Birational geometry, symplectic varieties, and moduli spaces

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This project addresses three central problems in algebraic geometry:Can one compute the ample cone of a polarized holomorphic-symplectic variety from its Hodge structure? What is the right functorial definition for compact moduli spaces of higher-dimensional varieties (and what might they be good for)? To what extent does the birational geometry of moduli spaces govern the behavior of related Geometric Invariant Theory problems, and vice versa? These questions are intertwined in intricate and beautiful ways: Intersection-theoretic constructions govern curve classes on both the moduli space of stable curves and holomorphic-symplectic varieties. The Torelli Theorem for K3 surfaces is the starting point for their moduli theory; the lack of such a result for higher dimensional holomorphic-symplectic manifolds is a major impetus for analyzing their ample cones. The elusive dream of a geometric compactification for the moduli space of K3 surfaces animates work on the interplay between Geometric Invariant Theory and moduli spaces.Algebraic geometry is the study of geometric objects defined by polynomial equations, which are called varieties. Examples of varieties include circles, ellipses, parabolas, spheres, etc. A fundamental problem is to classify all the varieties of a given type. One approach is to analyze all the varieties defined by polynomials of given degree, e.g., the conic sections studied in high school analytic geometry. Here the type of the variety is expressed in algebraic terms. Alternately, one can study all the varieties sharing common geometric characteristics, e.g., those with given numerical invariants. This project addresses the interplay between the algebraic and geometric quantities, and how these govern the behavior of families of varieties.