Integer cohomology of compact Hyperkähler manifolds
|
|
- Jayson Thompson
- 5 years ago
- Views:
Transcription
1 Integer cohomology of compact Hyperkähler manifolds Simon Kapfer University of Augsburg Géométrie Algébrique en Liberté Trieste, 27 June 204 Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds
2 Motivation Let X be a compact Hyperkähler manifold of complex dimension 2m or, equivalently, an IHS manifold. Why should we be interested in H (X, Z)? It feels more geometric. Comparing H (X, Z) with H (X, C) gives us information about X, e.g. on projectivity. We obtain restrictions to possible automorphisms of our manifold X.... Question Which constructions in H (X, R/C) carry over to H (X, Z)? Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds / 0
3 Beauville Bogomolov form As an example, consider the quadratic Beauville Bogomolov form q X : H (X, R) R Theorem (Fujiki) q X (α) m = c X α2m for some c R. Corollary q X can be renormalized to yield a primitive integral quadratic form: H (X, Z) Z. Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds 2 / 0
4 Hodge numbers for K3 surfaces For S a compact Hyperkähler manifold of complex dimension two, i.e. a K3 surface, we have h p,q (S) h k (S, Z) and the intersection pairing on H 2 is isomorphic to U 3 E 8 ( ) 2, where U and E 8 are the bilinear forms corresponding to the hyperbolic resp. E 8 lattice. Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds 3 / 0
5 Hodge numbers for Hilbert schemes Denote S [n] the Hilbert scheme of n points on the K3 surface S. Then, the Hodge decomposition for S [2] is given by: h p,q (S [2] ) h k (S [2], Z) and there are formulae for all S [n] due to Göttsche. Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds 4 / 0
6 Betti numbers k S [] S [2] S [3] S [4] S [5] S [6] For k fixed, these numbers stabilize for n big enough. Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds 5 / 0
7 Cohomology of Hilbert schemes Theorem (Nakajima) For each m and each α H j (S, Q), there is an operator a m (α) : H i (S [n], Q) H i+j+2m 2 (S [n+m], Q) and these operators, applied to H (S [0], Q) = Q, span the entire cohomology of all Hilbert schemes S [n]. Construction of the operators: Use the incidence scheme I S [n] S S [n+m] and define, using Poincaré duality, a m (α)β P.D. = p 3 ((p β p 2α) [I]). Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds 6 / 0
8 Integral basis for H (S [n], Z) This also works in integer cohomology! Theorem (Qin, Wang) Let H 0 (S, Z) be the canonical generator. The operators z λ a λ () : H i (S [n], Z) H i+2m 2k (S [n+m], Z) a m (α) : H i (S [n], Z) H i+j+2m 2k (S [n+m], Z) span the integer cohomologies H (S [n], Z). Here, the composition of several operators a mi () is denoted via a partition λ and z λ denotes some constant depending on λ. Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds 7 / 0
9 Algebra generated by H 2 (S [n], C) Theorem (Verbitsky) The subalgebra generated by H 2 (S [n], C) in H (S [n], C) is equal to Sym H 2 (S [n], C) α n+ q(α) = 0, where q is the Beauville-Bogomolov Form. In fact, this holds for any compact Hyperkähler manifold of complex dimension 2n. What about cohomology with integral coefficients? Lehn, Sorger and Vasserot developed formulae for the cup product on H (S [n], Q). We can use them also for computations in H (S [n], Z). Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds 8 / 0
10 Ring structure of H (S [n], Q) Rough idea of the algebraic model: To any composition of operators a m (α )... a mk (α k ) one associates a conjugacy class of the symmetric group S n, given by the partition λ = (m,..., m k ) The cup product is built with the product in S n : E.g. ( 2 3)(4 5) ( 4) = ( ), but ( 2 3)(4 5) ( 2) = ()(2 3)(4 5). If two cycles ν i, ν j are joined together by a transposition, multiply the corresponding classes, e.g. ( 2 3) αi (4 5) αj ( 4) αl = ( ) αi α j α l If a cycle is split in two by a transposition, use a map adjoint to the multiplication in H (S). Sum up all such possibilities and put the constants in the right way. Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds 9 / 0
11 Algebra generated by H 2 (S [n], Z) For integer cohomology, we get torsion for small n, e.g. H 4 (S [2], Z) = Sym 2 H 2 (S [2], Z) Z H 4 (S [3], Z) = Sym 2 H 2 (S [3], Z) Z 23 Z 3 H 4 (S [n], Z) = Sym 2 H 2 (S [n], Z) Z 24, n 4. Question Is there a geometric interpretation, e.g. for n = 3? Thank you for your attention! Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds 0 / 0
12 References S. Boissière, M. Nieper-Wisskirchen, and A. Sarti. Smith theory and irreducible holomorphic symplectic manifolds. arxiv e-prints, April 202. M. Gross, D. Huybrechts, and D. Joyce. Calabi-Yau Manifolds and Related Geometries: Lectures at a Summer School in Nordfjordeid, Norway, June, 200. Universitext (979). Springer Berlin Heidelberg, M. Lehn and C. Sorger. The cup product of the Hilbert scheme for K3 surfaces. arxiv Mathematics e-prints, December Zhenbo Qin and Weiqiang Wang. Integral operators and integral cohomology classes of Hilbert schemes. Mathematische Annalen, Simon Kapfer, University of Augsburg Integer cohomology of compact Hyperkähler manifolds
THE FOURIER TRANSFORM FOR CERTAIN HYPERKÄHLER FOURFOLDS. Contents Introduction 2
THE FOURIER TRANSFORM FOR CERTAIN HYPERKÄHLER FOURFOLDS MINGMIN SHEN AND CHARLES VIAL Abstract. Using a codimension-1 algebraic cycle obtained from the Poincaré line bundle, Beauville defined the Fourier
More informationHyperkähler 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 informationOn 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
More informationPullbacks 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
More informationA NEW FAMILY OF SYMPLECTIC FOURFOLDS
A NEW FAMILY OF SYMPLECTIC FOURFOLDS OLIVIER DEBARRE This is joint work with Claire Voisin. 1. Irreducible symplectic varieties It follows from work of Beauville and Bogomolov that any smooth complex compact
More informationUNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS
UNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS OLIVIER DEBARRE Abstract. Using Verbitsky s Torelli theorem, we show the existence of various isomorphisms between certain hyperkähler fourfolds. This
More informationarxiv: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
More informationAnalogs of Hodge Riemann relations
Analogs of Hodge Riemann relations in algebraic geometry, convex geometry and linear algebra V. Timorin State University of New York at Stony Brook March 28, 2006 Hodge Riemann form Let X be a compact
More informationKähler manifolds and variations of Hodge structures
Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic
More informationAlgebraic cycles on some special hyperkähler varieties
Rend. Mat. Appl. (7). Volume 38, (2017), 243 276 RENDICONTI DI MATEMATICA E DELLE SUE APPLICAZIONI Algebraic cycles on some special hyperkähler varieties Robert Laterveer Abstract. This note contains some
More informationHODGE THEORY AND LAGRANGIAN PLANES ON GENERALIZED KUMMER FOURFOLDS
HODGE THEORY AND LAGRANGIAN PLANES ON GENERALIZED KUMMER FOURFOLDS BRENDAN HASSETT AND YURI TSCHINKEL 1. Introduction Suppose X is a smooth projective complex variety. Let N 1 (X, Z) H 2 (X, Z) and N 1
More informationComplex manifolds, Kahler metrics, differential and harmonic forms
Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on
More informationThe 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 informationIV. 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 informationIntroduction To K3 Surfaces (Part 2)
Introduction To K3 Surfaces (Part 2) James Smith Calf 26th May 2005 Abstract In this second introductory talk, we shall take a look at moduli spaces for certain families of K3 surfaces. We introduce the
More information30 Surfaces and nondegenerate symmetric bilinear forms
80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces
More informationAlgebraic geometry versus Kähler geometry
Algebraic geometry versus Kähler geometry Claire Voisin CNRS, Institut de mathématiques de Jussieu Contents 0 Introduction 1 1 Hodge theory 2 1.1 The Hodge decomposition............................. 2
More informationIntroduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway
Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007 Outline Introduction Curves Surfaces Curves on surfaces
More informationHilbert scheme intersection numbers, Hurwitz numbers, and Gromov-Witten invariants
Contemporary Mathematics Hilbert scheme intersection numbers, Hurwitz numbers, and Gromov-Witten invariants Wei-Ping Li 1, Zhenbo Qin 2, and Weiqiang Wang 3 Abstract. Some connections of the ordinary intersection
More informationFANO VARIETIES AND EPW SEXTICS
FNO VRIETIES ND EPW SEXTICS OLIVIER DEBRRE bstract. We explore a connection between smooth projective varieties X of dimension n with an ample divisor H such that H n = 10 and K X = (n 2)H and a class
More informationThe generalized Hodge and Bloch conjectures are equivalent for general complete intersections
The generalized Hodge and Bloch conjectures are equivalent for general complete intersections Claire Voisin CNRS, Institut de mathématiques de Jussieu 0 Introduction Recall first that a weight k Hodge
More informationHYPERKÄHLER 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 informationOn the Chow Ring of Certain Algebraic Hyper-Kähler Manifolds
Pure and Applied Mathematics Quarterly Volume 4, Number 3 (Special Issue: In honor of Fedor Bogomolov, Part 2 of 2 ) 613 649, 2008 On the Chow Ring of Certain Algebraic Hyper-Kähler Manifolds Claire Voisin
More informationarxiv:math/ v2 [math.ag] 14 Jan 2002
Cohomology Ring of Crepant Resolutions of Orbifolds Yongbin Ruan 1 Department of Mathematics, Hong Kong University of Science and Technology and Department of Mathematics, University of Wisconsin-Madison
More informationHyperbolic geometry of the ample cone. of a hyperkähler manifold Ekaterina Amerik 1, Misha Verbitsky 2
Hyperbolic geometry of the ample cone Contents of a hyperkähler manifold Ekaterina Amerik 1, Misha Verbitsky 2 Abstract Let M be a compact hyperkähler manifold with maximal holonomy (IHS). The group H
More informationChern numbers and Hilbert Modular Varieties
Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point
More informationLinear 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 informationStable bundles with small c 2 over 2-dimensional complex tori
Stable bundles with small c 2 over 2-dimensional complex tori Matei Toma Universität Osnabrück, Fachbereich Mathematik/Informatik, 49069 Osnabrück, Germany and Institute of Mathematics of the Romanian
More informationModuli Spaces of Hyperkähler Manifolds and Mirror Symmetry
Moduli Spaces of Hyperkähler Manifolds and Mirror Symmetry Daniel Huybrechts Institut de Mathématiques Jussieu, Université Paris 7, France Lectures given at the School and Conference on Intersection Theory
More informationJapanese-European symposium on Symplectic Varieties and Moduli Spaces third edition
Japanese-European symposium on Symplectic Varieties and Moduli Spaces third edition Date 27th (Mon.) 31st (Fri.), Aug., 2018. Venue: Morito Memorial Hall (conference room 1), Kagurazaka campus, Tokyo University
More informationUseful theorems in complex geometry
Useful theorems in complex geometry Diego Matessi April 30, 2003 Abstract This is a list of main theorems in complex geometry that I will use throughout the course on Calabi-Yau manifolds and Mirror Symmetry.
More informationFANO VARIETIES OF CUBIC FOURFOLDS CONTAINING A PLANE. 1. Introduction
FANO VARIETIES OF CUBIC FOURFOLDS CONTAINING A PLANE EMANUELE MACRÌ AND PAOLO STELLARI Abstract. We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to
More informationGeneralized Tian-Todorov theorems
Generalized Tian-Todorov theorems M.Kontsevich 1 The classical Tian-Todorov theorem Recall the classical Tian-Todorov theorem (see [4],[5]) about the smoothness of the moduli spaces of Calabi-Yau manifolds:
More informationIDEAL LATTICES AND AUTOMORPHISMS OF HYPERKÄHLER MANIFOLDS
IDEAL LATTICES AND AUTOMORPHISMS OF HYPERKÄHLER MANIFOLDS SAMUEL BOISSIÈRE, CHIARA CAMERE, GIOVANNI MONGARDI, AND ALESSANDRA SARTI Abstract. We prove that there exists a holomorphic symplectic manifold
More informationRadon Transforms and the Finite General Linear Groups
Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 1-1-2004 Radon Transforms and the Finite General Linear Groups Michael E. Orrison Harvey Mudd
More informationStable 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 informationLecture 10 - Representation Theory II: Heuristics
Lecture 10 - Representation Theory II: Heuristics February 15, 2013 1 Weights 1.1 Weight space decomposition We switch notation from last time. We let Λ indicate a highest weight, and λ to be an arbitrary
More informationSymplectic Moduli Spaces
Symplectic Moduli Spaces M. Lehn Fachbereich Mathematik, Johannes-Gutenberg-Universität Mainz, Germany Lectures given at the School and Conference on Intersection Theory and Moduli Trieste, 9-27 September
More informationVARIETIES WITH P-UNITS
VARIETIES WITH P-UNITS ANDREAS KRUG Abstract. We study the class of compact Kähler manifolds with trivial canonical bundle and the property that the cohomology of the trivial line bundle is generated by
More informationarxiv: v1 [math.ag] 28 Sep 2016
LEFSCHETZ CLASSES ON PROJECTIVE VARIETIES JUNE HUH AND BOTONG WANG arxiv:1609.08808v1 [math.ag] 28 Sep 2016 ABSTRACT. The Lefschetz algebra L X of a smooth complex projective variety X is the subalgebra
More informationη = (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
More informationENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5
ENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) OSAMU FUJINO Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5 1. Preliminaries Let us recall the basic notion of the complex geometry.
More informationIntermediate Jacobians and Abel-Jacobi Maps
Intermediate Jacobians and Abel-Jacobi Maps Patrick Walls April 28, 2012 Introduction Let X be a smooth projective complex variety. Introduction Let X be a smooth projective complex variety. Intermediate
More informationLecture VI: Projective varieties
Lecture VI: Projective varieties Jonathan Evans 28th October 2010 Jonathan Evans () Lecture VI: Projective varieties 28th October 2010 1 / 24 I will begin by proving the adjunction formula which we still
More informationarxiv: v2 [math.ag] 7 Apr 2015
ON THE MOTIVE OF SOME HYPERKÄHLER VARIETIES CHARLES VIAL arxiv:1406.1073v2 [math.ag] 7 Apr 2015 Abstract. We show that the motive of the Hilbert scheme of length-n subschemes on a K3 surface or on an abelian
More informationarxiv: v3 [math.cv] 3 Oct 2016
The Toledo invariant, and Seshadri constants of fake projective planes arxiv:1601.03733v3 [math.cv] 3 Oct 2016 Luca F. Di Cerbo Abdus Salam International Centre for Theoretical Physics - ICTP ldicerbo@ictp.it
More informationCOMPACT 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
More informationShifted Symplectic Derived Algebraic Geometry and generalizations of Donaldson Thomas Theory
Shifted Symplectic Derived Algebraic Geometry and generalizations of Donaldson Thomas Theory Lecture 2 of 3:. D-critical loci and perverse sheaves Dominic Joyce, Oxford University KIAS, Seoul, July 2018
More informationEXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION
EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION JAN CHRISTIAN ROHDE Introduction By string theoretical considerations one is interested in Calabi-Yau manifolds since Calabi-Yau 3-manifolds
More informationKleine AG: Travaux de Shimura
Kleine AG: Travaux de Shimura Sommer 2018 Programmvorschlag: Felix Gora, Andreas Mihatsch Synopsis This Kleine AG grew from the wish to understand some aspects of Deligne s axiomatic definition of Shimura
More informationCHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents
CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally
More informationLAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS
LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere
More informationEigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups
Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Shrawan Kumar Talk given at AMS Sectional meeting held at Davidson College, March 2007 1 Hermitian eigenvalue
More informationSurjectivity in Honda-Tate
Surjectivity in Honda-Tate Brian Lawrence May 5, 2014 1 Introduction Let F q be a finite field with q = p a elements, p prime. Given any simple Abelian variety A over F q, we have seen that the characteristic
More informationOn embedding all n-manifolds into a single (n + 1)-manifold
On embedding all n-manifolds into a single (n + 1)-manifold Jiangang Yao, UC Berkeley The Forth East Asian School of Knots and Related Topics Joint work with Fan Ding & Shicheng Wang 2008-01-23 Problem
More information1. Quadratic lattices A quadratic lattice is a free abelian group M of finite rank equipped with a symmetric bilinear form.
Borcherds products learning seminar Quadratic lattices, Hermitian symmetric domains, and vector-valued modular forms Lecture 2 Igor Dolgachev February 12, 2016 1. Quadratic lattices A quadratic lattice
More informationNormalization of a Poisson algebra is Poisson
Normalization of a Poisson algebra is Poisson D. Kaledin Introduction It is well-known that functions on a smooth symplectic manifold M are equipped with a canonical skew-linear operation called the Poisson
More informationA global Torelli theorem for singular symplectic varieties
A global Torelli theorem for singular symplectic varieties Benjamin Bakker and Christian Lehn Abstract. We systematically study the moduli theory of singular symplectic varieties which have a resolution
More informationHomogeneous 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
More informationAlgebraic 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
More informationTopics in Geometry: Mirror Symmetry
MIT OpenCourseWare http://ocw.mit.edu 8.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:
More informationOn the motive of some hyperkähler varieties
J. reine angew. Math., Ahead of Print Journal für die reine und angewandte Mathematik DOI 10.1515/crelle-2015-0008 De Gruyter 2015 On the motive of some hyperkähler varieties By Charles Vial at Cambridge
More informationHomotopy and geometric perspectives on string topology
Homotopy and geometric perspectives on string topology Ralph L. Cohen Stanford University August 30, 2005 In these lecture notes I will try to summarize some recent advances in the new area of study known
More informationMirror symmetry, Langlands duality and the Hitchin system I
Mirror symmetry, Langlands duality and the Hitchin system I Tamás Hausel Royal Society URF at University of Oxford http://www.maths.ox.ac.uk/ hausel/talks.html April 200 Simons lecture series Stony Brook
More informationCOUNTING REAL RATIONAL CURVES ON K3 SURFACES
COUNTING REAL RATIONAL CURVES ON K3 SURFACES VIATCHESLAV KHARLAMOV AND RAREŞ RĂSDEACONU Abstract. We provide a real analog of the Yau-Zaslow formula counting rational curves on K3 surfaces. But man is
More informationStability of coisotropic fibrations on holomorphic symplectic manifolds. 1. Introduction
Stability of coisotropic fibrations on holomorphic symplectic manifolds Christian Lehn and Gianluca Pacienza Abstract. We investigate the stability of fibers of coisotropic fibrations on holomorphic symplectic
More informationHODGE NUMBERS OF COMPLETE INTERSECTIONS
HODGE NUMBERS OF COMPLETE INTERSECTIONS LIVIU I. NICOLAESCU 1. Holomorphic Euler characteristics Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p
More informationON THE PRIMITIVITY OF BIRATIONAL TRANSFORMATIONS OF IRREDUCIBLE SYMPLECTIC MANIFOLDS
ON THE PRIMITIVITY OF BIRATIONAL TRANSFORMATIONS OF IRREDUCIBLE SYMPLECTIC MANIFOLDS FEDERICO LO BIANCO ABSTRACT. Let f : X 99K X be a bimeromorphic transformation of a complex irreducible symplectic manifold
More informationThese slides available at joyce/talks.html
Kuranishi (co)homology: a new tool in symplectic geometry. II. Kuranishi (co)homology Dominic Joyce Oxford University, UK work in progress based on arxiv:0707.3572 v5, 10/08 summarized in arxiv:0710.5634
More informationHodge Theory of Maps
Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar
More informationarxiv:math/ v1 [math.ag] 24 Nov 1998
Hilbert schemes of a surface and Euler characteristics arxiv:math/9811150v1 [math.ag] 24 Nov 1998 Mark Andrea A. de Cataldo September 22, 1998 Abstract We use basic algebraic topology and Ellingsrud-Stromme
More informationDarboux theorems for shifted symplectic derived schemes and stacks
Darboux theorems for shifted symplectic derived schemes and stacks Lecture 1 of 3 Dominic Joyce, Oxford University January 2014 Based on: arxiv:1305.6302 and arxiv:1312.0090. Joint work with Oren Ben-Bassat,
More informationF. LAYTIMI AND D.S. NAGARAJ
REMARKS ON RAMANUJAM-KAWAMATA-VIEHWEG VANISHING THEOREM arxiv:1702.04476v1 [math.ag] 15 Feb 2017 F. LAYTIMI AND D.S. NAGARAJ Abstract. In this article weproveageneralresult on anef vector bundle E on a
More informationSymplectic 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 informationProblems on Minkowski sums of convex lattice polytopes
arxiv:08121418v1 [mathag] 8 Dec 2008 Problems on Minkowski sums of convex lattice polytopes Tadao Oda odatadao@mathtohokuacjp Abstract submitted at the Oberwolfach Conference Combinatorial Convexity and
More informationCohomologies and Elliptic Operators on Symplectic Manifolds
AMS/IP Studies in Advanced Mathematics Volume 51, 2012 Cohomologies and Elliptic Operators on Symplectic Manifolds Li-Sheng Tseng Abstract. In joint work with S.-T. Yau, we construct new cohomologies of
More informationAPPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY
APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY In this appendix we begin with a brief review of some basic facts about singular homology and cohomology. For details and proofs, we refer to [Mun84]. We then
More informationConjectures on counting associative 3-folds in G 2 -manifolds
in G 2 -manifolds Dominic Joyce, Oxford University Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, First Annual Meeting, New York City, September 2017. Based on arxiv:1610.09836.
More informationModuli of K3 surfaces and irreducible symplectic manifolds
Moduli of K3 surfaces and irreducible symplectic manifolds V. Gritsenko, K. Hulek, and G.K. Sankaran Abstract. The name K3 surfaces was coined by A. Weil in 1957 when he formulated a research programme
More informationQUALIFYING EXAM IN ALGEBRA August 2011
QUALIFYING EXAM IN ALGEBRA August 2011 1. There are 18 problems on the exam. Work and turn in 10 problems, in the following categories. I. Linear Algebra 1 problem II. Group Theory 3 problems III. Ring
More informationThe 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
More informationK3 Surfaces and Lattice Theory
K3 Surfaces and Lattice Theory Ichiro Shimada Hiroshima University 2014 Aug Singapore 1 / 26 Example Consider two surfaces S + and S in C 3 defined by w 2 (G(x, y) ± 5 H(x, y)) = 1, where G(x, y) := 9
More informationTorus Knots and q, t-catalan Numbers
Torus Knots and q, t-catalan Numbers Eugene Gorsky Stony Brook University Simons Center For Geometry and Physics April 11, 2012 Outline q, t-catalan numbers Compactified Jacobians Arc spaces on singular
More informationMAT 545: Complex Geometry Fall 2008
MAT 545: Complex Geometry Fall 2008 Notes on Lefschetz Decomposition 1 Statement Let (M, J, ω) be a Kahler manifol. Since ω is a close 2-form, it inuces a well-efine homomorphism L: H k (M) H k+2 (M),
More informationFAMILIES OF RATIONAL CURVES ON HOLOMORPHIC SYMPLECTIC VARIETIES. 1. Introduction
FAMILIES OF RATIONAL CURVES ON HOLOMORPHIC SYMPLECTIC VARIETIES FRANÇOIS CHARLES AND GIANLUCA PACIENZA Abstract. We study families of rational curves on certain irreducible holomorphic symplectic varieties.
More informationTAUTOLOGICAL CLASSES ON MODULI SPACES OF HYPERKÄHLER MANIFOLDS
TAUTOLOGICAL CLASSES ON MODULI SPACES OF HYPERKÄHLER MANIFOLDS NICOLAS BERGERON AND ZHIYUAN LI Abstract. We study algebraic cycles on moduli spaces F h of h-polarized hyperkähler manifolds. Following previous
More informationAn 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
More information(www.math.uni-bonn.de/people/harder/manuscripts/buch/), files chap2 to chap
The basic objects in the cohomology theory of arithmetic groups Günter Harder This is an exposition of the basic notions and concepts which are needed to build up the cohomology theory of arithmetic groups
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Extensions of profinite duality groups Alexander Schmidt and Kay Wingberg Preprint Nr. 25/2008 Extensions of profinite duality groups Alexander Schmidt and Kay Wingberg
More informationThe Grothendieck-Katz Conjecture for certain locally symmetric varieties
The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-
More informationFibrations on generalized Kummer varieties. Martin G. Gulbrandsen
Fibrations on generalized Kummer varieties Martin G. Gulbrandsen Contents Introduction 5 Chapter 1. Symplectic varieties and their fibrations 7 1.1. Symplectic varieties 7 1.2. Regular and rational fibrations
More informationarxiv: v2 [math.dg] 6 Nov 2014
A SHARP CUSP COUNT FOR COMPLEX HYPERBOLIC SURFACES AND RELATED RESULTS GABRIELE DI CERBO AND LUCA F. DI CERBO arxiv:1312.5368v2 [math.dg] 6 Nov 2014 Abstract. We derive a sharp cusp count for finite volume
More informationQuaternionic 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
More informationNon-Geometric Calabi- Yau Backgrounds
Non-Geometric Calabi- Yau Backgrounds CH, Israel and Sarti 1710.00853 A Dabolkar and CH, 2002 Duality Symmetries Supergravities: continuous classical symmetry, broken in quantum theory, and by gauging
More informationEXPLICIT QUANTIZATION OF THE KEPLER MANIFOLD
proceedings of the american mathematical Volume 77, Number 1, October 1979 society EXPLICIT QUANTIZATION OF THE KEPLER MANIFOLD ROBERT J. BLATTNER1 AND JOSEPH A. WOLF2 Abstract. Any representation ir of
More informationHalf the sum of positive roots, the Coxeter element, and a theorem of Kostant
DOI 10.1515/forum-2014-0052 Forum Math. 2014; aop Research Article Dipendra Prasad Half the sum of positive roots, the Coxeter element, and a theorem of Kostant Abstract: Interchanging the character and
More informationRIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY
RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY NICK MCCLEEREY 0. Complex Differential Forms Consider a complex manifold X n (of complex dimension n) 1, and consider its complexified tangent bundle T C
More informationPower structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points
arxiv:math/0407204v1 [math.ag] 12 Jul 2004 Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points S.M. Gusein-Zade I. Luengo A. Melle Hernández Abstract
More informationKac-Moody Algebras. Ana Ros Camacho June 28, 2010
Kac-Moody Algebras Ana Ros Camacho June 28, 2010 Abstract Talk for the seminar on Cohomology of Lie algebras, under the supervision of J-Prof. Christoph Wockel Contents 1 Motivation 1 2 Prerequisites 1
More informationStable bases for moduli of sheaves
Columbia University 02 / 03 / 2015 Moduli of sheaves on P 2 Let M v,w be the moduli space of deg v torsion-free sheaves of rank w on P 2, which compactifies the space of instantons Moduli of sheaves on
More information