# Stable bundles on CP 3 and special holonomies

Size: px
Start display at page:

Transcription

1 Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26,

2 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M is a Riemannian structure g and a triple of complex structures I, J, K, satisfying quaternionic relations I J = J I = K, such that g is Kähler for I, J, K. REMARK: A hyperkähler manifold has three symplectic forms ω I := g(i, ), ω J := g(j, ), ω K := g(k, ). REMARK: This is equivalent to I = J = K = 0: the parallel translation along the connection preserves I, J, K. DEFINITION: Let M be a Riemannian manifold, x M a point. The subgroup of GL(T x M) generated by parallel translations (along all paths) is called the holonomy group of M. REMARK: A hyperkähler manifold can be defined as a manifold which has holonomy in Sp(n) (the group of all endomorphisms preserving I, J, K). 2

3 Holomorphically symplectic manifolds DEFINITION: A holomorphically symplectic manifold is a complex manifold equipped with non-degenerate, holomorphic (2, 0)-form. REMARK: Hyperkähler manifolds are holomorphically symplectic. Indeed, Ω := ω J + 1 ω K is a holomorphic symplectic form on (M, I). THEOREM: (Calabi-Yau) A compact, Kähler, holomorphically symplectic manifold admits a unique hyperkähler metric in any Kähler class. EXAMPLE: An even-dimensional complex torus. REMARK: Take a symmetric square Sym 2 T, with a natural action of T, and let T [2] be a blow-up of a singular divisor. Then T [2] is naturally isomorphic to the Kummer surface T/±1. DEFINITION: A K3 surface is a complex 2-manifold obtained as a deformation of a Kummer surface. REMARK: A K3 surface is always hyperkähler. Any hyperkähler manifold of real dimension 4 is isomorphic to a torus or a K3 surface. 3

4 Complexification of a manifold DEFINITION: Let M be a complex manifold, equipped with an anticomplex involution ι. The fixed point set M R of ι is called a real analytic manifold, and a germ of M in M R is called a complexification of M R. QUESTION: What is a complexification of a Kähler manifold (considered as real analytic variety)? THEOREM: (D. Kaledin, B. Feix) Let M be a real analytic Kähler manifold, and M C its complexification. Then M C admits a hyperkähler structure, determined uniquely and functorially by the Kähler structure on M. QUESTION: What is a complexification of a hyperkähler manifold? THIS IS THE MAIN SUBJECT OF TODAY S TALK. (A joint work with Marcos Jardim). 4

5 Twistor space DEFINITION: Induced complex structures on a hyperkähler manifold are complex structures of form S 2 = {L := ai +bj +ck, a 2 +b 2 +c 2 = 1.} They are usually non-algebraic. Indeed, if M is compact, for generic a, b, c, (M, L) has no divisors (Fujiki). DEFINITION: A twistor space Tw(M) of a hyperkähler manifold is a complex manifold obtained by gluing these complex structures into a holomorphic family over CP 1. More formally: Let Tw(M) := M S 2. Consider the complex structure I m : T m M T m M on M induced by J S 2 H. Let I J denote the complex structure on S 2 = CP 1. The operator I Tw = I m I J : T x Tw(M) T x Tw(M) satisfies I Tw = Id. It defines an almost complex structure on Tw(M). This almost complex structure is known to be integrable (Obata, Salamon) EXAMPLE: If M = H n, Tw(M) = Tot(O(1) n ) = CP 2n+1 \CP 2n 1 REMARK: For M compact, Tw(M) never admits a Kähler structure. 5

6 Rational curves on Tw(M). REMARK: The twistor space has many rational curves. In fact, it is rationally connected (Campana). DEFINITION: Denote by Sec(M) the space of holomorphic sections π of the twistor fibration Tw(M) CP 1. DEFINITION: For each point m M, one has a horizontal section C m := {m} CP 1 of π. The space of horizontal sections is denoted Sec hor (M) Sec(M) REMARK: The space of horizontal sections of π is identified with M. The normal bundle NC m = O(1) dim M. Therefore, some neighbourhood of Sec hor (M) Sec(M) is a smooth manifold of dimension 2 dim M. DEFINITION: A twistor section C Tw(M) is called regular, if NC = O(1) dim M. CLAIM: For any I J CP n, consider the evaluation map Sec(M) E I,J (M, I) (M, J), s s(i) s(j). Then E I,J is an isomorphism around the set Sec 0 (M) of regular sections. 6

7 Complexification of a hyperkähler manifold. ι REMARK: Consider an anticomplex involution Tw(M) Tw(M) mapping (m, t) to (m, i(t)), where i : CP 1 CP 1 is a central symmetry. Then Sec hor (M) = M is a component of the fixed set of ι. COROLLARY: Sec(M) is a complexification of M. QUESTION: What are geometric structures on Sec(M)? Answer 1: For compact M, Sec(M) is holomorphically convex (Stein if dim M = 2). Answer 2: The space Sec 0 (M) admits a holomorphic, torsion-free connection with holonomy Sp(n, C) acting on C 2n C 2. This is the special holonomy mentioned in the title of the talk. REMARK: Merkulov, Schwachhöfer: classification of irreducible special holonomy. Sp(n, C)-action on C 2n C 2 is non-irreducible. 7

8 Mathematical instantons DEFINITION: A mathematical instanton on CP 3 is a stable rank 2 bundle B with c 1 (B) = 0 and H 1 (B( 1)) = 0. A framed instanton is a mathematical instanton equipped with a trivialization of B l for some fixed line l = CP 1 CP 3. REMARK: The space M c of framed instantons with c 2 = c is a principal SL(2)-bundle over the space of all mathematical instantons trivial on l. DEFINITION: An instanton on CP 2 is a stable bundle B with c 1 (B) = 0. A framed instanton is an instanton equipped with a trivialization B x for some fixed point x CP 2. THEOREM: (Atiyah-Drinfeld-Hitchin-Manin) The space M r,c of framed instantons on CP 2 is smooth, connected, hyperkähler. THEOREM: (Jardim V.) The space M c of framed mathematical instantons on CP 3 is naturally identified with the space of twistor sections Sec(M 2,c ). 8

9 The space of instantons on CP 3 CONJECTURE: The space of mathematical instantons is smooth and connected. THEOREM: (Grauert-Müllich, Hauzer-Langer) Every mathematical instanton on CP 3 is trivial on some line l CP 3. COROLLARY: The space of mathematical instantons is covered by Zariski open, dense subvarieties which take form M c /SL(2, C). COROLLARY: To prove that the space of mathematical instantons is smooth and connected it would suffice to prove it for M c. THEOREM: (Jardim V.) The space M c is smooth and connected. REMARK: To prove that M r,c is smooth, one could use hyperkähler reduction. To prove that M c is smooth and connected, we develop trihyperkähler reduction, which is a reduction defined on manifolds with holonomy in Sp(n, C) acting C 2n C 2. We prove that M c is a trihyperkähler quotient of a vector space by a reductive group action, hence smooth. 9

10 Holomorphic 3-webs. DEFINITION: Let M be a complex manifold, and S 1, S 2, S 3 integrable, pairwise transversal holomorphic sub-bundles in T M, of dimension 1 2 dim M. Then (S 1, S 2, S 3 ) is called a holomorphic 3-web on M. REMARK: On smooth manifolds, the theory of 3-webs is due to Chern and Blaschke (1930-ies). THEOREM: (Ph. D. thesis of Chern, 1936) Let S 1, S 2, S 3 be a holomorphic 3-web on a complex manifold M. Then there exists a unique holomorphic connection on M which preserves the sub-bundles S i, and such that its torsion T satisfies T (S 1, S 2 ) = 0. 10

11 Holomorphic SL(2)-webs. DEFINITION: A holomorphic 3-web on a complex manifold M is called an SL(2)-web if the projection operators P i,j of T M to S i along S j generate the standard action of Mat(2) on C 2 C n, for any nilpotent v Mat(2), the bundle v(t M) T M is involutive. REMARK: The set of v Mat(2) with rk v = 1 satisfies PV = CP 1, hence the sub-bundles v(t M) T M are parametrized by CP 1. An SL(2)-web is determined by a set of sub-bundles S t T M, t CP 1, which are pairwise transversal and involutive. EXAMPLE: Consider a hyperkähler manifold M. Let I CP 1, and ev I : Sec 0 (M) (M, I) be an evaluation map putting S Sec 0 (M) to S(I). Then ker Dev I, I CP 1 is an SL(2)-web. THEOREM: (Jardim V.) Let S t T M, t CP 1 be an SL(2)-web on M, and t 1, t 2, t 3 CP 1 distinct points. Then the Chern connection of a 3-web S t1, S t2, S t3 is a torsion-free affine holomorphic connection with holonomy in GL(n, C) acting on C 2n = C n C 2, and independent from the choice of t i. 11

12 Trisymplectic manifolds DEFINITION: Let Ω be a 3-dimensional space of holomorphic symplectic 2-forms on a manifold. Suppose that Ω contains a non-degenerate 2-form For each non-zero degenerate Ω Ω, one has rk Ω = 1 2 dim V. Then Ω is called a trisymplectic structure on M. REMARK: The bundles ker Ω are involutive, because Ω is closed. REMARK: This notion is similar to hypersymplectic structures (which are a triple of closed forms on a real manifold with the same rank condition). THEOREM: (Jardim V.) For any trisymplectic structure on M, the bundles ker Ω T M define an SL(2)-web. Moreover, the Chern connection of this SL(2)-web preserves all forms in Ω. REMARK: In this case, the Chern connection has holonomy in Sp(n, C) acting on C 2n C 2. 12

13 Trisymplectic structure on Sec 0 (M) EXAMPLE: Consider a hyperkähler manifold M. Let I CP 1, and ev I : Sec 0 (M) (M, I) be an evaluation map putting S Sec 0 (M) to S(I). Denote by Ω I the holomorphic symplectic form on (M, I). Then ev I Ω I, I CP 1 generate a trisymplectic structure. COROLLARY: Sec 0 (M) is equipped with a holomorphic, torsion-free connection with holonomy in Sp(n, C). 13

14 Hyperkähler reduction DEFINITION: Let G be a compact Lie group acting on a hyperkähler manifold M by hyperkähler isometries. A hyperkähler moment map is a G-equivariant smooth map µ : M g R 3 such that dµ i (v), ξ = ω i (ξ, v), for every v T M, ξ g and i = 1, 2, 3, where ω i is one the the Kähler forms associated with the hyperkähler structure. DEFINITION: The quotient manifold M /G := µ 1 (ξ 1, ξ 2, ξ 3 )/G is called the hyperkähler quotient of M. THEOREM: (Hitchin, Karlhede, Lindström, Roček)The quotient M /G is hyperkaehler. 14

15 Trihyperkähler reduction DEFINITION: A trisymplectic moment map µ C : M g R Ω takes vectors Ω Ω, g g = Lie(G) and maps them to a holomorphic function f O M, such that df = Ω g, where Ω g denotes the contraction of Ω and the vector field g DEFINITION: Let (M, Ω, S t ) be a trisymplectic structure on a complex manifold M. Assume that M is equipped with an action of a compact Lie group G preserving Ω, and an equivariant trisymplectic moment map µ C : M g R Ω. Let µ 1 C (0) be the corresponding level set of the moment map. Consider the action of the complex Lie group G C on µ 1 C (c). Assume that it is proper and free. Then the quotient µ 1 C (c)/g C is a smooth manifold called the trisymplectic quotient of (M, Ω, S t ), denoted by M /G. THEOREM: Suppose that the restriction of Ω to g T M is non-degenerate. Then M /G trisymplectic. 15

16 Hyperholomorphic connections REMARK: Let M be a hyperkähler manifold. The group SU(2) of unitary quaternions acts on Λ (M) multiplicatively. DEFINITION: A hyperholomorphic connection on a vector bundle B over M is a Hermitian connection with SU(2)-invariant curvature Θ Λ 2 (M) End(B). REMARK: Since the invariant 2-forms satisfy Λ 2 (M) SU(2) = I CP 1 Λ 1,1 I (M), a hyperholomorphic connection defines a holomorphic structure on B for each I induced by quaternions. REMARK: Let M be a compact hyperkähler manifold. Then SU(2) preserves harmonic forms, hence acts on cohomology. 16

17 Hyperholomorphic bundles and twistor sections THEOREM: (V., 1995) Let B be a stable bundle on a compact hyperkähler manifold with c 1 (B) and c 2 (B) SU(2)-invariant. Then B admits a unique hyperholomorphic connection. DEFINITION: A stable bundle with c 1 (B) and c 2 (B) SU(2)-invariant is called hyperholomorphic. COROLLARY: The space of deformations of a hyperholomorphic bundle is a hyperkähler manifold. COROLLARY: A hyperholomorphic bundle can be lifted to a holomorphic bundle B on a twistor space. THEOREM: (Kaledin V., 1996) The space Sec 0 (Def(B)) admits an open embedding to a space Def(B) of deformations of B on Tw(M), and its image is Zariski dense. REMARK: Let M 2,c be the space of framed instantons on C 2. The above theorem gives an embedding from Sec 0 (M 2,c ) to the space of holomorphic bundles on Tw(C 2 ) = CP 3 \CP 1. 17

18 Hyperholomorphic bundles and mathematical instantons REMARK: Using the monad description of mathematical instantons, we prove that that the map Sec 0 (M 2,c ) M c to the space of mathematical instantons is an isomorphism (Frenkel-Jardim, Jardim-V.). REMARK: The smoothness of the space Sec 0 (M 2,c ) = M c follows from the trihyperkähler reduction procedure: THEOREM: Let M be flat hyperkähler manifold, and G a compact Lie group acting on M by hyperkähler automorphisms. Suppose that the hyperkähler moment map exists, and the hyperkähler quotient M /G is smooth. Then there exists an open embedding Sec 0 (M) /G Ψ Sec 0 (M /G), which is compatible with the trisymplectic structures on Sec 0 (M) /G and Sec 0 (M /G). THEOREM: If M is the quiver space which gives M /G = M 2,c, Ψ gives an isomorphism Sec 0 (M) /G = Sec 0 (M /G). 18

### 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

### Algebraic geometry over quaternions

Algebraic geometry over quaternions Misha Verbitsky November 26, 2007 Durham University 1 History of algebraic geometry. 1. XIX centrury: Riemann, Klein, Poincaré. Study of elliptic integrals and elliptic

### 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

### 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)

### Morse theory and stable pairs

Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction

### Minimal surfaces in quaternionic symmetric spaces

From: "Geometry of low-dimensional manifolds: 1", C.U.P. (1990), pp. 231--235 Minimal surfaces in quaternionic symmetric spaces F.E. BURSTALL University of Bath We describe some birational correspondences

### H-projective structures and their applications

1 H-projective structures and their applications David M. J. Calderbank University of Bath Based largely on: Marburg, July 2012 hamiltonian 2-forms papers with Vestislav Apostolov (UQAM), Paul Gauduchon

### 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:

### Two simple ideas from calculus applied to Riemannian geometry

Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University

### 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

### 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.

### Multiplicity of singularities is not a bi-lipschitz invariant

Multiplicity of singularities is not a bi-lipschitz invariant Misha Verbitsky Joint work with L. Birbrair, A. Fernandes, J. E. Sampaio Geometry and Dynamics Seminar Tel-Aviv University, 12.12.2018 1 Zariski

### η = (e 1 (e 2 φ)) # = e 3

Research Statement My research interests lie in differential geometry and geometric analysis. My work has concentrated according to two themes. The first is the study of submanifolds of spaces with riemannian

### Twistor constructions of quaternionic manifolds and asymptotically hyperbolic Einstein Weyl spaces. Aleksandra Borówka. University of Bath

Twistor constructions of quaternionic manifolds and asymptotically hyperbolic Einstein Weyl spaces submitted by Aleksandra Borówka for the degree of Doctor of Philosophy of the University of Bath Department

### Deformations of trianalytic subvarieties nal version, Oct Deformations of trianalytic subvarieties of. hyperkahler manifolds.

Deformations of trianalytic subvarieties of hyperkahler manifolds. Misha Verbitsky, 1 verbit@thelema.dnttm.rssi.ru, verbit@math.ias.edu Contents Let M be a compact complex manifold equipped with a hyperkahler

### 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

### Moduli Space of Higgs Bundles Instructor: Marco Gualtieri

Moduli Space of Higgs Bundles Instructor: Lecture Notes for MAT1305 Taught Spring of 2017 Typeset by Travis Ens Last edited January 30, 2017 Contents Lecture Guide Lecture 1 [11.01.2017] 1 History Hyperkähler

### arxiv:alg-geom/ v1 29 Jul 1993

Hyperkähler embeddings and holomorphic symplectic geometry. Mikhail Verbitsky, verbit@math.harvard.edu arxiv:alg-geom/9307009v1 29 Jul 1993 0. ntroduction. n this paper we are studying complex analytic

### 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

### Holonomy groups. Thomas Leistner. School of Mathematical Sciences Colloquium University of Adelaide, May 7, /15

Holonomy groups Thomas Leistner School of Mathematical Sciences Colloquium University of Adelaide, May 7, 2010 1/15 The notion of holonomy groups is based on Parallel translation Let γ : [0, 1] R 2 be

### 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

### Homological mirror symmetry via families of Lagrangians

Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants

### Marsden-Weinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds

Marsden-Weinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds Chenchang Zhu Nov, 29th 2000 1 Introduction If a Lie group G acts on a symplectic manifold (M, ω) and preserves the

### 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

### Del Pezzo surfaces and non-commutative geometry

Del Pezzo surfaces and non-commutative geometry D. Kaledin (Steklov Math. Inst./Univ. of Tokyo) Joint work with V. Ginzburg (Univ. of Chicago). No definitive results yet, just some observations and questions.

### Some new torsional local models for heterotic strings

Some new torsional local models for heterotic strings Teng Fei Columbia University VT Workshop October 8, 2016 Teng Fei (Columbia University) Strominger system 10/08/2016 1 / 30 Overview 1 Background and

### 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

### Pullbacks of hyperplane sections for Lagrangian fibrations are primitive

Pullbacks of hyperplane sections for Lagrangian fibrations are primitive Ljudmila Kamenova, Misha Verbitsky 1 Dedicated to Professor Claire Voisin Abstract. Let p : M B be a Lagrangian fibration on a hyperkähler

### 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.

### 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

### Multi-moment maps. CP 3 Journal Club. Thomas Bruun Madsen. 20th November 2009

Multi-moment maps CP 3 Journal Club Thomas Bruun Madsen 20th November 2009 Geometry with torsion Strong KT manifolds Strong HKT geometry Strong KT manifolds: a new classification result Multi-moment maps

### 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,

### 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

### Holonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012

Holonomy groups Thomas Leistner Mathematics Colloquium School of Mathematics and Physics The University of Queensland October 31, 2011 May 28, 2012 1/17 The notion of holonomy groups is based on Parallel

### Geometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type

Geometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type Paul Gauduchon Golden Sands, Bulgaria September, 19 26, 2011 1 Joint

### The geometry of Landau-Ginzburg models

Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror

### The Strominger Yau Zaslow conjecture

The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex

### 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

### Periodic monopoles and difference modules

Periodic monopoles and difference modules Takuro Mochizuki RIMS, Kyoto University 2018 February Introduction In complex geometry it is interesting to obtain a correspondence between objects in differential

### COMPACT HYPERKÄHLER MANIFOLDS: GENERAL THEORY. Contents 1. Introduction Yau s Theorem and its implications

COMPACT HYPEKÄHLE MANIFOLDS: GENEAL THEOY KIEAN G. O GADY SAPIENA UNIVESITÀ DI OMA Contents 1. Introduction 1 2. Yau s Theorem and its implications 1 3. The local period map and the B-B quadratic form

### LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in

### Ω Ω /ω. To these, one wants to add a fourth condition that arises from physics, what is known as the anomaly cancellation, namely that

String theory and balanced metrics One of the main motivations for considering balanced metrics, in addition to the considerations already mentioned, has to do with the theory of what are known as heterotic

### LOCAL STRUCTURE OF HYPERKÄHLER SINGULARITIES IN O GRADY S EXAMPLES

MOSCOW MATHEMATICAL JOURNAL Volume 7, Number 4, October December 2007, Pages 653 672 LOCAL STRUCTURE OF HYPERKÄHLER SINGULARITIES IN O GRADY S EXAMPLES D. KALEDIN AND M. LEHN To Victor Ginzburg, on the

### Projective parabolic geometries

Projective parabolic geometries David M. J. Calderbank University of Bath ESI Wien, September 2012 Based partly on: Hamiltonian 2-forms in Kähler geometry, with Vestislav Apostolov (UQAM), Paul Gauduchon

### The exceptional holonomy groups and calibrated geometry

Proceedings of 12 th Gökova Geometry-Topology Conference pp. 110 139 Published online at GokovaGT.org The exceptional holonomy groups and calibrated geometry Dominic Joyce Dedicated to the memory of Raoul

### 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

### An Invitation to Geometric Quantization

An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012 What is quantization? Quantization is a process of associating a classical mechanical system to

### 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)

### 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

### Holomorphic Symplectic Manifolds

Holomorphic Symplectic Manifolds Marco Andreatta joint project with J. Wiśniewski Dipartimento di Matematica Universitá di Trento Holomorphic Symplectic Manifolds p.1/35 Definitions An holomorphic symplectic

### Titles and Abstracts

29 May - 2 June 2017 Laboratoire J.A. Dieudonné Parc Valrose - Nice Titles and Abstracts Tamas Hausel : Mirror symmetry with branes by equivariant Verlinde formulae I will discuss an agreement of equivariant

### 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

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### Compact Riemannian manifolds with exceptional holonomy

Unspecified Book Proceedings Series Compact Riemannian manifolds with exceptional holonomy Dominic Joyce published as pages 39 65 in C. LeBrun and M. Wang, editors, Essays on Einstein Manifolds, Surveys

### 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

### arxiv:math/ v3 [math.dg] 5 Mar 2003

arxiv:math/0302219v3 [math.dg] 5 Mar 2003 Contents Vanishing theorems for locally conformal hyperkähler manifolds Misha Verbitsky, 1 verbit@maths.gla.ac.uk, verbit@mccme.ru Abstract Let M be a compact

### Fundamental Lemma and Hitchin Fibration

Fundamental Lemma and Hitchin Fibration Gérard Laumon CNRS and Université Paris-Sud May 13, 2009 Introduction In this talk I shall mainly report on Ngô Bao Châu s proof of the Langlands-Shelstad Fundamental

### Geometry of symmetric R-spaces

Geometry of symmetric R-spaces Makiko Sumi Tanaka Geometry and Analysis on Manifolds A Memorial Symposium for Professor Shoshichi Kobayashi The University of Tokyo May 22 25, 2013 1 Contents 1. Introduction

### UNIVERSAL UNFOLDINGS OF LAURENT POLYNOMIALS AND TT STRUCTURES. Claude Sabbah

UNIVERSAL UNFOLDINGS OF LAURENT POLYNOMIALS AND TT STRUCTURES AUGSBURG, MAY 2007 Claude Sabbah Introduction Let f : (C ) n C be a Laurent polynomial, that I assume to be convenient and non-degenerate,

### Rational Curves On K3 Surfaces

Rational Curves On K3 Surfaces Jun Li Department of Mathematics Stanford University Conference in honor of Peter Li Overview of the talk The problem: existence of rational curves on a K3 surface The conjecture:

### 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

### Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1

Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines

### Quaternionic Complexes

Quaternionic Complexes Andreas Čap University of Vienna Berlin, March 2007 Andreas Čap (University of Vienna) Quaternionic Complexes Berlin, March 2007 1 / 19 based on the joint article math.dg/0508534

### arxiv:math/ v2 [math.ag] 1 Jul 2004

Hyperholomorphic connections on coherent sheaves and stability arxiv:math/0107182v2 [math.ag] 1 Jul 2004 Contents Misha Verbitsky, 1 verbit@thelema.dnttm.ru, verbit@mccme.ru Abstract Let M be a hyperkähler

### CALIBRATED FIBRATIONS ON NONCOMPACT MANIFOLDS VIA GROUP ACTIONS

DUKE MATHEMATICAL JOURNAL Vol. 110, No. 2, c 2001 CALIBRATED FIBRATIONS ON NONCOMPACT MANIFOLDS VIA GROUP ACTIONS EDWARD GOLDSTEIN Abstract In this paper we use Lie group actions on noncompact Riemannian

### Flat complex connections with torsion of type (1, 1)

Flat complex connections with torsion of type (1, 1) Adrián Andrada Universidad Nacional de Córdoba, Argentina CIEM - CONICET Geometric Structures in Mathematical Physics Golden Sands, 20th September 2011

### 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

### Generalized complex geometry and topological sigma-models

Generalized complex geometry and topological sigma-models Anton Kapustin California Institute of Technology Generalized complex geometry and topological sigma-models p. 1/3 Outline Review of N = 2 sigma-models

### Uniform K-stability of pairs

Uniform K-stability of pairs Gang Tian Peking University Let G = SL(N + 1, C) with two representations V, W over Q. For any v V\{0} and any one-parameter subgroup λ of G, we can associate a weight w λ

### Compact 8-manifolds with holonomy Spin(7)

Compact 8-manifolds with holonomy Spin(7) D.D. Joyce The Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB. Published as Invent. math. 123 (1996), 507 552. 1 Introduction In Berger s classification

### HARMONIC MAPS INTO GRASSMANNIANS AND A GENERALIZATION OF DO CARMO-WALLACH THEOREM

Proceedings of the 16th OCU International Academic Symposium 2008 OCAMI Studies Volume 3 2009, pp.41 52 HARMONIC MAPS INTO GRASSMANNIANS AND A GENERALIZATION OF DO CARMO-WALLACH THEOREM YASUYUKI NAGATOMO

### LECTURE 2: SYMPLECTIC VECTOR BUNDLES

LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over

### Hard Lefschetz Theorem for Vaisman manifolds

Hard Lefschetz Theorem for Vaisman manifolds Antonio De Nicola CMUC, University of Coimbra, Portugal joint work with B. Cappelletti-Montano (Univ. Cagliari), J.C. Marrero (Univ. La Laguna) and I. Yudin

### ARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES

ASIAN J. MATH. c 2008 International Press Vol. 12, No. 3, pp. 289 298, September 2008 002 ARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES RAUL QUIROGA-BARRANCO Abstract.

### 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

### Homogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky

Homogeneous para-kähler Einstein manifolds Dmitri V. Alekseevsky Hamburg,14-18 July 2008 1 The talk is based on a joint work with C.Medori and A.Tomassini (Parma) See ArXiv 0806.2272, where also a survey

### Solvable Lie groups and the shear construction

Solvable Lie groups and the shear construction Marco Freibert jt. with Andrew Swann Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel 19.05.2016 1 Swann s twist 2 The shear construction The

### arxiv: v6 [math.cv] 8 Feb 2018

GROUP ACTIONS, NON-KÄHLER COMPLEX MANIFOLDS AND SKT STRUCTURES arxiv:1603.04629v6 [math.cv] 8 Feb 2018 MAINAK PODDAR AND AJAY SINGH THAKUR Abstract. We give a construction of integrable complex structures

### arxiv:math/ v2 [math.ag] 13 Jul 2001

arxiv:math/0012008v2 [math.ag] 13 Jul 2001 Symplectic resolutions: deformations and birational maps D. Kaledin Abstract We study projective birational maps of the form π : X Y, where Y is a normal irreducible

### An Introduction to Kuga Fiber Varieties

An Introduction to Kuga Fiber Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 28, 2012 Notation G a Q-simple

### 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.

### September 27, :51 WSPC/INSTRUCTION FILE biswas-loftin. Hermitian Einstein connections on principal bundles over flat affine manifolds

International Journal of Mathematics c World Scientific Publishing Company Hermitian Einstein connections on principal bundles over flat affine manifolds Indranil Biswas School of Mathematics Tata Institute

### Exotic nearly Kähler structures on S 6 and S 3 S 3

Exotic nearly Kähler structures on S 6 and S 3 S 3 Lorenzo Foscolo Stony Brook University joint with Mark Haskins, Imperial College London Friday Lunch Seminar, MSRI, April 22 2016 G 2 cones and nearly

### 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

### 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

### Symmetric Spaces. Andrew Fiori. Sept McGill University

McGill University Sept 2010 What are Hermitian? A Riemannian manifold M is called a Riemannian symmetric space if for each point x M there exists an involution s x which is an isometry of M and a neighbourhood

### A NOTE ON RIGIDITY OF 3-SASAKIAN MANIFOLDS

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 10, Pages 3027 3034 S 0002-9939(99)04889-3 Article electronically published on April 23, 1999 A NOTE ON RIGIDITY OF 3-SASAKIAN MANIFOLDS

### Modern Geometric Structures and Fields

Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface

### On the cohomology ring of compact hyperkähler manifolds

On the cohomology ring of compact hyperkähler manifolds Tom Oldfield 9/09/204 Introduction and Motivation The Chow ring of a smooth algebraic variety V, denoted CH (V ), is an analogue of the cohomology

### Structure theorems for compact Kähler manifolds

Structure theorems for compact Kähler manifolds Jean-Pierre Demailly joint work with Frédéric Campana & Thomas Peternell Institut Fourier, Université de Grenoble I, France & Académie des Sciences de Paris

### Techniques of computations of Dolbeault cohomology of solvmanifolds

.. Techniques of computations of Dolbeault cohomology of solvmanifolds Hisashi Kasuya Graduate School of Mathematical Sciences, The University of Tokyo. Hisashi Kasuya (Graduate School of Mathematical

### arxiv: v1 [math.dg] 21 Nov 2017

JGSP 25 (2012) 1 11 arxiv:1711.07948v1 [math.dg] 21 Nov 2017 TWISTOR SPACES AND COMPACT MANIFOLDS ADMITTING BOTH KÄHLER AND NON-KÄHLER STRUCTURES LJUDMILA KAMENOVA Communicated by Ivailo Mladenov Abstract.

### Conjectures in Kahler geometry

Conjectures in Kahler geometry S.K. Donaldson Abstract. We state a general conjecture about the existence of Kahler metrics of constant scalar curvature, and discuss the background to the conjecture 1.

### 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

### THE VORTEX EQUATION ON AFFINE MANIFOLDS. 1. Introduction

THE VORTEX EQUATION ON AFFINE MANIFOLDS INDRANIL BISWAS, JOHN LOFTIN, AND MATTHIAS STEMMLER Abstract. Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show

### Harmonic bundle and pure twistor D-module

Harmonic bundle and pure twistor D-module Takuro Mochizuki RIMS, Kyoto University 2012 June Harmonic bundle A harmonic bundle is a Higgs bundle with a pluri-harmonic metric. A Higgs field of a holomorphic

### a double cover branched along the smooth quadratic line complex

QUADRATIC LINE COMPLEXES OLIVIER DEBARRE Abstract. In this talk, a quadratic line complex is the intersection, in its Plücker embedding, of the Grassmannian of lines in an 4-dimensional projective space

### Comparison for infinitesimal automorphisms. of parabolic geometries

Comparison techniques for infinitesimal automorphisms of parabolic geometries University of Vienna Faculty of Mathematics Srni, January 2012 This talk reports on joint work in progress with Karin Melnick

### ERRATA FOR INTRODUCTION TO SYMPLECTIC TOPOLOGY

ERRATA FOR INTRODUCTION TO SYMPLECTIC TOPOLOGY DUSA MCDUFF AND DIETMAR A. SALAMON Abstract. These notes correct a few typos and errors in Introduction to Symplectic Topology (2nd edition, OUP 1998, reprinted