX is ample, there exists a unique (up to a constantmultiple) Kahler-Einstein metric on X with negative Riccicurvature.
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
In the framed situation:
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).
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
In the framed situation:
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).
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
In the framed situation:
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).
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
In the framed situation:
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
Observation:
KD = (KX ⊗ [D])|D is ample,
so there is a unique (up to a constant multiple)Kahler-Einstein metric on D.
Theorem (Schumacher ’98)
ωPoin, when restricted to
Dσ0 := σ = σ0,
converges to ωD locally uniformly as σ0 → 0.
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
Observation:
KD = (KX ⊗ [D])|D is ample,
so there is a unique (up to a constant multiple)Kahler-Einstein metric on D.
Theorem (Schumacher ’98)
ωPoin, when restricted to
Dσ0 := σ = σ0,
converges to ωD locally uniformly as σ0 → 0.
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
Choose local coordinates (σ, z2, . . . , zn) near a pointof D. Let
I gσσ etc. be the coefficients of ωPoin and
I gσσ etc. be the entries of the inverse matrix.
Theorem (Schumacher ’02)
I gσσ ∼ |σ|2 log2(1/|σ|2),
I gσk, g lσ = O(|σ| log1−α(1/|σ|2)
), k, l = 2, . . . , n,
I gkk ∼ 1, k = 2, . . . , n and
I g lk → 0 as σ → 0, k, l = 2, . . . , n, k 6= l.
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
Choose local coordinates (σ, z2, . . . , zn) near a pointof D. Let