Foundations of Moduli Theory

University Of Washington, Seattle WA

Investigators

Abstract

This project investigates moduli spaces in algebraic geometry and allied fields. Algebraic geometry studies the geometry of spaces defined by polynomial equations. In addition to being one of the most ancient subjects in mathematics, algebraic geometry is also at the forefront of research in modern mathematics. Its abstract foundations are vital in the study of modern number theory, algebraic topology, representation theory, and combinatorics. At the same time, it provides powerful tools in applied mathematics, with applications including cryptography, theory of computation, convex optimization, computer graphics, statistics, and machine learning. This project aims to study one of the most fundamental questions in algebraic geometry, namely classifying algebraic varieties. A moduli space is itself an algebraic variety whose points are in one-to-one correspondence with the algebraic varieties that are being classified. Moduli spaces provide us with rich information about the geometric objects being classified and moreover have deep applications in numerous other fields of mathematics, both pure and applied. The research objectives are twofold: to develop abstract foundational tools in moduli theory and then apply these tools to study specific moduli spaces. This project will also involve the training of graduate students in moduli theory research. The investigator has developed a new approach to construct projective moduli spaces of objects with positive dimensional automorphism groups. While the moduli space of stable curves classifies objects with finite automorphism groups, there are many other moduli spaces of interest that do not share this feature. Examples include the moduli of vector bundles or sheaves, the moduli of Bridgeland semistable complexes, and the moduli of K-semistable varieties. Recent developments have provided necessary and sufficient conditions for an algebraic stack to admit a good moduli space in characteristic 0. This result has already been applied to construct new projective moduli spaces of Bridgeland semistable objects and K-semistable Fano varieties. This approach rests on local structure theorems for algebraic stacks which, at the moment, are limited to characteristic 0. This project aims to extend these results to positive and mixed characteristics. At the same time, the project aims to apply recent advances to study specific moduli spaces of varieties such as modular descriptions of log canonical models of the moduli space of curves. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.