Geometry of KSBA spaces

The award supports research in algebraic geometry, a central branch of mathematics which aims to understand, both practically and conceptually, solutions of systems of polynomial equations in many variables. The particular focus of this project is on the study of families of algebraic varieties and the way these varieties deform and break up. Such studies found important applications in other fields of mathematics, such as number theory and topology, as well as in string theory in physics. Graduate students will be involved in and supported by this project. The PI will work on several projects concerning geometry and enumerative geometry of KSBA spaces, a generalization of Deligne-Mumford's moduli spaces of curves to higher dimensions. This includes a detailed study of the KSBA moduli spaces of anticanonical surfaces; KSBA compactifications of moduli of Calabi-Yau hyperplane arrangements; KSBA compactifications of moduli of certain 3-dimensional Calabi-Yau varieties; KSBA compactifications of moduli spaces of K3 surfaces with an automorphism; and the study of the kappa classes (generalized MMM classes) on various KSBA moduli spaces. 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.