Back to Algebraic Geometry Seminar: Fall 2020
Algebraic Geometry Seminar
Date: Tuesday, Oct 27, 2020
Time: 3:30PM - 4:30PM
Location: Zoom
Devlin Mallory
University of Michigan
Title |
D-simplicity and bigness of the tangent bundle of Fano varieties |
Abstract |
Let $R$ be a singular ring, and $D_R$ the ring of differential operators on $R$. Despite decades of study and many fruitful applications, $D_R$ remains quite mysterious. In this talk, we will discuss connections between the algebraic description of differential operators on singular rings and the global geometry of Fano varieties. A recurring question in the literature is whether “nice” properties of differential operators (e.g., if R is a simple $D_R$-module, or “D-simple” for short) are implied by the “mildness” of the singularity. Specifically, we focus on the question of whether a Gorenstein ring with klt singularities must be D-simple. We answer this question in the negative, by giving several explicit examples, which arise as homogeneous coordinate rings of smooth Fano varieties $X$. The property of this ring being D-simple can be translated into the question of positivity (specifically, bigness) of the tangent bundle of $X$. I will discuss results on the bigness of the tangent bundle of del Pezzo surfaces and recent work of Höring, Liu, and Shao which completed the surface case; perhaps more importantly, I will point out how much is left to discover about bigness of the tangent bundle in higher dimensions. |
Algebraic Geometry