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The complex hyperbolic geometry of the moduli space of cubic surfaces
Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer
Extensions of mixed Hodge structures
According to Deligne, the cohomology groups of a complex algebraic variety carry a generalized Hodge structure, or, in precise terms, a mixed Hodge structure [2]. The purpose of this paper is to
Period Mappings and Period Domains
Part I. Basic Theory of the Period Map: 1. Introductory examples 2. Cohomology of compact Kahler manifolds 3. Holomorphic invariants and cohomology 4. Cohomology of manifolds varying in a family 5.
Harmonic mappings of Kähler manifolds to locally symmetric spaces
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A Defect Relation for Equidimensional Holomorphic Mappings Between Algebraic Varieties
0. Introduction 1. Notations, terminology, and sign conventions (a) Line bundles and Chern classes (b) Currents and forms in C0 2. Construction of a volume form 3. A second main theorem for
Infinitesimal variations of Hodge structure and the global Torelli problem
The aim of this report is to introduce the infinitesimal variation of Hodge structure as a tool for studying the global Torelli problem. Although our techniques have so far borne fruit only in
The Moduli Space of Cubic Threefolds As a Ball Quotient
The moduli space of cubic threefolds in $\mathbb{C}P^4$, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. The
The Millennium Prize Problems
A history of prizes in mathematics by J. Gray The Birch and Swinnerton-Dyer conjecture by A. Wiles The Hodge conjecture by P. Deligne Existence and smoothness of the Navier-Stokes equation by C. L.
Quadratic presentations and nilpotent Kähler groups
It has been known for at least thirty years that certain nilpotent groups cannot be Kahler groups, i.e., fundamental groups of compact Kahler manifolds. The best known examples are lattices in the
Cubic surfaces with special periods
We show that the vector of period ratios of a cubic surface is rational over Q(ω), where ω = exp(2πi/3) if and only if the associate abelian variety is isogeneous to a product of Fermat elliptic