Morse theory and stable pairs

Size: px
Start display at page:

Download "Morse theory and stable pairs"

Transcription

1 Richard A. SCGAS 2010

2 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado)

3 Outline Introduction 1 Introduction 2 3

4 Outline Introduction 1 Introduction 2 3

5 Cohomology of Kähler and hyperkähler quotients The goal is to compute the equivariant cohomology of symplectic (Kähler or hyperkähler) reductions. By the Kempf-Ness, Guillemin-Sternberg theorem, examples arise in geometric invariant theory. Kirwan, Atiyah-Bott: In the symplectic case there is a perfect" Morse stratification. HyperKähler case still unknown.

6 Infinite dimensional examples: Higgs bundles (Hitchin) Stable pairs (Bradlow) Quiver varieties (Nakajima) These involve symplectic reduction in the presence of singularities. Key points: this poses no (additional) analytic difficulties. Singularities can cause the Morse stratification to lose perfection." Computations of cohomology are (sometimes) still possible.

7 Application to representation varieties M = a closed Riemann surface g 2 π = π 1 (M, ) G = a compact connected Lie group G C = its complexification (e.g. G = U(n), G C = GL(n, C)) Representation varieties: Hom(π, G)/G moduli of G-bundles Hom(π, G C ) // G C moduli of G-Higgs bundles

8 Application to representation varieties Theorem (Daskalopoulos-Weitsman--Wilkin 09) The Poincaré polynomial is given by P SL(2,C) t (Hom(π, SL(2, C)) = (1 + t3 ) 2g (1 + t) 2g t 2g+2 (1 t 2 )(1 t 4 ) t 4g 4 + t2g+2 (1 + t) 2g (1 t 2 )(1 t 4 ) + (1 t)2g t 4g 4 4(1 + t 2 ) + (1 + t)2g t 4g 4 ( 2g 2(1 t 2 ) t t ) + (3 2g) ( 2 ) (22g 1)t 4g 4 (1 + t) 2g 2 + (1 t) 2g 2 2

9 Outline Introduction 1 Introduction 2 3

10 Symplectic reduction (X, ω) symplectic manifold (compact) G compact, connected Lie group, acting symplectically µ : X g a moment map (dµ ξ ( ) = ω(ξ, )) the Marsden-Weinstein quotient µ 1 (0)/G is a symplectic variety What is the cohomology of µ 1 (0)/G?

11 Example Introduction Take S 2 with the action of S 1 by rotation in the xy-plane. A moment map is given by the height: µ(x, y, z) = z + const. If µ = z, then µ 1 (0) is the equator with a free action of S 1. If µ = z + 1, then µ 1 (0) is a point with a trivial S 1 action. In both cases, µ 1 (0)/S 1 is a point.

12 Equivariant cohomology Z topological space with an action by G Classifying space: EG BG is a contractible, principal G-bundle Equivariant cohomology: H G (Z ) = H (Z G EG) If the action is free: H G (Z ) = H (Z /G) If the action is trivial: H G (Z ) = H (Z ) H (BG)

13 Perfect equivariant Morse theory S 1 acts freely on the sphere S, so BS 1 = CP Therefore P t (BS 1 ) = 1 + t 2 + t 4 + = 1 1 t 2 Example of the sphere: Pt S1 (S 2 ) = P t (H (S 2 ) H (BS 1 )) = 1 + t2 1 t 2 Sum over critical points of f = µ 2 : P S1 t (S 2 ) = p j crit. pt. t λ j P S1 t (p j )

14 Example Introduction f = µ 2, µ = z: two critical points of index 2, plus the minimum: 1 + t2 1 t 2 + t2 1 t 2 = 1 + t2 1 t 2 f = µ 2, µ = z + 1: two critical points, one of index 2 and one of index zero: 1 1 t 2 + t2 1 t 2 = 1 + t2 1 t 2

15 Kirwan, Atiyah-Bott For the general case, study the gradient flow f = µ 2. Critical sets η β are characterized in terms of isotropy in G. Gradient flow smooth stratification X = β I S β ; with normal bundles ν β. The corresponding long exact sequence splits H G (S β, α<β S α ) H G (S β) H G ( α<βs α ) Compute change at each step from H G (S β, α<β S α )

16 Two key steps Morse-Bott Lemma: H G (S β, α<β S α ) H G (ν β, ν β \ {0}) H λ β G (η β ) Atiyah-Bott Lemma: criterion for multiplication by the equivariant Euler class to be injective. H p G (S β, α<β S α ) H p G (S β) = H p G (ν β, ν β \ {0}) H p G (η β)

17 Perfect equivariant Morse theory Theorem (Kirwan, Atiyah-Bott) P G t (µ 1 (0)) = P G t (X) β t λ β P G t (η β ) Theorem (Kirwan surjectivity) The map on cohomology HG (X) H G (µ 1 (0)) induced from inclusion µ 1 (0) X is surjective.

18 Vector bundles on Riemann surfaces M a Riemann surface. A = {unitary connections A on hermitian bundle E M} G = gauge group of unitary endomorphisms of E. µ : A Lie(G) is given by A F A µ 2 = Yang-Mills functional

19 Minimum is the space of projectively flat connections (i.e. representation variety) 1 FA = const. I The flow converges and the Morse stratification is smooth (Daskalopoulos) Higher critical sets correspond to split Yang-Mills connections, i.e. representations to smaller groups. For example, E = L 1 L 2, d = deg L 1 > deg L 2 : η d = Jac(M) Jac(M) Morse-Bott lemma: Negative directions given by H 0,1 (L 1 L 2); λ d = dim is constant.

20 Theorem (Atiyah-Bott, Daskalopoulos) P SU(2) t (Hom(π, SU(2)) = P(BG) d=0 t λ d Pt S1 (Jac d (M)) = (1 + t3 ) 2g t 2g+2 (1 + t) 2g (1 t 2 )(1 t 4 )

21 Outline Introduction 1 Introduction 2 3

22 Holomorphic pairs B pairs = {(A, Φ) : Φ Ω 0 (E), A Φ = 0} Higher rank version of Sym d (C) Moduli space of Bradlow pairs corresponds to solutions of the τ-vortex equations: 1 FA + ΦΦ = τ I This is the moment map for the action on B pairs A Ω 0 (E).

23 Higgs bundles B higgs = {(A, Φ) : Φ Ω 1,0 (End E), A Φ = 0} Dimensional reduction of anti-self dual equations. Moduli space corresponds to solutions of the Hitchin equations: F A + [Φ, Φ ] = 0 Homeomorphic to the space of flat GL(n, C) connections (Corlette-Donaldson). Hyperkähler structure.

24 Singularities Singularities because of the jump in dim ker A. Kuranishi model: {Slice} H 1 (deformation complex) Negative directions: ν β is the intersection of negative directions with the image of the slice. Morse-Bott isomorphism: Need to define a deformation retraction.

25 Critical Higgs bundle ( ) Φ1 0 E = L 1 L 2, A = A 1 A 2, Φ = 0 Φ 2 Negative directions ν: (a, ϕ) strictly lower triangular. a H 0,1 (L 1 L 2), ϕ H 1,0 (L 1 L 2) deg(l 1 L 2) < 0 deg(l 1 L 2 K M ) is not necessarily negative Can still prove HG (X d, X d 1 ) HG (ν d, ν d \ {0})

26 Critical pair The set of pairs (A, 0), where A is minimal Yang-Mills is a critical set of B pairs. ν = H 0 (E) If d > 4g 4, then H 0 (E) is constant in dimension, and the Morse-Bott lemma holds. If d 4g 4, H 0 (E) jumps in dimension; describes Brill-Noether loci. Can still compute the contribution from this critical set.

27 Theorem (Daskalopoulos-Weitsman--Wilkin) For the case of Higgs bundles, Kirwan surjectivity holds for GL(2, C) but fails for SL(2, C). Theorem (-Wilkin) For stable pairs, Kirwan surjectivity holds, even though the Morse stratification fails to be perfect.

28 Higher degree embedding Theorem (MacDonald) The embedding Sym d M Sym d+1 M of symmetric products of Riemann surfaces induces a surjection in cohomology. Theorem (-Wilkin) The same result holds for rank 2 semistable pairs. These are important in showing splitting of the long exact sequences.

29 Conclusion More examples, general construction? Proof of the Morse-Bott lemma in general? (Kuranishi model) Hyperkähler reduction using the sum of the squares of the moment map? Finite dimensional hyperkähler example where Kirwan surjectivity fails?

MORSE THEORY, HIGGS FIELDS, AND YANG-MILLS-HIGGS FUNCTIONALS

MORSE THEORY, HIGGS FIELDS, AND YANG-MILLS-HIGGS FUNCTIONALS MORSE THEORY, HIGGS FIELDS, AND YANG-MILLS-HIGGS FUNCTIONALS STEVEN B. BRADLOW AND GRAEME WILKIN Dedicated to Professor Dick Palais on the occasion of his 80th birthday 1. Introduction In classical Morse

More information

The Yang-Mills equations over Klein surfaces

The Yang-Mills equations over Klein surfaces The Yang-Mills equations over Klein surfaces Chiu-Chu Melissa Liu & Florent Schaffhauser Columbia University (New York City) & Universidad de Los Andes (Bogotá) Seoul ICM 2014 Outline 1 Moduli of real

More information

IGA Lecture I: Introduction to G-valued moment maps

IGA Lecture I: Introduction to G-valued moment maps IGA Lecture I: Introduction to G-valued moment maps Adelaide, September 5, 2011 Review: Hamiltonian G-spaces Let G a Lie group, g = Lie(G), g with co-adjoint G-action denoted Ad. Definition A Hamiltonian

More information

Stable bundles on CP 3 and special holonomies

Stable bundles on CP 3 and special holonomies Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M

More information

Higgs Bundles and Character Varieties

Higgs Bundles and Character Varieties Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character

More information

Linear connections on Lie groups

Linear connections on Lie groups Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)

More information

Moment map flows and the Hecke correspondence for quivers

Moment map flows and the Hecke correspondence for quivers and the Hecke correspondence for quivers International Workshop on Geometry and Representation Theory Hong Kong University, 6 November 2013 The Grassmannian and flag varieties Correspondences in Morse

More information

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:

More information

arxiv:alg-geom/ v2 9 Jun 1993

arxiv:alg-geom/ v2 9 Jun 1993 GROMOV INVARIANTS FOR HOLOMORPHIC MAPS FROM RIEMANN SURFACES TO GRASSMANNIANS arxiv:alg-geom/9306005v2 9 Jun 1993 Aaron Bertram 1 Georgios Daskalopoulos Richard Wentworth 2 Dedicated to Professor Raoul

More information

2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d

2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d ON THE BRILL-NOETHER PROBLEM FOR VECTOR BUNDLES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Abstract. On an arbitrary compact Riemann surface, necessary and sucient conditions are found for the

More information

Morse Theory for the Space of Higgs Bundles

Morse Theory for the Space of Higgs Bundles Morse Theory for the Space of Higgs Bundles Graeme Wilkin Abstract The purpose of this paper is to prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on

More information

Gravitating vortices, cosmic strings, and algebraic geometry

Gravitating vortices, cosmic strings, and algebraic geometry Gravitating vortices, cosmic strings, and algebraic geometry Luis Álvarez-Cónsul ICMAT & CSIC, Madrid Seminari de Geometria Algebraica UB, Barcelona, 3 Feb 2017 Joint with Mario García-Fernández and Oscar

More information

Donaldson Invariants and Moduli of Yang-Mills Instantons

Donaldson Invariants and Moduli of Yang-Mills Instantons Donaldson Invariants and Moduli of Yang-Mills Instantons Lincoln College Oxford University (slides posted at users.ox.ac.uk/ linc4221) The ASD Equation in Low Dimensions, 17 November 2017 Moduli and Invariants

More information

Res + X F F + is defined below in (1.3). According to [Je-Ki2, Definition 3.3 and Proposition 3.4], the value of Res + X

Res + X F F + is defined below in (1.3). According to [Je-Ki2, Definition 3.3 and Proposition 3.4], the value of Res + X Theorem 1.2. For any η HH (N) we have1 (1.1) κ S (η)[n red ] = c η F. Here HH (F) denotes the H-equivariant Euler class of the normal bundle ν(f), c is a non-zero constant 2, and is defined below in (1.3).

More information

HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

More information

Pacific Journal of Mathematics

Pacific Journal of Mathematics Pacific Journal of Mathematics THE MODULI OF FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES Eugene Z. Xia Volume 195 No. 1 September 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 1, 2000 THE MODULI OF

More information

THE YANG-MILLS EQUATIONS OVER KLEIN SURFACES

THE YANG-MILLS EQUATIONS OVER KLEIN SURFACES THE YANG-MILLS EQUATIONS OVER KLEIN SURFACES CHIU-CHU MELISSA LIU AND FLORENT SCHAFFHAUSER Abstract. Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit

More information

Cohomology of Quotients in Symplectic and Algebraic Geometry. Frances Clare Kirwan

Cohomology of Quotients in Symplectic and Algebraic Geometry. Frances Clare Kirwan Cohomology of Quotients in Symplectic and Algebraic Geometry Frances Clare Kirwan CONTENTS 1 Contents 1 Introduction 2 2 The moment map 11 3 Critical points for the square of the moment map 19 4 The square

More information

Flat connections on 2-manifolds Introduction Outline:

Flat connections on 2-manifolds Introduction Outline: Flat connections on 2-manifolds Introduction Outline: 1. (a) The Jacobian (the simplest prototype for the class of objects treated throughout the paper) corresponding to the group U(1)). (b) SU(n) character

More information

Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

More information

A DIRECT EXISTENCE PROOF FOR THE VORTEX EQUATIONS OVER A COMPACT RIEMANN SURFACE

A DIRECT EXISTENCE PROOF FOR THE VORTEX EQUATIONS OVER A COMPACT RIEMANN SURFACE A DIRECT EXISTENCE PROOF FOR THE VORTEX EQUATIONS OVER A COMPACT RIEMANN SURFACE OSCAR GARCIA-PRADA ABSTRACT We give a direct proof of an existence theorem for the vortex equations over a compact Riemann

More information

Morse Theory and Applications to Equivariant Topology

Morse Theory and Applications to Equivariant Topology Morse Theory and Applications to Equivariant Topology Morse Theory: the classical approach Briefly, Morse theory is ubiquitous and indomitable (Bott). It embodies a far reaching idea: the geometry and

More information

arxiv: v2 [math.ag] 5 Jun 2018

arxiv: v2 [math.ag] 5 Jun 2018 A Brief Survey of Higgs Bundles 1 21/03/2018 Ronald A. Zúñiga-Rojas 2 Centro de Investigaciones Matemáticas y Metamatemáticas CIMM Universidad de Costa Rica UCR San José 11501, Costa Rica e-mail: ronald.zunigarojas@ucr.ac.cr

More information

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces

More information

The Topology of Higgs Bundle Moduli Spaces

The Topology of Higgs Bundle Moduli Spaces The Topology of Higgs Bundle Moduli Spaces Peter Beier Gothen This thesis is submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy at the University of Warwick. The

More information

Cohomology of the Mumford Quotient

Cohomology of the Mumford Quotient Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten

More information

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory

More information

Hamiltonian Toric Manifolds

Hamiltonian Toric Manifolds Hamiltonian Toric Manifolds JWR (following Guillemin) August 26, 2001 1 Notation Throughout T is a torus, T C is its complexification, V = L(T ) is its Lie algebra, and Λ V is the kernel of the exponential

More information

MORSE THEORY OF THE MOMENT MAP FOR REPRESENTATIONS OF QUIVERS MEGUMI HARADA AND GRAEME WILKIN

MORSE THEORY OF THE MOMENT MAP FOR REPRESENTATIONS OF QUIVERS MEGUMI HARADA AND GRAEME WILKIN MORSE THEORY OF THE MOMENT MAP FOR REPRESENTATIONS OF QUIVERS MEGUMI HARADA AND GRAEME WILKIN ABSTRACT. The results of this paper concern the Morse theory of the norm-square of the moment map on the space

More information

REAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES PAUL NORBURY AND MARC SANDERS Abstract. Let M(k; SO(n)) be the moduli space of ba

REAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES PAUL NORBURY AND MARC SANDERS Abstract. Let M(k; SO(n)) be the moduli space of ba REAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES PAUL NORBURY AND MARC SANDERS Abstract. Let M(k; SO(n)) be the moduli space of based gauge equivalence classes of SO(n) instantons on

More information

Stratified Symplectic Spaces and Reduction

Stratified Symplectic Spaces and Reduction Stratified Symplectic Spaces and Reduction Reyer Sjamaar Eugene Lerman Mathematisch Instituut der Rijksuniversiteit te Utrecht Current addresses: R. Sjamaar, Dept. of Mathematics, MIT, Cambridge, MA 02139

More information

arxiv:math/ v2 [math.sg] 19 Feb 2003

arxiv:math/ v2 [math.sg] 19 Feb 2003 THE KIRWAN MAP, EQUIVARIANT KIRWAN MAPS, AND THEIR KERNELS ariv:math/0211297v2 [math.sg] 19 eb 2003 LISA C. JEREY AND AUGUSTIN-LIVIU MARE Abstract. or an arbitrary Hamiltonian action of a compact Lie group

More information

CONFORMAL LIMITS AND THE BIA LYNICKI-BIRULA

CONFORMAL LIMITS AND THE BIA LYNICKI-BIRULA CONFORMAL LIMITS AND THE BIA LYNICKI-BIRULA STRATIFICATION OF THE SPACE OF λ-connections BRIAN COLLIER AND RICHARD WENTWORTH Abstract. The Bia lynicki-birula decomposition of the space of λ-connections

More information

Cohomology jump loci of quasi-projective varieties

Cohomology jump loci of quasi-projective varieties Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)

More information

Lecture on Equivariant Cohomology

Lecture on Equivariant Cohomology Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove

More information

EQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms

EQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms EQUIVARIANT COHOMOLOGY MARTINA LANINI AND TINA KANSTRUP 1. Quick intro Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be

More information

Delzant s Garden. A one-hour tour to symplectic toric geometry

Delzant s Garden. A one-hour tour to symplectic toric geometry Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem

More information

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be

More information

Constructing compact 8-manifolds with holonomy Spin(7)

Constructing compact 8-manifolds with holonomy Spin(7) Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June 2017. Based on Invent. math. 123 (1996), 507 552; J. Diff. Geom.

More information

Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang

Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions

More information

Geometry of the momentum map

Geometry of the momentum map Geometry of the momentum map Gert Heckman Peter Hochs February 18, 2005 Contents 1 The momentum map 1 1.1 Definition of the momentum map.................. 2 1.2 Examples of Hamiltonian actions..................

More information

Math. Res. Lett. 13 (2006), no. 1, c International Press 2006 ENERGY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ.

Math. Res. Lett. 13 (2006), no. 1, c International Press 2006 ENERGY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ. Math. Res. Lett. 3 (2006), no., 6 66 c International Press 2006 ENERY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ Katrin Wehrheim Abstract. We establish an energy identity for anti-self-dual connections

More information

Lecture 1: Introduction

Lecture 1: Introduction Lecture 1: Introduction Jonathan Evans 20th September 2011 Jonathan Evans () Lecture 1: Introduction 20th September 2011 1 / 12 Jonathan Evans () Lecture 1: Introduction 20th September 2011 2 / 12 Essentially

More information

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9

More information

k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula 20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim

More information

LECTURE 4: SYMPLECTIC GROUP ACTIONS

LECTURE 4: SYMPLECTIC GROUP ACTIONS LECTURE 4: SYMPLECTIC GROUP ACTIONS WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic circle actions We set S 1 = R/2πZ throughout. Let (M, ω) be a symplectic manifold. A symplectic S 1 -action on (M, ω) is

More information

arxiv:math/ v3 [math.sg] 22 Sep 2003

arxiv:math/ v3 [math.sg] 22 Sep 2003 THE KIRWAN MAP, EQUIVARIANT KIRWAN MAPS, AND THEIR KERNELS arxiv:math/0211297v3 [math.sg] 22 Sep 2003 LISA C. JEREY AND AUGUSTIN-LIVIU MARE Abstract. Consider a Hamiltonian action of a compact Lie group

More information

INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY

INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY YOUNG-HOON KIEM 1. Definitions and Basic Properties 1.1. Lie group. Let G be a Lie group (i.e. a manifold equipped with differentiable group operations mult

More information

CHARACTERISTIC CLASSES

CHARACTERISTIC CLASSES 1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact

More information

Enumerative Geometry: from Classical to Modern

Enumerative Geometry: from Classical to Modern : from Classical to Modern February 28, 2008 Summary Classical enumerative geometry: examples Modern tools: Gromov-Witten invariants counts of holomorphic maps Insights from string theory: quantum cohomology:

More information

Vortex equations in abelian gauged σ-models

Vortex equations in abelian gauged σ-models DATP-2004-64 arxiv:math/0411517v2 [math.dg] 10 Nov 2005 Vortex equations in abelian gauged σ-models J.. Baptista Department of Applied athematics and Theoretical Physics University of Cambridge June 2004

More information

Symplectic geometry of deformation spaces

Symplectic geometry of deformation spaces July 15, 2010 Outline What is a symplectic structure? What is a symplectic structure? Denition A symplectic structure on a (smooth) manifold M is a closed nondegenerate 2-form ω. Examples Darboux coordinates

More information

APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD. 1. Introduction

APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD. 1. Introduction APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD INDRANIL BISWAS, JOHN LOFTIN, AND MATTHIAS STEMMLER Abstract. Given a flat Higgs vector bundle (E,, ϕ) over a compact

More information

Manifolds with holonomy. Sp(n)Sp(1) SC in SHGAP Simon Salamon Stony Brook, 9 Sep 2016

Manifolds with holonomy. Sp(n)Sp(1) SC in SHGAP Simon Salamon Stony Brook, 9 Sep 2016 Manifolds with holonomy Sp(n)Sp(1) SC in SHGAP Simon Salamon Stony Brook, 9 Sep 2016 The list 1.1 SO(N) U( N 2 ) Sp( N 4 )Sp(1) SU( N 2 ) Sp( N 4 ) G 2 (N =7) Spin(7) (N =8) All act transitively on S N

More information

The Based Loop Group of SU(2) Lisa Jeffrey. Department of Mathematics University of Toronto. Joint work with Megumi Harada and Paul Selick

The Based Loop Group of SU(2) Lisa Jeffrey. Department of Mathematics University of Toronto. Joint work with Megumi Harada and Paul Selick The Based Loop Group of SU(2) Lisa Jeffrey Department of Mathematics University of Toronto Joint work with Megumi Harada and Paul Selick I. The based loop group ΩG Let G = SU(2) and let T be its maximal

More information

SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

More information

Cohomology jump loci of local systems

Cohomology jump loci of local systems Cohomology jump loci of local systems Botong Wang Joint work with Nero Budur University of Notre Dame June 28 2013 Introduction Given a topological space X, we can associate some homotopy invariants to

More information

Riemannian Curvature Functionals: Lecture III

Riemannian Curvature Functionals: Lecture III Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli

More information

Lecture 19: Equivariant cohomology I

Lecture 19: Equivariant cohomology I Lecture 19: Equivariant cohomology I Jonathan Evans 29th November 2011 Jonathan Evans () Lecture 19: Equivariant cohomology I 29th November 2011 1 / 13 Last lecture we introduced something called G-equivariant

More information

HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES INTRODUCTION

HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES INTRODUCTION ËÁ Ì ÖÛÒ ËÖĐÓÒÖ ÁÒØÖÒØÓÒÐ ÁÒ ØØÙØ ÓÖ ÅØÑØÐ ÈÝ ÓÐØÞÑÒÒ ¹½¼¼ ÏÒ Ù ØÖ ÀØÒßÃÓÝ ÓÖÖ ÓÒÒ ÉÙÚÖ Ò ÎÓÖØ ÄÙ ÐÚÖÞßÓÒ ÙÐ Ç Ö ÖßÈÖ ÎÒÒ ÈÖÖÒØ ËÁ ½¾ ¾¼¼ µ ÂÒÙÖÝ ½ ¾¼¼ ËÙÓÖØ Ý Ø Ù ØÖÒ ÖÐ ÅÒ ØÖÝ Ó ÙØÓÒ ËÒ Ò ÙÐØÙÖ ÚÐÐ Ú

More information

The derived category of a GIT quotient

The derived category of a GIT quotient September 28, 2012 Table of contents 1 Geometric invariant theory 2 3 What is geometric invariant theory (GIT)? Let a reductive group G act on a smooth quasiprojective (preferably projective-over-affine)

More information

TITLE AND ABSTRACT OF HALF-HOUR TALKS

TITLE AND ABSTRACT OF HALF-HOUR TALKS TITLE AND ABSTRACT OF HALF-HOUR TALKS Speaker: Jim Davis Title: Mapping tori of self-homotopy equivalences of lens spaces Abstract: Jonathan Hillman, in his study of four-dimensional geometries, asked

More information

Intersection of stable and unstable manifolds for invariant Morse functions

Intersection of stable and unstable manifolds for invariant Morse functions Intersection of stable and unstable manifolds for invariant Morse functions Hitoshi Yamanaka (Osaka City University) March 14, 2011 Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and

More information

K-stability and Kähler metrics, I

K-stability and Kähler metrics, I K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates

More information

A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM CONTENTS

A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM CONTENTS A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM TIMOTHY E. GOLDBERG ABSTRACT. This is a handout for a talk given at Bard College on Tuesday, 1 May 2007 by the author. It gives careful versions

More information

On the Virtual Fundamental Class

On the Virtual Fundamental Class On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview Donaldson-Thomas theory: counting invariants

More information

NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD

NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD INDRANIL BISWAS Abstract. Our aim is to review some recent results on holomorphic principal bundles over a compact Kähler manifold.

More information

LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. Contents

LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. Contents LOCALIZATION AND SPECIALIZATION FOR HAMILTONIAN TORUS ACTIONS. MILENA PABINIAK Abstract. We consider a Hamiltonian action of n-dimensional torus, T n, on a compact symplectic manifold (M, ω) with d isolated

More information

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

More information

Atiyah classes and homotopy algebras

Atiyah classes and homotopy algebras Atiyah classes and homotopy algebras Mathieu Stiénon Workshop on Lie groupoids and Lie algebroids Kolkata, December 2012 Atiyah (1957): obstruction to existence of holomorphic connections Rozansky-Witten

More information

Torus actions and Ricci-flat metrics

Torus actions and Ricci-flat metrics Department of Mathematics, University of Aarhus November 2016 / Trondheim To Eldar Straume on his 70th birthday DFF - 6108-00358 Delzant HyperKähler G2 http://mscand.dk https://doi.org/10.7146/math.scand.a-12294

More information

The Topology and Geometry of Hyperkähler Quotients. Jonathan Michael Fisher

The Topology and Geometry of Hyperkähler Quotients. Jonathan Michael Fisher The Topology and Geometry of Hyperkähler Quotients by Jonathan Michael Fisher A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics

More information

The moduli spaces of symplectic vortices on an orbifold Riemann surface Hironori Sakai

The moduli spaces of symplectic vortices on an orbifold Riemann surface Hironori Sakai The moduli spaces of symplectic vortices on an orbifold Riemann surface Hironori Sakai Higher Structures in Algebraic Analysis Feb 13, 2014 Mathematisches Institut, WWU Münster http://sakai.blueskyproject.net/

More information

Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current

Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current Author, F., and S. Author. (2015) Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current, International Mathematics Research Notices, Vol. 2015, Article ID rnn999, 7 pages. doi:10.1093/imrn/rnn999

More information

12 Geometric quantization

12 Geometric quantization 12 Geometric quantization 12.1 Remarks on quantization and representation theory Definition 12.1 Let M be a symplectic manifold. A prequantum line bundle with connection on M is a line bundle L M equipped

More information

GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH. Dedicated to Professor Karen K. Uhlenbeck, on the occasion of her 60th birthday.

GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH. Dedicated to Professor Karen K. Uhlenbeck, on the occasion of her 60th birthday. CONVERGENCE PROPERTIES OF THE YANG-MILLS FLOW ON KÄHLER SURFACES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Dedicated to Professor Karen K. Uhlenbeck, on the occasion of her 60th birthday. Abstract.

More information

Infinitesimal Einstein Deformations. Kähler Manifolds

Infinitesimal Einstein Deformations. Kähler Manifolds on Nearly Kähler Manifolds (joint work with P.-A. Nagy and U. Semmelmann) Gemeinsame Jahrestagung DMV GDM Berlin, March 30, 2007 Nearly Kähler manifolds Definition and first properties Examples of NK manifolds

More information

LAGRANGIAN BOUNDARY CONDITIONS FOR ANTI-SELF-DUAL INSTANTONS AND THE ATIYAH-FLOER CONJECTURE. Katrin Wehrheim

LAGRANGIAN BOUNDARY CONDITIONS FOR ANTI-SELF-DUAL INSTANTONS AND THE ATIYAH-FLOER CONJECTURE. Katrin Wehrheim LAGRANGIAN BOUNDARY CONDITIONS FOR ANTI-SELF-DUAL INSTANTONS AND THE ATIYAH-FLOER CONJECTURE Katrin Wehrheim 1. Introduction The purpose of this survey is to explain an approach to the Atiyah-Floer conjecture

More information

AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY

AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY 1 AN INTRODUCTION TO THE MASLOV INDEX IN SYMPLECTIC TOPOLOGY Andrew Ranicki and Daniele Sepe (Edinburgh) http://www.maths.ed.ac.uk/ aar Maslov index seminar, 9 November 2009 The 1-dimensional Lagrangians

More information

Cohomology and Vector Bundles

Cohomology and Vector Bundles Cohomology and Vector Bundles Corrin Clarkson REU 2008 September 28, 2008 Abstract Vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes

More information

Quantising noncompact Spin c -manifolds

Quantising noncompact Spin c -manifolds Quantising noncompact Spin c -manifolds Peter Hochs University of Adelaide Workshop on Positive Curvature and Index Theory National University of Singapore, 20 November 2014 Peter Hochs (UoA) Noncompact

More information

The symplectic structure on moduli space (in memory of Andreas Floer)

The symplectic structure on moduli space (in memory of Andreas Floer) The symplectic structure on moduli space (in memory of Andreas Floer) Alan Weinstein Department of Mathematics University of California Berkeley, CA 94720 USA (alanw@math.berkeley.edu) 1 Introduction The

More information

ALF spaces and collapsing Ricci-flat metrics on the K3 surface

ALF spaces and collapsing Ricci-flat metrics on the K3 surface ALF spaces and collapsing Ricci-flat metrics on the K3 surface Lorenzo Foscolo Stony Brook University Recent Advances in Complex Differential Geometry, Toulouse, June 2016 The Kummer construction Gibbons

More information

EXISTENCE THEORY FOR HARMONIC METRICS

EXISTENCE THEORY FOR HARMONIC METRICS EXISTENCE THEORY FOR HARMONIC METRICS These are the notes of a talk given by the author in Asheville at the workshop Higgs bundles and Harmonic maps in January 2015. It aims to sketch the proof of the

More information

NilBott Tower of Aspherical Manifolds and Torus Actions

NilBott Tower of Aspherical Manifolds and Torus Actions NilBott Tower of Aspherical Manifolds and Torus Actions Tokyo Metropolitan University November 29, 2011 (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29,

More information

ON ALGEBRAIC ASPECTS OF THE MODULI SPACE OF FLAT CONNECTIONS

ON ALGEBRAIC ASPECTS OF THE MODULI SPACE OF FLAT CONNECTIONS ON ALGEBRAIC ASPECTS OF THE MODULI SPACE OF FLAT CONNECTIONS VICTORIA HOSKINS 1. Overview These notes are based on two seminar talks given in a seminar on moduli of flat connections organised by Ivan Contreras

More information

A Crash Course of Floer Homology for Lagrangian Intersections

A Crash Course of Floer Homology for Lagrangian Intersections A Crash Course of Floer Homology for Lagrangian Intersections Manabu AKAHO Department of Mathematics Tokyo Metropolitan University akaho@math.metro-u.ac.jp 1 Introduction There are several kinds of Floer

More information

HAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY

HAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY HAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY TIMOTHY E. GOLDBERG These are notes for a talk given in the Lie Groups Seminar at Cornell University on Friday, September 25, 2009. In retrospect, perhaps

More information

1. Classifying Spaces. Classifying Spaces

1. Classifying Spaces. Classifying Spaces Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.

More information

The Calabi Conjecture

The Calabi Conjecture The Calabi Conjecture notes by Aleksander Doan These are notes to the talk given on 9th March 2012 at the Graduate Topology and Geometry Seminar at the University of Warsaw. They are based almost entirely

More information

arxiv: v1 [math.ag] 24 Sep 2010

arxiv: v1 [math.ag] 24 Sep 2010 arxiv:1009.4827v1 [math.ag] 24 Sep 2010 Edinburgh Lectures on Geometry, Analysis and Physics Sir Michael Atiyah Notes by Thomas Köppe Contents Preface iii 1 From Euclidean 3-space to complex matrices 1

More information

LECTURE 26: THE CHERN-WEIL THEORY

LECTURE 26: THE CHERN-WEIL THEORY LECTURE 26: THE CHERN-WEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a k-tensor T k V is called symmetric if

More information

Patrick Iglesias-Zemmour

Patrick Iglesias-Zemmour Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries

More information

Quadratic differentials as stability conditions. Tom Bridgeland (joint work with Ivan Smith)

Quadratic differentials as stability conditions. Tom Bridgeland (joint work with Ivan Smith) Quadratic differentials as stability conditions Tom Bridgeland (joint work with Ivan Smith) Our main result identifies spaces of meromorphic quadratic differentials on Riemann surfaces with spaces of stability

More information

THE MAPPING CLASS GROUP ACTS REDUCIBLY ON SU(n)-CHARACTER VARIETIES

THE MAPPING CLASS GROUP ACTS REDUCIBLY ON SU(n)-CHARACTER VARIETIES THE MAPPING CLASS GROUP ACTS REDUCIBLY ON SU(n)-CHARACTER VARIETIES WILLIAM M. GOLDMAN Abstract. When G is a connected compact Lie group, and π is a closed surface group, then Hom(π, G)/G contains an open

More information

DIFFERENTIAL GEOMETRY AND THE QUATERNIONS. Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013

DIFFERENTIAL GEOMETRY AND THE QUATERNIONS. Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013 DIFFERENTIAL GEOMETRY AND THE QUATERNIONS Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013 16th October 1843 ON RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 1 SHIING-SHEN CHERN Introduction.

More information

Math 550 / David Dumas / Fall Problems

Math 550 / David Dumas / Fall Problems Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;

More information

IV. Birational hyperkähler manifolds

IV. Birational hyperkähler manifolds Université de Nice March 28, 2008 Atiyah s example Atiyah s example f : X D family of K3 surfaces, smooth over D ; X smooth, X 0 has one node s. Atiyah s example f : X D family of K3 surfaces, smooth over

More information

ALF spaces and collapsing Ricci-flat metrics on the K3 surface

ALF spaces and collapsing Ricci-flat metrics on the K3 surface ALF spaces and collapsing Ricci-flat metrics on the K3 surface Lorenzo Foscolo Stony Brook University Simons Collaboration on Special Holonomy Workshop, SCGP, Stony Brook, September 8 2016 The Kummer construction

More information

Fundamental Lemma and Hitchin Fibration

Fundamental Lemma and Hitchin Fibration Fundamental Lemma and Hitchin Fibration Gérard Laumon CNRS and Université Paris-Sud September 21, 2006 In order to: compute the Hasse-Weil zeta functions of Shimura varieties (for example A g ), prove

More information