Special quaternionic manifolds from variations of Hodge structures. A. Baarsma, J. Stienstra, T. van der Aalst, S. Vandoren (to appear)
|
|
- Clemence Griffin
- 5 years ago
- Views:
Transcription
1 Special quaternionic manifolds from variations of Hodge structures A. Baarsma, J. Stienstra, T. van der Aalst, S. Vandoren (to appear)
2 A conversation Physicist: Quaternion-Kähler manifolds from T-dualizing vector multiplets into hypermultiplets in N=2 D=4 supergravity Mathematician: Excuse me? Physicist: Compactify type IIA strings on a CY 3 manifold, look at complex structure deformations, and add RR-fields + dilaton Mathematician: What are you talking about?
3 Two years later Mathematician: Aha! Start with variations of Hodge structures of CY type, construct bundle of Weil intermediate Jacobians, and add punctured discs. Is that it? Physicist: What are you talking about?
4 Aim I will present a construction of quaternion- Kähler manifolds, discovered by physicists (Cecotti, Ferrara and Girardello, 91 the c-map ), but now translated into purely mathematical terms. [Related work by: M. Roček, C. Vafa, S.V. 06; N. Hitchin, 09; V. Cortés, 98; V. Cortés, T Mohaupt, H. Xu, arxiv: (see Sunday-talk!)]
5 Quaternion-Kähler manfiolds QK manifolds (M,g,J) of dimension 4n have holonomy Sp(n)xSp(1). They are Einstein.[Supersymmetry : R<0!!] Quaternionic structure J i J j ij 1 ijk J k Not integrable: Sp(1) connection p r J r p r J r r R(p) gj
6 Content Variations of CY-Hodge structures Quaternionic vector bundles Hyperbolic spaces & Heisenberg groups Quaternion-Kähler manifolds Weil intermediate Jacobians The full picture
7 CY-Hodge structures Let X be a compact Kähler manifold; (complexified) cohomology groups H k (X,C) Hodge decomposition H k (X, ) p q k H p,q (X) k=3 (CY 3, dim H 3,0 =1): H H 3,0 H 2,1 H 1,2 H 0,3 H
8 CY-Hodge structures Weil operator (complex structure on H 3 (X,R)) C H p,q i p q H p,q H q, p Non-degenerate alternating form Q(H p,q, H p ',q ' ) 0 Q(a,Ca) 0 a H \ {0} Q(a,b) X a b if (p,q) (q', p')
9 CY-Hodge structures Hodge filtration F: F q p q H p,3 q H p,q F p F q Instead of Hodge decomposition, we work with the filtration 0 F 4 F 3 F 2 F 1 F 0 H 3 (X, )
10 Variation of Hodge structures Family of compact Kähler manifolds (think of moduli space M of complex structure deformations of CY 3 ) Family of Hodge structures. Hodge bundle (Bryant & Griffiths, 83): H M Variation of Hodge structure on base manifold M: (M, H, GM,F,Q)
11 Variation of Hodge structures Griffiths transversality: GM (F p ) 1 (F p 1 ) Essentially, this means that : F 3 TM F 2 / F 3 H 2,1 X (X) [ X ] is a homomorphism. When M is such that this is an isomorphism, M is called (projective) special Kähler. [Cortés, Freed]
12 Special Kähler manifolds Hermitian form on H R (indefinite): h 2iQ(, ) Restriction to F 3 =H 3,0 is positive definite and defines Kähler form and (Weil- Petersson) metric: log(h(, )) K WP log(h(, ))
13 Weil versus Griffiths Griffiths: is positive definite on H 3,0 +H 1,2 and negative definite on H 0,3 +H 2,1. [ Q(a,Ca)>0 ] Weil metric: h 2iQ(, ) h W (.,. ) Q( C.,. ) iq(.,. ) is positve definite on entire H H 3,0 H 2,1 H 1,2 H 0,3 C : i i i i
14 Quaternionic Vector Bundles Let M be a special Kähler manifold. We can construct a vector bundle over M: For each fibre above to the quaternionic algebra: Q Q Q Q M F 3 t M,Q t is isomorphic ( f, ) (g, ) a ( fg h(, ), f g )
15 Quaternionic Vector Bundles Define now the vector bundle W F 3* F 2 F 2 Hom(F 3,F 2 ) F 2 Hom(H 3,0, H 3,0 H 2,1 ) H 1,2 H 0,3.1 TM H 1,2 H 0,3 Right-action of the quaternions on W (fibre-wise): W Q W
16 Quaternionic Vector Bundles Define metric g W :W W g (w,w ) Re(h(, )h W (, ) h W (, )) W w h(., ) F 3* 1,2 1,2 1,2 1,2 Metric compatible with quaternionic action: g W (w 1. q,w 2. q) q 2 g W (w 1,w 2 )
17 Further plan Consider the fibres Hodge bundle M. H H,t ; t M of the Statement 1: We can enlarge each fibre to become complex hyperbolic space. Statement 2: Bundle of hyperbolic spaces over M is a quaternion-kähler manifold.
18 Complex hyperbolic space Representation in terms of Siegel domain: S {(z,w) n Im(w) 1 2 z 2 } Siegel domain associated to a Hodge structure: S W {(a, ) H Im( ) 1 2 hw (a,a)}
19 Complex hyperbolic space Bergmann metric h B on S W : Kähler, with Kähler potential K B (a, ) log(2im( ) h W (a,a)) Siegel domain S W is diffeomorphic to the extended Heisenberg group EHeis H Group multiplication (a,, ) (a', ', ') (a a', ' Q(a,a'), ')
20 Bundle of hyperbolic spaces Let (M, H, GM,F,Q) be a special Kähler manifold, Hodge bundle Define the product manifold H M M QK H Projection map Fibres ˆ : M QK M EHeis t ˆ 1 (t) H,t
21 Metric We can construct a metric on M QK : Tangent bundle g QK ˆ * g WP g B TM QK ˆ * TM ker(d ˆ )
22 Quaternionic vector bundles Reminder: vector bundles over M: Q M F 3 W F 3* F 2 F 2 W Q W Pull back to vector bundles over M QK : ˆQ ˆ * Q Wˆ ˆ * W One can construct an isomorphism : ˆ W TM QK
23 Quaternionic Geometry Pull back the metric g W on Theorem: W to Wˆ g QK ( w 1, w 2 ) g ˆ W (w 1,w 2 ) w 1,2 ˆ Wˆt ; W t Also after the pull back, ˆQ acts linearly on W ˆ ; TM, so ˆQ QK provides subbundle of End(TM QK ), hence a quaternionic structure. One can show that M QK is QK!
24 Summary of further results We can construct bundle of intermediate Jacobians Jac H / H M Combined with Weil complex structure and Weil metric, this becomes an abelian variety (torus).
25 Summary of further results One can take quotient of Siegel domain EHeis( ) H Heis( ) Bundle of punctured disks over Jac over M For fixed, one gets circle bundle over the Weil intermediate Jacobian: contact geometry, Sasaki structure
26 The full picture (A. Baarsma)
The exact quantum corrected moduli space for the universal hypermultiplet
The exact quantum corrected moduli space for the universal hypermultiplet Bengt E.W. Nilsson Chalmers University of Technology, Göteborg Talk at "Miami 2009" Fort Lauderdale, December 15-20, 2009 Talk
More informationLecture II: Geometric Constructions Relating Different Special Geometries I
Lecture II: Geometric Constructions Relating Different Special Geometries I Vicente Cortés Department of Mathematics University of Hamburg Winter School Geometry, Analysis, Physics Geilo (Norway), March
More informationGeometry 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 informationNon-perturbative strings from automorphic forms
Bengt E.W. Nilsson Chalmers University of Technology, Göteborg Talk at the International Conference on "Strings, M-theory and Quantum Gravity" Ascona, July 25-29, 2010 Talk based on: two papers with Ling
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 informationGeometry 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 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 informationHolonomy 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
More informationOn 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 informationElementary Deformations HK/QK. Oscar MACIA. Introduction [0] Physics [8] Twist [20] Example [42]
deformations and the HyperKaehler/Quaternionic Kaehler correspondence. (Joint work with Prof. A.F. SWANN) Dept. Geometry Topology University of Valencia (Spain) June 26, 2014 What do we have 1 2 3 4 Berger
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 informationTwo 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 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 informationGeometry of the Moduli Space of Curves
Harvard University January, 2007 Moduli spaces and Teichmüller spaces of Riemann surfaces have been studied for many years, since Riemann. They have appeared in many subjects of mathematics, from geometry,
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 informationGeometric structures on the Figure Eight Knot Complement. ICERM Workshop
Figure Eight Knot Institut Fourier - Grenoble Sep 16, 2013 Various pictures of 4 1 : K = figure eight The complete (real) hyperbolic structure M = S 3 \ K carries a complete hyperbolic metric M can be
More informationHERMITIAN SYMMETRIC DOMAINS. 1. Introduction
HERMITIAN SYMMETRI DOMAINS BRANDON LEVIN 1. Introduction Warning: these are rough notes based on the two lectures in the Shimura varieties seminar. After the unitary Hermitian symmetric spaces example,
More informationPeriod 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 informationGeneralising Calabi Yau Geometries
Generalising Calabi Yau Geometries Daniel Waldram Stringy Geometry MITP, 23 September 2015 Imperial College, London with Anthony Ashmore, to appear 1 Introduction Supersymmetric background with no flux
More informationInfinitesimal Einstein Deformations. Kähler Manifolds
on Nearly Kähler Manifolds (joint work with P.-A. Nagy and U. Semmelmann) Gemeinsame Jahrestagung DMV GDM Berlin, March 30, 2007 Nearly Kähler manifolds Definition and first properties Examples of NK manifolds
More informationHolonomy 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
More informationString Theory and Generalized Geometries
String Theory and Generalized Geometries Jan Louis Universität Hamburg Special Geometries in Mathematical Physics Kühlungsborn, March 2006 2 Introduction Close and fruitful interplay between String Theory
More informationFormality 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 informationRelating DFT to N=2 gauged supergravity
Relating DFT to N=2 gauged supergravity Erik Plauschinn LMU Munich Chengdu 29.07.2016 based on... This talk is based on :: Relating double field theory to the scalar potential of N=2 gauged supergravity
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 informationAN 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 informationD3-instantons, Mock Theta series and Twistors
D3-instantons, Mock Theta series and Twistors Boris Pioline CERN & LPTHE String Math 2012 based on arxiv:1207.1109 with S. Alexandrov and J. Manschot Boris Pioline (CERN & LPTHE) D3-instantons and mock
More informationHeterotic Flux Compactifications
Heterotic Flux Compactifications Mario Garcia-Fernandez Instituto de Ciencias Matemáticas, Madrid String Pheno 2017 Virginia Tech, 7 July 2017 Based on arxiv:1611.08926, and joint work with Rubio, Tipler,
More informationThe 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 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 informationMirror 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 informationHodge 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 informationGeometric 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
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationF-theory effective physics via M-theory. Thomas W. Grimm!! Max Planck Institute for Physics (Werner-Heisenberg-Institut)! Munich
F-theory effective physics via M-theory Thomas W. Grimm Max Planck Institute for Physics (Werner-Heisenberg-Institut) Munich Ahrenshoop conference, July 2014 1 Introduction In recent years there has been
More informationTorus actions and Ricci-flat metrics
Department of Mathematics, University of Aarhus November 2016 / Trondheim To Eldar Straume on his 70th birthday DFF - 6108-00358 Delzant HyperKähler G2 http://mscand.dk https://doi.org/10.7146/math.scand.a-12294
More informationElements of Topological M-Theory
Elements of Topological M-Theory (with R. Dijkgraaf, S. Gukov, C. Vafa) Andrew Neitzke March 2005 Preface The topological string on a Calabi-Yau threefold X is (loosely speaking) an integrable spine of
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 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 informationPietro Fre' SISSA-Trieste. Paolo Soriani University degli Studi di Milano. From Calabi-Yau manifolds to topological field theories
From Calabi-Yau manifolds to topological field theories Pietro Fre' SISSA-Trieste Paolo Soriani University degli Studi di Milano World Scientific Singapore New Jersey London Hong Kong CONTENTS 1 AN INTRODUCTION
More informationTHE 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 informationCohomology jump loci of quasi-projective varieties
Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)
More information7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the
More informationHARMONIC 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
More informationGeneralized N = 1 orientifold compactifications
Generalized N = 1 orientifold compactifications Thomas W. Grimm University of Wisconsin, Madison based on: [hep-th/0602241] Iman Benmachiche, TWG [hep-th/0507153] TWG Madison, Wisconsin, November 2006
More informationPROGRAM. Monday Tuesday Wednesday Thursday Friday. 11:00 Coffee Coffee Coffee Coffee Coffee. Inverso. 16:00 Coffee Coffee Coffee Coffee Coffee
PROGRAM Monday Tuesday Wednesday Thursday Friday 11:00 Coffee Coffee Coffee Coffee Coffee 11:30 Vicente Cortés Georgios Papadopoulos Calin Lazaroiu Alessandro Tomasiello Luis Álvarez Cónsul 15:00 Dietmar
More information2d-4d wall-crossing and hyperholomorphic bundles
2d-4d wall-crossing and hyperholomorphic bundles Andrew Neitzke, UT Austin (work in progress with Davide Gaiotto, Greg Moore) DESY, December 2010 Preface Wall-crossing is an annoying/beautiful phenomenon
More informationCHARACTERISTIC CLASSES
1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact
More informationMAPPING CLASS ACTIONS ON MODULI SPACES. Int. J. Pure Appl. Math 9 (2003),
MAPPING CLASS ACTIONS ON MODULI SPACES RICHARD J. BROWN Abstract. It is known that the mapping class group of a compact surface S, MCG(S), acts ergodically with respect to symplectic measure on each symplectic
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 informationMoment Maps and Toric Special Holonomy
Department of Mathematics, University of Aarhus PADGE, August 2017, Leuven DFF - 6108-00358 Delzant HyperKähler G 2 Outline The Delzant picture HyperKähler manifolds Hypertoric Construction G 2 manifolds
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 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 informationPart III- Hodge Theory Lecture Notes
Part III- Hodge Theory Lecture Notes Anne-Sophie KALOGHIROS These lecture notes will aim at presenting and explaining some special structures that exist on the cohomology of Kähler manifolds and to discuss
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 informationarxiv: v3 [hep-th] 22 Sep 2014
Prepared for submission to JHEP IPhT-t14/056 Spin(7-manifolds in compactifications to four dimensions arxiv:1405.3698v3 [hep-th] 22 Sep 2014 Mariana Graña, a C. S. Shahbazi a and Marco Zambon b a Institut
More informationOn Flux Quantization in F-Theory
On Flux Quantization in F-Theory Raffaele Savelli MPI - Munich Bad Honnef, March 2011 Based on work with A. Collinucci, arxiv: 1011.6388 Motivations Motivations The recent attempts to find UV-completions
More informationGeneralized Contact Structures
Generalized Contact Structures Y. S. Poon, UC Riverside June 17, 2009 Kähler and Sasakian Geometry in Rome in collaboration with Aissa Wade Table of Contents 1 Lie bialgebroids and Deformations 2 Generalized
More informationBlack holes, instantons and twistors
Black holes, instantons and twistors Boris Pioline LPTHE, Paris Based on work with Günaydin, Neitzke, Waldron, Alexandrov, Saueressig, Vandoren,... ULB-VUB-KUL Seminar, Nov 19, 2008, Leuven Boris Pioline
More informationTechniques 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
More informationGeneralized 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 information1 Moduli spaces of polarized Hodge structures.
1 Moduli spaces of polarized Hodge structures. First of all, we briefly summarize the classical theory of the moduli spaces of polarized Hodge structures. 1.1 The moduli space M h = Γ\D h. Let n be an
More informationREVISTA 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 informationContributors. Preface
Contents Contributors Preface v xv 1 Kähler Manifolds by E. Cattani 1 1.1 Complex Manifolds........................... 2 1.1.1 Definition and Examples.................... 2 1.1.2 Holomorphic Vector Bundles..................
More informationPoisson modules and generalized geometry
arxiv:0905.3227v1 [math.dg] 20 May 2009 Poisson modules and generalized geometry Nigel Hitchin May 20, 2009 Dedicated to Shing-Tung Yau on the occasion of his 60th birthday 1 Introduction Generalized complex
More information1: Lie groups Matix groups, Lie algebras
Lie Groups and Bundles 2014/15 BGSMath 1: Lie groups Matix groups, Lie algebras 1. Prove that O(n) is Lie group and that its tangent space at I O(n) is isomorphic to the space so(n) of skew-symmetric matrices
More informationConstructing 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 informationDIFFERENTIAL GEOMETRY AND THE QUATERNIONS. Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013
DIFFERENTIAL GEOMETRY AND THE QUATERNIONS Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013 16th October 1843 ON RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 1 SHIING-SHEN CHERN Introduction.
More informationDEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE
DEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE ANGELA ORTEGA (NOTES BY B. BAKKER) Throughout k is a field (not necessarily closed), and all varieties are over k. For a variety X/k, by a basepoint
More informationMany of the exercises are taken from the books referred at the end of the document.
Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the
More informationMinimal 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 informationTwistor 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
More informationOn Spectrum and Arithmetic
On Spectrum and Arithmetic C. S. Rajan School of Mathematics, Tata Institute of Fundamental Research, Mumbai rajan@math.tifr.res.in 11 August 2010 C. S. Rajan (TIFR) On Spectrum and Arithmetic 11 August
More informationNOTES ON FIBER BUNDLES
NOTES ON FIBER BUNDLES DANNY CALEGARI Abstract. These are notes on fiber bundles and principal bundles, especially over CW complexes and spaces homotopy equivalent to them. They are meant to supplement
More informationDouble Field Theory and Stringy Geometry
Double Field Theory and Stringy Geometry String/M Theory Rich theory, novel duality symmetries and exotic structures Fundamental formulation? Supergravity limit - misses stringy features Infinite set of
More information1 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 informationGeneralized Hitchin-Kobayashi correspondence and Weil-Petersson current
Author, F., and S. Author. (2015) Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current, International Mathematics Research Notices, Vol. 2015, Article ID rnn999, 7 pages. doi:10.1093/imrn/rnn999
More informationIntroduction to Poincare Conjecture and the Hamilton-Perelman program
Introduction to Poincare Conjecture and the Hamilton-Perelman program David Glickenstein Math 538, Spring 2009 January 20, 2009 1 Introduction This lecture is mostly taken from Tao s lecture 2. In this
More informationG 2 manifolds and mirror symmetry
G 2 manifolds and mirror symmetry Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics First Annual Meeting, New York, 9/14/2017 Andreas Braun University of Oxford based on [1602.03521]
More informationTopics 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 informationPeriodic 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
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 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 informationUniformization in several complex variables.
Uniformization in several complex variables. Grenoble. April, 16, 2009. Transmitted to IMPA (Rio de Janeiro). Philippe Eyssidieux. Institut Fourier, Université Joseph Fourier (Grenoble 1). 1 The uniformization
More informationMirror symmetry for G 2 manifolds
Mirror symmetry for G 2 manifolds based on [1602.03521] [1701.05202]+[1706.xxxxx] with Michele del Zotto (Stony Brook) 1 Strings, T-duality & Mirror Symmetry 2 Type II String Theories and T-duality Superstring
More informationSimon Salamon. Turin, 24 April 2004
G 2 METRICS AND M THEORY Simon Salamon Turin, 24 April 2004 I Weak holonomy and supergravity II S 1 actions and triality in six dimensions III G 2 and SU(3) structures from each other 1 PART I The exceptional
More informationHodge Structures and Shimura Data
Hodge Structures and Shimura Data It is interesting to understand when the example of GL 2 (R) acting on the Hermitian symmetric space C R or Sp 2g (R) acting on H g, or U(p, q) acting on X (from the last
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 informationT-duality : a basic introduction
T-duality : a basic introduction Type IIA () Type IIB 10th Geometry and Physics conference, Quantum Geometry Anogia, 6-10 August 2012 Mathai Varghese Collaborators and reference joint work with: - Peter
More informationMirrored K3 automorphisms and non-geometric compactifications
Mirrored K3 automorphisms and non-geometric compactifications Dan Israe l, LPTHE New Frontiers in String Theory, Kyoto 2018 Based on: D.I. and Vincent Thie ry, arxiv:1310.4116 D.I., arxiv:1503.01552 Chris
More informationDIFFERENTIAL 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 informationDolbeault 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 informationA relative version of Kostant s theorem
A relative version of Kostant s theorem 1 University of Vienna Faculty of Mathematics Srni, January 2015 1 supported by project P27072 N25 of the Austrian Science Fund (FWF) This talk reports on joint
More informationA Supergravity Dual for 4d SCFT s Universal Sector
SUPERFIELDS European Research Council Perugia June 25th, 2010 Adv. Grant no. 226455 A Supergravity Dual for 4d SCFT s Universal Sector Gianguido Dall Agata D. Cassani, G.D., A. Faedo, arxiv:1003.4283 +
More informationHolomorphic 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
More informationGeometry of moduli spaces
Geometry of moduli spaces 20. November 2009 1 / 45 (1) Examples: C: compact Riemann surface C = P 1 (C) = C { } (Riemann sphere) E = C / Z + Zτ (torus, elliptic curve) 2 / 45 (2) Theorem (Riemann existence
More informationExercises 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
More informationELEMENTARY DEFORMATIONS AND THE HYPERKAEHLER / QUATERNIONIC KAEHLER CORRESPONDENCE
ELEMENTARY DEFORMATIONS AND THE HYPERKAEHLER / QUATERNIONIC KAEHLER CORRESPONDENCE Oscar Macia (U. Valencia) (in collaboration with Prof. A.F. Swann) Aarhus, DK March 2, 2017 1 Planning Introduction and
More informationAn 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
More informationShafarevich 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