Stability and Hermitian-Einstein metrics for vector bundles on framed manifolds [Elektronische Ressource] / vorgelegt von Matthias Stemmler

Matthias Stemmler

Description

Stability and Hermitian-Einsteinmetrics for vector bundles on framedmanifoldsDissertationzur Erlangung des Doktorgradesder Naturwissenschaften (Dr. rer. nat.)demFachbereich Mathematik und Informatikder Philipps-Universitat Marburgvorgelegt vonMatthias Stemmleraus Homberg (Efze)Marburg, im Dezember 2009Vom Fachbereich Mathematik und Informatikder Philipps-Universitat Marburgals Dissertation angenommen am: 14.12.2009Erstgutachter: Prof. Dr. G. SchumacherZweitgutachter: Prof. Dr. Th. BauerTag der mundlic hen Prufung: 20.01.2010Contents1 Introduction 32 Poincare metrics and quasi-coordinates 112.1 De nition and existence of Poincare metrics . . . . . . . . . . . . . . . . . . . . . 112.2 Quasi-coordinates and H older spaces . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 A K ahler-Einstein Poincare metric . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Square-integrability for the Poincare metric . . . . . . . . . . . . . . . . . . . . . 253 Stability and Hermitian-Einstein metrics 313.1 Review of the compact case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Adaptation for the framed case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Solution of the heat equation 474.1 Existence for nite times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Convergence in in nite time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.

Sujets

• Mathematics

Informations

• Publié par: philipps-universitat_marburg
• Publié le: 01 janvier 2009
• Nombre de lectures: 9
• Langue: English

Extrait

Stability metrics for

and Hermitian-Einstein vector bundles on framed manifolds

Dissertation

zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.)

dem Fachbereich Mathematik und Informatik derPhilipps-Universita ¨ tMarburg vorgelegt von

Matthias Stemmler

aus Homberg (Efze)

Marburg, im Dezember 2009

Vom Fachbereich Mathematik und Informatik derPhilipps-Universit ¨ atMarburg als Dissertation angenommen am: 14.12.2009

Erstgutachter: Prof. Dr. G. Schumacher Zweitgutachter: Prof. Dr. Th. Bauer

Tagdermu ¨ ndlichenPru ¨ fung:20.01.2010

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5 Further aspects

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Solution of the heat equation 4.1 Existence for finite times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Convergence in infinite time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Regularity of weakly holomorphic subbundles . . . . . . . . . . . . . . . . . . . .

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Deutsche Zusammenfassung

Contents

Danksagung

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Bibliography

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Introduction

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Poincar´emetricsandquasi-coordinates 2.1DefinitionandexistenceofPoincare´metrics.............. 2.2 Quasi-coordinates and Holder spaces . . . . . . . . . . . . . . . . . . ¨ 2.3AK¨ahler-EinsteinPoincaremetric................... ´ 2.4Square-integrabilityforthePoincar´emetric..............

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Stability and Hermitian-Einstein metrics 3.1 Review of the compact case . . . . . . . . 3.2 Adaptation for the framed case . . . . . .

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Lebenslauf

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1 Introduction

This thesis is a contribution to algebraic geometry using transcendental methods. The so-called Kobayashi-Hitchin correspondence, which has been known since the 80s of the 20th century, establishes a connection between algebraic geometry and analysis by giving a relation between the algebraic-geometric notion ofstabilityof a holomorphic vector bundle on an (in the classical case)compactKa¨hlermanifoldandtheanalyticnotionofaHermitian-Einstein metricin such a vector bundle. The notion of stability considered here is the one introduced by Takemoto in [Ta72], which is also known asslope-stabilityorMumford-Takemoto stabilitymoaptc¨KG.vinecaifoldahlerman (X g) of complex dimensionn The, it can be formulated as follows.g-degreeof a torsion-free coherent analytic sheafFonXis defined as

deg g (F) = Z X c 1 (F)∧ω n−1 

wherec 1 (F) denotes the first Chern class ofFandωnufeemadlatnmrofthislhreKea¨foht metricg. IfFis non-trivial, the ratio

deg g (F) = µ g (F (nkra ) F)

of theg-degree ofFand its rank is called theg-slopeofF. A torsion-free coherent analytic sheafEonXis then calledg-semistableif

µ g (F)6µ g (E)

holds for every coherent subsheafFofEwith 0<rank(F). If, moreover, the strict inequality

µ g (F)< µ g (E)

holds for every coherent subsheafFofEwith 0<rank(F)<rank(E), thenEis calledg-stable. The notion of stability can be applied to a holomorphic vector bundleEonXby considering its sheafE=O X (E  Every) of holomorphic sections.stable holomorphic vector bundle on a compactK¨ahlermanifoldissimple the only holomorphic sections of its endomorphism, i. e. bundle are the homotheties. A Hermitian metrichinEis called ag-Hermitian-Einstein metric if √ −1Λ g F h =λ h id E with a real constantλ h , where− √ 1Λ g is the contraction withω,F h is the curvature form of the Chern connection of the Hermitian holomorphic vector bundle (E h) and id E is the identity

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1

Introduction

endomorphism ofE. In this case,λ h is called theEinstein factorofhand (E h) is called a g-Hermitian-Einstein vector bundleT.dolifanrmleahK¨ctorinfansteheEitnehdnosedepnoyl (X g) and the vector bundleE fact, we have. In

λ h = (n− 2π 1) µ ! g vo (E l g ( ) X) 

where vol g (X) is the volume ofXwith respect tog notion of a Hermitian-Einstein metric. The wasintroducedbyS.Kobayashiin[Kb80]asageneralizationofthenotionofaKa¨hler-Einstein metricinthetangentbundleofacompactK¨ahlermanifold. The Kobayashi-Hitchin correspondence states that an irreducible holomorphic vector bundle admits ag-Hermitian-Einstein metric if and only if it isg-stable. The proof of theg-stability of an irreducibleg-HietsniE-naitimreundleisdnvectorbabayhs[ieuot.SoKubL¨ke82Kbnd]a [Lue83]. The other implication, namely the existence of ag-Hermitian-Einstein metric in ag- stable holomorphic vector bundle, was shown for compact Riemann surfaces by Donaldson in [Do83], who gave a new proof of a famous theorem of Narasimhan and Seshadri [NS65]. He later proved the statement for projective-algebraic surfaces in [Do85] and, more generally, for projective-algebraic manifolds of arbitrary dimension in [Do87]. Finally, Uhlenbeck and Yau treatedthegeneralcaseofacompactK¨ahlermanifoldin[UY86](seealso[UY89]).Allproofs are based on the fact that, given a smooth Hermitian metrich 0 inE(the so-calledbackground metric), any Hermitian metrichinEcan be written ash=h 0 f, i. e.

h(s t) =h 0 (f(s) t)

for all sectionssandtofE, wherefis a smooth endomorphism ofEwhich is positive definite and self-adjoint with respect toh 0  observes that. Onehis ag-Hermitian-Einstein metric if and only iff in his proof, considerssatisfies a certain non-linear partial differential equation. Donaldson, an evolution equation of the heat conduction type involving a real parametert he obtains. After a solution defined for all non-negative values oft, he shows the convergence of the solution as tthe stability of the vector bundle and an induction argument on thegoes to infinity by using dimension of the complex manifold. The limit is a solution of the partial differential equation and thus yields the desired Hermitian-Einstein metric. Uhlenbeck and Yau, in their proof, consider a perturbed version of the partial differential equation depending on a real parameterε. They show that it has solutions for every small positiveεthese solutions converge in a good sense . If asεzero, the limit yields a Hermitian-Einstein metric. If the solutions are, however,approaches divergent, this produces a coherent subsheaf contradicting the stability of the vector bundle. The Kobayashi-Hitchin correspondence has been subject to many generalizations and adap- tations for additional structures on the holomorphic vector bundle and the underlying complex manifold.LiandYauprovedageneralizationfornon-Ka¨hlermanifoldsin[LY87],whichwas independently proved for the surface case by Buchdahl in [Bu88]. Hitchin [Hi87] and Simpson [Si88] introduced the notion of aHiggs bundleon a complex manifoldX, which is a pair (E θ) consisting of a holomorphic vector bundleEand a bundle mapθ:E→E⊗Ω 1 X  generalized. They the notions of stability and Hermitian-Einstein metrics to Higgs bundles and proved a Kobayashi- Hitchin correspondence under the integrability condition 0 =θ∧θ:E→E⊗Ω 2 X . Bando and Siu

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extended the notion of a Hermitian-Einstein metric to the case of reflexive sheaves in [BaS94] and proved a Kobayashi-Hitchin correspondence for this situation. The two generalizations for Higgs bundles and reflexive sheaves were recently combined into a generalization forHiggs sheavesby Biswas and Schumacher in [BsS09]. Moreover, the Kobayashi-Hitchin correspondence has been considered for the situation of aholomorphic pairis a holomorphic vector bundle to-, which gether with a global holomorphic section as introduced by Bradlow in [Br94], and aholomorphic triplewhich is a pair of two holomorphic vector bundles together with a global holomorphic, homomorphismbetweenthemasintroducedbyBradlowandGarcı´a-Pradain[BG96]. In this thesis, we consider the situation of aframed manifold.

Definition 1.1.

(i) Aframed manifoldis a pair (X D) consisting of a compact complex manifoldX smooth divisorDinX.

and a

(ii) A framed manifold (X D) is calledcanonically polarizedif the line bundleK X ⊗[D] is ample, whereK X denotes the canonical line bundle ofXand [D] is the line bundle associated to the divisorD.

The notion of a framed manifold, which is also referred to as alogarithmic pair, is introduced e. g. in [Sch98a] and [Sch98b] (see also [ST04]) in analogy to the concept of aframed vector bundle(cf. [Le93], simple example of a canonically polarized framed [Lue93] and [LOS93]). A manifold is (P n  V), whereP n is then-dimensional complex-projective space andVis a smooth hypersurface inP n of degree>n a canonically polarized framed manifold ( Given+ 2.X D), oneobtainsaspecialKa¨hlermetriconthecomplementX 0 :=X\DofDinX.

Theorem 1.2(R. Kobayashi, [Ko84]).If(X D)is a canonically polarized framed manifold, thereisaunique(uptoaconstantmultiple)completeKa¨hler-EinsteinmetriconX 0 with negative Ricci curvature.

This is an analogue to the classical theorem of Yau saying that every compact complex mani- foldwithamplecanonicalbundlepossessesaunique(uptoaconstantmultiple)K¨ahler-Einstein metric with negative Ricci curvature, cf. [Yau78b]. The metric from theorem 1.2, which is of Poincare´-typegrowthnearthedivisorDand will therefore be referred to as theracnioPe´emrtci, isanaturalchoicewhenlookingforasuitableKa¨hlermetriconX 0 . In [Ko84], R. Kobayashi introduces special “coordinate systems” onX 0 calledquasi-coordi- natessevensenrtianacetdtopaetlldayrewieraesehT.emr´rietPohecainahtstnO.cyase X 0 iththePoogetherwteiricosniac´rmeftbounded geometry. This concept has also been investigated by Cheng and Yau in [CY80] and by Tian and Yau in [TY87]. It will be of great importancefortheresultsofthisthesisthattheasymptoticbehaviourofthePoincare´metricis well-known. In fact, in [Sch98a], Schumacher gives an explicit description of its volume form in terms of the quasi-coordinates.

Theorem 1.3(Schumacher, [Sch98a], theorem 2).There is a number0< α61such that for allk∈ {01 . . .}andβ∈(01)vehtmulorofetfom,icostfehofmrhePoincar´emetri ν ||σ|| 2 glo 2 2 Ω (1/||σ|| 2 )   log 1 + α (1/||σ|| 2 )  withν∈ C k,β (X 0 )

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Introduction

whereΩis a smooth volume form onX,σis a canonical section of[D],||∙||is the norm induced by a Hermitian metric in[D]andC k,β (X 0 )¨HlotsehiofpacedersC k,β functions with respect to the quasi-coordinates.

Moreover,in[Sch98a],SchumachershowsthatthefundamentalformofthePoincar´emetric convergestoaK¨ahler-EinsteinmetriconDlocally uniformly when restricted to coordinate directions parallel toD this, one obtains the following result on the asymptotics of the. From Poincar´emetric.Letσbe a canonical section of [Dbe regarded as a local coordinate], which can in a neighbourhood of a pointp∈D we can choose local coordinates (. Thenσ z 2  . . .  z n ) nearp such that ifg σ¯σ ,g σ¯ ehufdnmaneatflromofthePoincar´emirtectenodec.ettfostneicffieoceht ¯ andg σσ  the entries of the corresponding inverse matrix, we have the followingetc. denote statement from [Sch02].

Proposition 1.4.With0< α61from theorem 1.3, we have

(i)

(ii)

g σ¯σ ∼ |σ| 2 log 2 (1/|σ| 2 ), g σ¯i  g ¯σ =O  |σ|log 1−α (1/|σ| 2 )  ,i j= 2 . . .  n,

(iii)g ı¯i ∼1,i= 2 . . .  nand

(iv)g ¯i →0asσ→0,i j= 2 . . .  n,i6=j.

We will use the above estimates in order to establish the relevant notions for a Kobayashi- Hitchin correspondence for vector bundles on framed manifolds. In order to do this, one can proceed in several directions. One way is to considerparabolic bundlesas introduced by Mehta and Seshadri in [MS80] on Riemann surfaces and generalized to higher-dimensional varieties by Maruyama and Yokogawa in [MY92] (see also [Bs95], [Bs97a], [Bs97b]). Let (X D) be a framed manifold andEa torsion-free coherent analytic sheaf onX. Aquasi-parabolic structureonE with respect toDis a filtration

E=F 1 (E)⊃ F 2 (E)⊃ ∙ ⊃ F ∙ ∙ l (E)⊃ F l+1 (E) =E(−D)

by coherent subsheaves, whereE(−D) is the image ofE ⊗ O X O X (−D) inE integer. Thelis called thelength of the filtration. Aparabolic structureis a quasi-parabolic structure together with a system ofparabolic weights{α 1  . . .  α l }such that 06α 1 < α 2 <∙ ∙ ∙< α l <1. The weightα i corresponds toF i (E sheaf). TheEtogether with these data is then called aparabolic sheafand denoted by (EF ∗  α ∗ ) or simply byE ∗ . Ifghaelas¨KirocmrteniX, the notion of g Given-stability can be adapted for parabolic sheaves. a parabolic sheaf (EF ∗  α ∗ ), let

E t =F i (E)(− btcD) for anyt∈R

wherebtcis the integral part oftandi∈ {1 . . .  l+ 1}is such that

α i−1 < t− btc6α i 

whereα 0 :=α l −1 andα l+1  filtration ( The:= 1.E t ) t∈R thus defined has the following properties.

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