Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds

Abstract

This proceedings note introduces aspects of the authors' work relating mirror symmetry and integral variations of Hodge structure. The emphasis is on their classification of the integral variations of Hodge structure which can underly families of Calabi-Yau threefolds over the thrice-punctured sphere with b^3 = 4, or equivalently h^{2,1} = 1, and the related issues of geometric realization of these variations. The presentation parallels that of the first author's talk at the BIRS workshop.

21 pages; To appear in "Mirror Symmetry V", Proceedings of BIRS workshop on Calabi-Yau Varieties and Mirror Symmetry, December 6-11, 2003; Video of the first author's talk at the workshop is available on the PIMS website at http://www.pims.math.ca/birs/workshops/2003/03w5061/doran/

References in corpus (7)

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces
• Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections
• Hypergeometric functions and mirror symmetry in toric varieties
• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• Collapsing D-Branes in Calabi-Yau Moduli Space: I
• Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Space

Cited by in corpus (41)

• Infinite Distances in Field Space and Massless Towers of States
• Supersymmetric partition functions on Riemann surfaces
• Arithmeticity of Certain Symplectic Hypergeometric Groups
• Topological String Theory on Compact Calabi-Yau: Modularity and Boundary Conditions
• Boundary Components of Mumford-Tate Domains
• Families of Quintic Calabi-Yau 3-Folds with Discrete Symmetries
• Conifolds and Tunneling in the String Landscape
• Dual Pairs of Gauged Linear Sigma Models and Derived Equivalences of Calabi-Yau threefolds
• Higher genus Gromov-Witten invariants of the Grassmannian, and the Pfaffian Calabi-Yau threefolds
• Arithmeticity of Four Hypergeometric Monodromy Groups associated to Calabi-Yau threefolds
• Eisenstein type series for Calabi-Yau varieties
• Calabi-Yau Threefolds Fibred by Mirror Quartic K3 Surfaces
• Families of lattice polarized K3 surfaces with monodromy
• Some monodromy groups of finite index in $S{p}_4(\mathbb{Z})$
• Finiteness and the Emergence of Dualities
• Algebraic Topology of Calabi-Yau Threefolds in Toric Varieties
• The 14th case VHS via K3 fibrations
• Examples of Calabi-Yau threefolds parametrised by Shimura varieties
• Global mirror symmetry for invertible simple elliptic singularities
• Analytic Continuation of Hypergeometric Functions in the Resonant Case
• Experimenting with symplectic hypergeometric monodromy groups
• Calabi--Yau Operators
• Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds
• Hodge classes associated to 1-parameter families of Calabi-Yau 3-folds
• Monodromy of an Inhomogeneous Picard-Fuchs Equation
• Modular-type functions attached to mirror quintic Calabi-Yau varieties
• Hodge Numbers from Picard-Fuchs Equations
• Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions
• The art of finding Calabi-Yau differential eq1uations
• On the mixed-twist construction and monodromy of associated Picard-Fuchs systems
• Distributions of extremal black holes in Calabi-Yau compactifications
• Mirror Symmetry and Projective Geometry of Fourier-Mukai Partners
• Two One-Parameter Special Geometries
• Differential equations satisfied by modular forms and K3 surfaces
• Homology supported in Lagrangian submanifolds in mirror quintic threefolds
• On the unipotence of autoequivalences of toric complete intersection Calabi-Yau categories
• A Strange Family of Calabi-Yau 3-folds
• Parabolic Higgs bundles, $t{t}^{\ast }$ connections and opers
• Modular-type functions attached to Calabi-Yau varieties: integrality properties
• Lattices generated by skeletons of reflexive polytopes
• Orthogonal Hypergeometric Groups with a Maximally Unipotent Monodromy