Moduli Theory of Sheaves Over Low-Dimensional Varieties

Northeastern University, Boston MA

Investigators

Abstract

This project is in the field of algebraic geometry, one of the oldest yet currently most active areas of mathematics. At its core, algebraic geometry is the study of geometric spaces cut out by solution sets of systems of polynomial equations. The subject goes back to Greek antiquity on the one hand, while its modern development provides the mathematical foundation for current efforts to understand the physics of the early universe. For theoretical physics, the most relevant area of algebraic geometry is moduli theory, which deals with the classificaiton and deformation properties of important geometric objects of the same type. The current project will investigate moduli theory of sheaves, objects which correspond to the notion of physical (gauge) fields, such as the electromagnetic field. Mathematical invariants associated with moduli spaces of sheaves calculate amplitudes of high-energy processes in particle physics. Studying the structure of these invariants therefore bears on essential questions in both geometry and high-energy physics; the completion of this project will thus improve our understanding of both mathematics and theoretical physics. The endeavor fits in the important current research goal of setting needed mathematical foundations to contemporary theoretical physics. This project will focus on the study of the geometry of moduli spaces of sheaves on low-dimensional varieties, including the important setting when the variety is allowed to vary in moduli. In particular, the virtual intersection theory of Grothendieck Quot schemes over K3 surfaces, in a relative setting, will be used to study the tautological Chow ring of the moduli space of quasipolarized K3 surfaces. Special Chow classes, such as the Chern classes of Verlinde sheaves over the moduli space, will also be investigated. More generally, the virtual intersection theory of higher-rank Grothendieck Quot schemes over an arbitrary surface will be used to deduce results on the structure of important invariants on fundamental spaces such as Hilbert schemes of points. The higher-rank Quot scheme geometry also bears interesting connections to questions in representation theory.