Varieties with trivial canonical class and Hodge theory

Algebraic geometry is the study of geometric objects defined by polynomial equations. It is a richly structured subject at the intersection of several areas of mathematics, combining tools and ideas from algebra, geometry, topology, representation theory, and number theory. This project focuses on one of the core questions in the field: the classification and structure of K-trivial varieties. These include elliptic curves, which underpin modern cryptography, and Calabi-Yau threefolds, which play a central role in string theory as geometric models of the universe. The questions explored in this research are fundamental to current developments in mathematics and connect with related disciplines such as physics and computer science, offering promising avenues for real-world applications. A key component of the project is the involvement of early-career researchers, especially doctoral students, who will contribute directly to the development of new techniques and their application to the problems under investigation. This project will address two foundational questions in the study of K-trivial varieties: the structure of hyper-Kaehler manifolds and the compactification of moduli spaces of strict Calabi-Yau manifolds. First, the classification of hyper-Kaehler manifolds is one of the most elusive aspects of contemporary algebraic geometry; the most recent breakthrough in this direction was made by O'Grady over a quarter century ago. Recent methods and perspectives enable significant progress. A central aspect of the project is the investigation of deep connections between hyper-Kaehler manifolds and representation theory. In particular, the PI aims to classify Kummer-type hyper-Kaehler manifolds and, more broadly, to understand the symplectic correspondences between hyper-Kaehler manifolds and abelian varieties. Second, for strict Calabi-Yau varieties, the absence of compact moduli spaces is a significant challenge. The project seeks to develop a Hodge-theoretic approach to compactifying Calabi-Yau moduli spaces, building on the PI's earlier work with Griffiths, Friedman, and others. The primary tools for addressing these questions are Hodge theory, birational geometry, representation theory, and the recently developed theory of higher Du Bois and higher rational singularities. 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.