Kęstutis Česnavičius, Problems about torsors over regular rings
Abstract: Problems about torsors, such as the Bass--Quillen and the Grothendieck--Serre conjectures, are captivating in that they force one to understand the fine local structure of a smooth variety or, more generally, of a regular local ring. In my lectures, I will discuss this interplay, outline the intervening geometric arguments, such as versions of the Gabber--Quillen presentation lemma, and will overview the problems that remain open. In part, I will base my lectures on the survey article arXiv:2201.06424.
Mircea Mustață, Hodge filtration on local cohomology and the Du Bois complex
Abstract: It has been known for a long time that perverse sheaves have a lot to say about singularities of algebraic varieties. Morihiko Saito's theory of mixed Hodge modules is an enhancement of this theory, built on filtered D-modules, that brings in the picture coherent information. It turns out that interesting algebraic invariants, such as local cohomology modules, underlie mixed Hodge modules and as such, they come endowed with a Hodge filtration. Each step of this filtration is a coherent sheaf and in many instances it encodes in a subtle way information about the singularities of the objects involved (these can be varieties or morphisms). These lectures will give an introduction to the Hodge filtration on localizations (and, more generally, on local cohomology) and its connection with several other invariants of singularities. Here is a rough plan of the lectures:
1. Introduction to D-modules, b-functions, and related invariants of singularities
2. A brief introduction to V-filtrations and the theory of mixed Hodge modules
3. The Hodge filtration on local cohomology
4. Higher Du Bois and rational hypersurface singularities
Kevin Tucker, Recent developments in singularity theory in mixed and positive characteristic algebraic geometry
Abstract: Standard "reduction to characteristic p" techniques have long been used to relate singularities defined via the Frobenius map in positive characteristic and those arising in complex algebraic geometry and the Minimal Model Program (MMP). In particular, log terminal and F-regular singularities, log canonical and F-pure singularities, rational and F-rational singularities, and Du Bois and F-injective singularities are either known or conjectured to correspond to one another via reduction. Exciting developments have recently made it possible to exploit these connections in the mixed characteristic setting as well, drawing on the (conjectured) characterization of F-regular rings as splinters in characteristic p > 0. A ring is a splinter if it is a direct summand of every finite cover, and Hochster’s direct summand conjecture (now a Theorem) is the modest assertion that a regular ring of any characteristic is a splinter. This conjecture was settled affirmatively by André in 2018 who proved the mixed characteristic case more than three decades after Hochster's verification of the conjecture in equal characteristic using Frobenius techniques. In these talks, I will discuss some recent works on splinter rings in mixed and positive characteristics. In particular, inspired by the result of Bhatt in 2020 on the Cohen-Macaulayness of the absolute integral closure, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This can be seen as a generalization of the theory of globally F-regular pairs from positive to mixed characteristic, and led to the successful development of the three dimensional MMP in mixed characteristics (0; p > 5).
Chenyang Xu, Algebraic K-stability theory of Fano varieties
Abstract: In this lecture series, I want to discuss the project of using higher dimensional geometry to understand K-stability of Fano varieties. As a result, it leads to the construction of a project moduli space parametrizing K-polystable Fano varieties, as well as the solution of Yau-Tian-Donaldson Conjecture in the full generality for Fano varieties. In the lecture series, I plan to cover
1. Valuative criterion of K-stability for Fano varieties
2. Higher rank finite generation theorem
3. Construction of the K-moduli space
4. If time permits, I will also discuss the local theory centered around the notion of the normalized volume of a singularity.
No knowledge of K-stability is assumed, but certain familiarity with higher dimensional geometry will be helpful. Here are two reference links:
https://web.math.princeton.edu/~chenyang/Kstabilitybook.pdf
https://web.math.princeton.edu/~chenyang/Kstabilitysurvey.pdf