When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety $X$, to what extent can $D^b(X)$ be used as an invariant to answer rationality questions? In particular, what properties of $D^b(X)$ are implied by $X$ being rational, stably rational, or having a rational point? On the other hand, is there a property of $D^b(X)$ that implies that $X$ is rational, stably rational, or has a rational point? In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Mattew Ballard, Alexander Duncan, and Patrick McFaddin.

Cohen-Macaulay rings play a central role in commutative algebra. Translating to algebraic geometry, we have Cohen-Macaulay schemes. Cohen-Macaulay schemes have many nice geometric properties. In this talk, I will explain how we can understand Cohen-Macaulay objects in the category of D-modules by using duality, especially for relative D-modules. Then, we will use them to study Bernstein-Sato polynomials/ideals. As an application, I will use Cohen-Macaulayness of relative D-modules to study the topological cohomology jumping loci of rank 1 local systems and prove a conjecture of Budur. This is joint work with Budur, Veer and Zhou.

The ACC conjecture for minimal log discrepancies (mlds) is one of the core conjectures in birational geometry, and is expected to play a crucial role in the proof of termination of flips, one of the most important conjectures of the minimal model program. In this talk, I will survey some of my recent progress on the ACC conjecture for mlds. Part of the talk is joint works with Jingjun Han, V. V. Shokurov, and Liudan Xiao.

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.

By work of McQuillan and Brunella, it is known that foliated surfaces of general type with only canonical foliation singularities admit a unique canonical model. It is then natural to investigate the moduli space parametrizing canonical models. One issue is that the condition being a canonical model is neither open nor closed. In this talk, I will introduce the generalized canonical models to fix this issue and study some properties (boundedness/separatedness/properness/local-closedness) of the moduli space of generalized canonical models.

Positivity of direct images of relative canonical bundles are important for the study of geometry of algebraic morphisms. In this talk, I would like to discuss a notion of metric positivity for coherent sheaves and prove that a large class of sheaves from Hodge theory, including direct images of relative canonical bundles, always satisfy the metric positivity. This result unifies and strengthens several results of positivity on the algebraic side (i.e. weak positivity). Based on joint work with Christian Schnell.

We study the stability of objects in the Kuznetsov component of the derived category of the double Veronese cone ramified over a cubic; it belongs to one of the five families of Fano threefolds of index two. There are three classes in the numerical Grothendieck group of the Kuznetsov component, minimal with respect to the Euler form, and an autoequivalence, which induces isomorphisms of the three corresponding moduli spaces. We determine the stable objects of minimal numerical classes and show that the moduli spaces consist of two components: one isomorphic to the Veronese double cone itself and the other to its Fano surface of lines. This is joint work with Franco Rota.