Topics in Geometry: Mirror Symmetry

Size: px
Start display at page:

Download "Topics in Geometry: Mirror Symmetry"

Transcription

1 MIT OpenCourseWare Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit:

2 MIRROR SYMMETRY: LECTURE 3 DENIS AUROU Last time, we say that a deformation of (, J) is given by () {s Ω 0, (, T ) s + 2 [s, s] = 0}/Diff() To first order, these are determined by Def (, J) = H (, T ), but extending these to higher order is obstructed by elements of H 2 (, T ). In the Calabi-Yau case, recall that: Theorem (Bogomolov-Tian-Todorov). For a compact Calabi-Yau (Ω n,0 = O ) with H 0 (, T ) = 0 (automorphisms are discrete), deformations of are unobstructed. Note that, if is a Calabi-Yau manifold, we have a natural isomorphism T = Ω n, v i v Ω, so (2) H 0 (, T ) = H n,0 () = H 0, and similarly (3) H (, T ) = H n,, H 2 (, T ) = H n,2. Hodge theory Given a Kähler metric, we have a Hodge operator and L 2 -adjoints (4) d = d, = and Laplacians (5) Δ = dd + d d, = + Every (d/)-cohomology class contains a unique harmonic form, and one can show that = Δ. We obtain 2 (6) H k (, C) dr = Ker (Δ : Ω k (, C) ) = Ker ( : Ω k ) = Ker ( : Ω p,q ) = H p,q () p+q=k p+q=k

3 2 DENIS AUROU The Hodge operator gives an isomorphism H p,q = H n p,n q. Complex conjugation gives H p,q = H q,p, giving us a Hodge diamond h n,n h n,n h 0,n h n,n h n,n.... (7) h, h 0,.... For a Calabi-Yau, we have h n,0 h,0 h 0,0 (8) H p,0 = H n,n p = H n p (, Ω n ) = H n p (, O = H n p,0 ) = H 0,n p Specifically, for a Calabi-Yau 3-fold with h,0 = 0, we have a reduced Hodge diamond 0 0 (9) 0 h, h 2, 0 0 h 2, h, Mirror symmetry says that there is another Calabi-Yau manifold whose Hodge diamond is the mirror image (or 90 degree rotation) of this one. There is another interpretation of the Kodaira-Spencer map H (, T ) = H n,. For = (, J t ) t S a family of complex deformations of (, J), c (K ) = c (T ) = 0 implies that Ω n (,J = under the assumption H O t) () = 0, so we don t have to worry about deforming outside the Calabi-Yau case. Then [Ω t ] H n,0 J () H n (, C). How does this depend on t? Given T 0 S, t t t Ω t Ω n,0 Ω n, by Griffiths transversality: (0) α t Ω p,q = α t Ω p,q + Ω p,q+ + Ω p+,q J t t

4 MIRROR SYMMETRY: LECTURE 3 3 Since Ωt t t=0 is d-closed (dω t = 0), ( Ωt t=0 ) (n,) is -closed, while t () ( Ω t t=0 ) (n,) + ( Ω t t=0 ) (n,) = 0 t t Thus, [( Ωt t=0 ) (n,) ] H n, (). t For fixed Ω 0, this is independent of the choice of Ω t. If we rescale f(t)ω t, f Ω (2) t (f(t)ω t ) = Ω t + f(t) t t t Taking t 0, the former term is (n, 0), while for the latter, f(0) scales linearly with Ω 0. (3) H n, () = H (, Ω n ) = H (, T ) and the two maps T 0 S H n, (), H (, T ) agree. Hence, for θ H (, T ) a first-order deformation of complex structure, θ Ω H (, Ω n T ) = H n, () and (the Gauss-Manin connection) [ θ Ω] (n,) H n, () are the same. We can iterate this to the third-order derivative: on a Calabi-Yau threefold, we have (4) θ, θ 2, θ 3 = Ω (θ θ 2 θ 3 Ω) = Ω ( θ θ2 θ3 Ω) where the latter wedge is of a (3, 0) and a (0, 3) form. 2. Pseudoholomorphic curves (reference: McDuff-Salamon) Let ( 2n, ω) be a symplectic manifold, J a compatible almost-complex structure, ω(, J ) the associated Riemannian metric. Furthermore, let (Σ, j) be a Riemann surface of genus g, z,..., z k Σ market points. There is a well-defined moduli space M g,k = {(Σ, j, z,..., z k )} modulo biholomorphisms of complex dimension 3g 3 + k (note that M 0,3 = {pt}). Definition. u : Σ is a J-holomorphic map if J du = du J, i.e. J u = (du + Jduj) = 0. For β H 2 (, Z), we obtain an associated moduli 2 space (5) M g,k (, J, β) = {(Σ, j, z,..., z k ), u : Σ u [Σ] = β, J u = 0}/ where is the equivalence given by φ below. u Σ, z,..., z k (6) φ = u Σ, z,..., z k This space is the zero set of the section J of E Map(Σ, ) β M g,k, where E is the (Banach) bundle defined by E u = W r,p (Σ, Ω 0, u T ). Σ

5 4 DENIS AUROU We can define a linearized operator D : W r+,p (Σ, u T ) T M g,k W r,p (Σ, Ω 0, U T ) Σ (7) D (v, j ) = 2 ( v + J vj + ( v J) du j + J du j ) = v + 2 ( v J)du j + 2 J du j This operator is Fredholm, with real index (8) index R D := 2d = 2 c (T ), β + n(2 2g) + (6g 6 + 2k) One can ask about transversality, i.e. whether we can ensure that D is onto at every solution. We say that u is regular if this is true at u: if so, M g,k (, J, β) is smooth of dimension 2d. Definition 2. We say that a map Σ is simple (or somewhere injective ) if z Σ s.t. du(z) = 0 and u (u(z)) = {z}. Note that otherwise u will factor through a covering Σ Σ. We set M g,k (, J, β) to be the moduli space of such simple curves. Theorem 2. Let J (, ω) be the set of compatible almost-complex structures on : then (9) J reg (, β) = {J J (, ω) every simple J-holomorphic curve in class β is regular} is a Baire subset in J (, ω), and for J J reg (, β), M (, J, β) is smooth g,k (as an orbifold, if M g,k is an orbifold) of real dimension 2d and carries a natural orientation. The main idea here is to view J u = 0 as an equation on Map(Σ, ) M g,k J (, ω) (u, j, J). Then D is easily seen to be surjective for simple maps. We have a universal moduli space MM π J J (, ω) given by a Fredholm map, and by Sard-Smale, a generic J is a regular value of π J. This universal moduli space is M = J J (,ω) M g,k(, J, β). For such J, M g,k(, J, β) is smooth of dimension 2d, and the tangent space is Ker (D ). For the orientability, we need an orientation on Ker (D ). If J is integrable, the D is C-linear (D = ), so Ker is a C-vector space. Moreover, J 0, J J reg (, β), a (dense set of choices of) path {J t } t [0,] s.t. t [0,] M g,k(, J t, β) is a smooth oriented cobordism. We still need compactness.

Mirror Symmetry: Introduction to the B Model

Mirror Symmetry: Introduction to the B Model Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds

More information

Topics in Geometry: Mirror Symmetry

Topics in Geometry: Mirror Symmetry MIT OpenCourseWare http://ocw.mit.edu 18.969 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 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

Useful theorems in complex geometry

Useful 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 information

On the BCOV Conjecture

On the BCOV Conjecture Department of Mathematics University of California, Irvine December 14, 2007 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called

More information

Nodal symplectic spheres in CP 2 with positive self intersection

Nodal symplectic spheres in CP 2 with positive self intersection Nodal symplectic spheres in CP 2 with positive self intersection Jean-François BARRAUD barraud@picard.ups-tlse.fr Abstract 1 : Let ω be the canonical Kähler structure on CP 2 We prove that any ω-symplectic

More information

Formality of Kähler manifolds

Formality of Kähler manifolds Formality of Kähler manifolds Aron Heleodoro February 24, 2015 In this talk of the seminar we like to understand the proof of Deligne, Griffiths, Morgan and Sullivan [DGMS75] of the formality of Kähler

More information

L19: Fredholm theory. where E u = u T X and J u = Formally, J-holomorphic curves are just 1

L19: Fredholm theory. where E u = u T X and J u = Formally, J-holomorphic curves are just 1 L19: Fredholm theory We want to understand how to make moduli spaces of J-holomorphic curves, and once we have them, how to make them smooth and compute their dimensions. Fix (X, ω, J) and, for definiteness,

More information

Kähler manifolds and variations of Hodge structures

Kähler 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 information

THE HODGE DECOMPOSITION

THE HODGE DECOMPOSITION THE HODGE DECOMPOSITION KELLER VANDEBOGERT 1. The Musical Isomorphisms and induced metrics Given a smooth Riemannian manifold (X, g), T X will denote the tangent bundle; T X the cotangent bundle. The additional

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

Overview of classical mirror symmetry

Overview of classical mirror symmetry Overview of classical mirror symmetry David Cox (notes by Paul Hacking) 9/8/09 () Physics (2) Quintic 3-fold (3) Math String theory is a N = 2 superconformal field theory (SCFT) which models elementary

More information

Geometry of the Calabi-Yau Moduli

Geometry of the Calabi-Yau Moduli Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51

More information

Lecture III: Neighbourhoods

Lecture III: Neighbourhoods Lecture III: Neighbourhoods Jonathan Evans 7th October 2010 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 1 / 18 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 2 / 18 In

More information

Complex manifolds, Kahler metrics, differential and harmonic forms

Complex 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 information

Classifying complex surfaces and symplectic 4-manifolds

Classifying complex surfaces and symplectic 4-manifolds Classifying complex surfaces and symplectic 4-manifolds UT Austin, September 18, 2012 First Cut Seminar Basics Symplectic 4-manifolds Definition A symplectic 4-manifold (X, ω) is an oriented, smooth, 4-dimensional

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

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

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,...

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,... J. Korean Math. Soc. 40 (2003), No. 4, pp. 667 680 HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES Takao Akahori Abstract. The mirror conjecture means originally the deep relation between complex

More information

AN ALMOST KÄHLER STRUCTURE ON THE DEFORMATION SPACE OF CONVEX REAL PROJECTIVE STRUCTURES

AN ALMOST KÄHLER STRUCTURE ON THE DEFORMATION SPACE OF CONVEX REAL PROJECTIVE STRUCTURES Trends in Mathematics Information Center for Mathematical ciences Volume 5, Number 1, June 2002, Pages 23 29 AN ALMOT KÄHLER TRUCTURE ON THE DEFORMATION PACE OF CONVEX REAL PROJECTIVE TRUCTURE HONG CHAN

More information

UNIVERSAL UNFOLDINGS OF LAURENT POLYNOMIALS AND TT STRUCTURES. Claude Sabbah

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,

More information

arxiv:alg-geom/ v1 29 Jul 1993

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

More information

FAKE PROJECTIVE SPACES AND FAKE TORI

FAKE PROJECTIVE SPACES AND FAKE TORI FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.

More information

Derived Differential Geometry

Derived Differential Geometry Derived Differential Geometry Lecture 13 of 14: Existence of derived manifold or orbifold structures on moduli spaces Dominic Joyce, Oxford University Summer 2015 These slides, and references, etc., available

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

THE MODULI SPACE OF TRANSVERSE CALABI YAU STRUCTURES ON FOLIATED MANIFOLDS

THE MODULI SPACE OF TRANSVERSE CALABI YAU STRUCTURES ON FOLIATED MANIFOLDS Moriyama, T. Osaka J. Math. 48 (211), 383 413 THE MODULI SPACE OF TRANSVERSE CALAI YAU STRUCTURES ON FOLIATED MANIFOLDS TAKAYUKI MORIYAMA (Received August 21, 29, revised December 4, 29) Abstract In this

More information

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu

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

Period Domains. Carlson. June 24, 2010

Period Domains. Carlson. June 24, 2010 Period Domains Carlson June 4, 00 Carlson - Period Domains Period domains are parameter spaces for marked Hodge structures. We call Γ\D the period space, which is a parameter space of isomorphism classes

More information

Two simple ideas from calculus applied to Riemannian geometry

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

More information

Intermediate Jacobians and Abel-Jacobi Maps

Intermediate 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 information

J-curves and the classification of rational and ruled symplectic 4-manifolds

J-curves and the classification of rational and ruled symplectic 4-manifolds J-curves and the classification of rational and ruled symplectic 4-manifolds François Lalonde Université du Québec à Montréal (flalonde@math.uqam.ca) Dusa McDuff State University of New York at Stony Brook

More information

Algebraic geometry versus Kähler geometry

Algebraic 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 information

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu

More information

Genus Zero Gromov-Witten Invariants

Genus Zero Gromov-Witten Invariants Genus Zero Gromov-Witten Invariants Roland Grinis June 11, 2012 Contents 1 Introduction 1 2 Moduli Spaces of Stable Maps and Curves 3 3 Kuranishi Structure on M 0,k (X, β; J) 9 4 The Virtual Moduli Cycle

More information

LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS

LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS Contents 1. Almost complex manifolds 1. Complex manifolds 5 3. Kähler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds Almost complex structures.

More information

Non-Kähler Calabi-Yau Manifolds

Non-Kähler Calabi-Yau Manifolds Non-Kähler Calabi-Yau Manifolds Shing-Tung Yau Harvard University String-Math 2011 University of Pennsylvania June 6, 2011 String and math have had a very close interaction over the past thirty years.

More information

Geometry of the Moduli Space of Curves and Algebraic Manifolds

Geometry of the Moduli Space of Curves and Algebraic Manifolds Geometry of the Moduli Space of Curves and Algebraic Manifolds Shing-Tung Yau Harvard University 60th Anniversary of the Institute of Mathematics Polish Academy of Sciences April 4, 2009 The first part

More information

The topology of symplectic four-manifolds

The topology of symplectic four-manifolds The topology of symplectic four-manifolds Michael Usher January 12, 2007 Definition A symplectic manifold is a pair (M, ω) where 1 M is a smooth manifold of some even dimension 2n. 2 ω Ω 2 (M) is a two-form

More information

Generalized Tian-Todorov theorems

Generalized 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 information

Hermitian vs. Riemannian Geometry

Hermitian vs. Riemannian Geometry Hermitian vs. Riemannian Geometry Gabe Khan 1 1 Department of Mathematics The Ohio State University GSCAGT, May 2016 Outline of the talk Complex curves Background definitions What happens if the metric

More information

ENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5

ENOKI 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 information

A NEW FAMILY OF SYMPLECTIC FOURFOLDS

A 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 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

Complexes of Hilbert C -modules

Complexes of Hilbert C -modules Complexes of Hilbert C -modules Svatopluk Krýsl Charles University, Prague, Czechia Nafpaktos, 8th July 2018 Aim of the talk Give a generalization of the framework for Hodge theory Aim of the talk Give

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

η = (e 1 (e 2 φ)) # = e 3

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

More information

The Strominger Yau Zaslow conjecture

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

More information

Shafarevich s Conjecture for CY Manifolds I

Shafarevich s Conjecture for CY Manifolds I Pure and Applied Mathematics Quarterly Volume 1, Number 1, 28 67, 2005 Shafarevich s Conjecture for CY Manifolds I Kefeng Liu, Andrey Todorov Shing-Tung Yau and Kang Zuo Abstract Let C be a Riemann surface

More information

Some brief notes on the Kodaira Vanishing Theorem

Some brief notes on the Kodaira Vanishing Theorem Some brief notes on the Kodaira Vanishing Theorem 1 Divisors and Line Bundles, according to Scott Nollet This is a huge topic, because there is a difference between looking at an abstract variety and local

More information

Moduli space of special Lagrangians in almost Kahler spaces

Moduli space of special Lagrangians in almost Kahler spaces Moduli space of special Lagrangians in almost Kahler spaces LI MA Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China Manuscript received on August 30, 2000; accepted for publication

More information

arxiv:math/ v2 [math.dg] 11 Jun 1998

arxiv:math/ v2 [math.dg] 11 Jun 1998 ariv:math/9805095v2 [math.dg] 11 Jun 1998 IDENTIFICATION OF TWO FROBENIUS MANIFOLDS IN MIRROR SYMMETRY HUAI-DONG CAO & JIAN ZHOU Abstract. We identify two Frobenius manifolds obtained from two different

More information

STABILITY OF THE ALMOST HERMITIAN CURVATURE FLOW

STABILITY OF THE ALMOST HERMITIAN CURVATURE FLOW STABILITY OF THE ALOST HERITIAN CURVATURE FLOW DANIEL J. SITH Abstract. The almost Hermitian curvature flow was introduced in [9] by Streets and Tian in order to study almost Hermitian structures, with

More information

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents

CHAPTER 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 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

Elliptic Regularity. Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n.

Elliptic Regularity. Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n. Elliptic Regularity Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n. 1 Review of Hodge Theory In this note I outline the proof of the following Fundamental

More information

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat

More information

Homological Mirror Symmetry and VGIT

Homological Mirror Symmetry and VGIT Homological Mirror Symmetry and VGIT University of Vienna January 24, 2013 Attributions Based on joint work with M. Ballard (U. Wisconsin) and Ludmil Katzarkov (U. Miami and U. Vienna). Slides available

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

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

arxiv: v1 [math.ag] 13 Mar 2019

arxiv: v1 [math.ag] 13 Mar 2019 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show

More information

CALIBRATED GEOMETRY JOHANNES NORDSTRÖM

CALIBRATED GEOMETRY JOHANNES NORDSTRÖM CALIBRATED GEOMETRY JOHANNES NORDSTRÖM Summer school on Differential Geometry, Ewha University, 23-27 July 2012. Recommended reading: Salamon, Riemannian Geometry and Holonomy Groups [11] http://calvino.polito.it/~salamon/g/rghg.pdf

More information

Hodge Structures. October 8, A few examples of symmetric spaces

Hodge Structures. October 8, A few examples of symmetric spaces Hodge Structures October 8, 2013 1 A few examples of symmetric spaces The upper half-plane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half space H

More information

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY

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

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

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

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

More information

Notes by Maksim Maydanskiy.

Notes by Maksim Maydanskiy. SPECTRAL FLOW IN MORSE THEORY. 1 Introduction Notes by Maksim Maydanskiy. Spectral flow is a general formula or computing the Fredholm index of an operator d ds +A(s) : L1,2 (R, H) L 2 (R, H) for a family

More information

The d-orbifold programme. Lecture 3 of 5: D-orbifold structures on moduli spaces. D-orbifolds as representable 2-functors

The d-orbifold programme. Lecture 3 of 5: D-orbifold structures on moduli spaces. D-orbifolds as representable 2-functors The d-orbifold programme. Lecture 3 of 5: D-orbifold structures on moduli spaces. D-orbifolds as representable 2-functors Dominic Joyce, Oxford University May 2014 For the first part of the talk, see preliminary

More information

MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY

MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract

More information

Lecture Fish 3. Joel W. Fish. July 4, 2015

Lecture Fish 3. Joel W. Fish. July 4, 2015 Lecture Fish 3 Joel W. Fish July 4, 2015 Contents 1 LECTURE 2 1.1 Recap:................................. 2 1.2 M-polyfolds with boundary..................... 2 1.3 An Implicit Function Theorem...................

More information

Minimal surfaces in quaternionic symmetric spaces

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

More information

Generalized complex geometry and topological sigma-models

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

More information

G. Bini - D. Iacono DIFFEOMORPHISM CLASSES OF CALABI-YAU VARIETIES

G. Bini - D. Iacono DIFFEOMORPHISM CLASSES OF CALABI-YAU VARIETIES Geom. Struc. on Riem. Man.-Bari Vol. 73/1, 3 4 (2015), 9 20 G. Bini - D. Iacono DIFFEOMORPHISM CLASSES OF CALABI-YAU VARIETIES Abstract. In this article we investigate diffeomorphism classes of Calabi-Yau

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

DIFFERENTIAL FORMS AND COHOMOLOGY

DIFFERENTIAL FORMS AND COHOMOLOGY DIFFERENIAL FORMS AND COHOMOLOGY ONY PERKINS Goals 1. Differential forms We want to be able to integrate (holomorphic functions) on manifolds. Obtain a version of Stokes heorem - a generalization of the

More information

RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY

RIEMANN 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 information

Symplectic geometry on moduli spaces of J-holomorphic curves

Symplectic geometry on moduli spaces of J-holomorphic curves Symplectic geometry on moduli spaces of J-holomorphic curves J. Coffey, L. Kessler, and Á. Pelayo Abstract Let M, ω be a symplectic manifold, and Σ a compact Riemann surface. We define a 2-form ω SiΣ on

More information

EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION

EXAMPLES 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 information

HODGE THEORY, SINGULARITIES AND D-MODULES

HODGE THEORY, SINGULARITIES AND D-MODULES Claude Sabbah HODGE THEORY, SINGULARITIES AND D-MODULES LECTURE NOTES (CIRM, LUMINY, MARCH 2007) C. Sabbah UMR 7640 du CNRS, Centre de Mathématiques Laurent Schwartz, École polytechnique, F 91128 Palaiseau

More information

The geometry of Landau-Ginzburg models

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

More information

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

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

More information

Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture)

Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) Building Geometric Structures: Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover

More information

manifold structure on Dolbeault cohomology27 toc contentsline subsection numberline

manifold structure on Dolbeault cohomology27 toc contentsline subsection numberline toc contentsline section numberline 1.A construction of formal Frobenius manifolds18 toc contentsline subsection numberline 1.1.Frobenius algebras and formal Frobenius manifolds18 toc contentsline subsection

More information

REVISTA DE LA REAL ACADEMIA DE CIENCIAS. Exactas Físicas Químicas y Naturales DE ZARAGOZA. Serie 2.ª Volumen 69

REVISTA DE LA REAL ACADEMIA DE CIENCIAS. Exactas Físicas Químicas y Naturales DE ZARAGOZA. Serie 2.ª Volumen 69 REVISTA DE LA REAL ACADEMIA DE CIENCIAS Exactas Físicas Químicas y Naturales DE ZARAGOZA Serie 2.ª Volumen 69 2014 ÍNDICE DE MATERIAS Special Hermitian metrics, complex nilmanifolds and holomorphic deformations

More information

Applications to the Beilinson-Bloch Conjecture

Applications to the Beilinson-Bloch Conjecture Applications to the Beilinson-Bloch Conjecture Green June 30, 2010 1 Green 1 - Applications to the Beilinson-Bloch Conjecture California is like Italy without the art. - Oscar Wilde Let X be a smooth projective

More information

Rational Curves On K3 Surfaces

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:

More information

DEFORMATIONS OF A STRONGLY PSEUDO-CONVEX DOMAIN OF COMPLEX DIMENSION 4

DEFORMATIONS OF A STRONGLY PSEUDO-CONVEX DOMAIN OF COMPLEX DIMENSION 4 TOPICS IN COMPLEX ANALYSIS BANACH CENTER PUBLICATIONS, VOLUME 31 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1995 DEFORMATIONS OF A STRONGLY PSEUDO-CONVEX DOMAIN OF COMPLEX DIMENSION 4

More information

DONAGI-MARKMAN CUBIC FOR HITCHIN SYSTEMS arxiv:math/ v2 [math.ag] 24 Oct 2006

DONAGI-MARKMAN CUBIC FOR HITCHIN SYSTEMS arxiv:math/ v2 [math.ag] 24 Oct 2006 DONAGI-MARKMAN CUBIC FOR HITCHIN SYSTEMS arxiv:math/0607060v2 [math.ag] 24 Oct 2006 DAVID BALDUZZI Abstract. The Donagi-Markman cubic is the differential of the period map for algebraic completely integrable

More information

Conjectures on counting associative 3-folds in G 2 -manifolds

Conjectures 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 information

L6: Almost complex structures

L6: Almost complex structures L6: Almost complex structures To study general symplectic manifolds, rather than Kähler manifolds, it is helpful to extract the homotopy-theoretic essence of having a complex structure. An almost complex

More information

MONODROMY INVARIANTS IN SYMPLECTIC TOPOLOGY

MONODROMY INVARIANTS IN SYMPLECTIC TOPOLOGY MONODROMY INVARIANTS IN SYMPLECTIC TOPOLOGY DENIS AUROUX This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides

More information

Solvability of the Dirac Equation and geomeric applications

Solvability of the Dirac Equation and geomeric applications Solvability of the Dirac Equation and geomeric applications Qingchun Ji and Ke Zhu May 21, KAIST We study the Dirac equation by Hörmander s L 2 -method. On surfaces: Control solvability of the Dirac equation

More information

FANO VARIETIES AND EPW SEXTICS

FANO 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 information

Compact Riemannian manifolds with exceptional holonomy

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

More information

J-holomorphic curves in symplectic geometry

J-holomorphic curves in symplectic geometry J-holomorphic curves in symplectic geometry Janko Latschev Pleinfeld, September 25 28, 2006 Since their introduction by Gromov [4] in the mid-1980 s J-holomorphic curves have been one of the most widely

More information

MAT 545: Complex Geometry Fall 2008

MAT 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 information

Broken pencils and four-manifold invariants. Tim Perutz (Cambridge)

Broken pencils and four-manifold invariants. Tim Perutz (Cambridge) Broken pencils and four-manifold invariants Tim Perutz (Cambridge) Aim This talk is about a project to construct and study a symplectic substitute for gauge theory in 2, 3 and 4 dimensions. The 3- and

More information

THE UNIFORMISATION THEOREM OF RIEMANN SURFACES

THE UNIFORMISATION THEOREM OF RIEMANN SURFACES THE UNIFORISATION THEORE OF RIEANN SURFACES 1. What is the aim of this seminar? Recall that a compact oriented surface is a g -holed object. (Classification of surfaces.) It can be obtained through a 4g

More information

1 Hermitian symmetric spaces: examples and basic properties

1 Hermitian symmetric spaces: examples and basic properties Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................

More information

Dolbeault cohomology and. stability of abelian complex structures. on nilmanifolds

Dolbeault cohomology and. stability of abelian complex structures. on nilmanifolds Dolbeault cohomology and stability of abelian complex structures on nilmanifolds in collaboration with Anna Fino and Yat Sun Poon HU Berlin 2006 M = G/Γ nilmanifold, G: real simply connected nilpotent

More information