New directions in birational geometry and singularity theory

In this project the PI will study questions in algebraic geometry and commutative algebra. Algebraic geometry is the study of algebraic varieties, which are solution sets for systems of polynomial equations. For example, in the xy-plane, the solutions for y=0 consist of all points along the x-axis, while the solutions for xy=0 consist of all points along both coordinate axes. Since the tangent line at the origin (0,0) is not defined for the algebraic variety defined by xy=0, we say that this variety has a singularity at the origin (0,0). A central focus in the proposed research is that studying singularities is indispensable even when the objects of interest are smooth manifolds, which are not singular. In another direction, certain analytic objects defined using possibly divergent power series are often unavoidable as well. This contrasts with the usual situation on smooth manifolds, where all smooth functions have Taylor series expansions that converge in some neighborhood of a point. The PI will study the analogues of these situations in the field of algebraic geometry -- in particular, birational geometry -- and its interactions with other fields of mathematics, such as commutative algebra, complex geometry, and arithmetic geometry. The project will also provide research training opportunities for students.

This project unifies and expands the classical boundaries of algebraic geometry and commutative algebra in multiple interconnected directions. First, the PI will expand the scope of birational geometry and the minimal model program to broader classes of singularities and categories of spaces. This work includes work on the relative minimal model program for (semi-)log canonical pairs on schemes and algebraic spaces. This work also extends to both complex and non-Archimedean analytic spaces (where functions are convergent power series) and to formal schemes (where functions are divergent power series). The PI will also extend Matsusaka's big theorem to families of varieties with normal or rational singularities, and work towards solving new cases of ACC conjecture for minimal log discrepancies, where divergent power series play a pivotal role. Second, the PI will develop new tools and techniques to construct embeddings of projective varieties in projective spaces. This work includes higher-dimensional cases of the PI's Fujita-type conjectures and a new notion of centers of F-purity, the latter of which will be applied to Iitaka's Conjecture on the subadditivity of Kodaira dimension.

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.

Award Number: 2502653
Principal Investigator: Takumi Murayama
Funds Obligated: $144,500
State: IN