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

Eva Elduque, Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules

Abstract: In previous work jointly with Geske, Herradón Cueto, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this talk, we will talk about the solution to this question, as well as some consequences and explicit computations. Joint work with Moisés Herradón Cueto.

Philip Engel, Compact moduli of K3 surfaces

Abstract: The moduli space of K3 surfaces of genus g is the quotient of a symmetric space of complex dimension 19 by the action of an arithmetic group. In this capacity, it has a natural infinite class of so-called "toroidal" compactifications. Is any one of these toroidal compactifications distinguished, in the sense that it parameterizes some generalized or "stable" K3 surfaces? I will describe joint work with V. Alexeev giving an affirmative answer to this question. I will also discuss some of the combinatorics involved in understanding (1) the boundary and (2) the stable K3 surfaces that appear there.

Lena Ji, Curve classes on conic bundle threefolds and applications to rationality

Abstract: Clemens–Griffiths introduced the classical intermediate Jacobian obstruction to rationality for complex threefolds in their proof of the irrationality of the cubic threefold. Recently, over non-closed fields, Hassett–Tschinkel refined this obstruction using torsors over the intermediate Jacobian, and for geometrically rational threefolds Benoist–Wittenberg defined a group scheme whose components are these torsors. In this talk, we identify this group scheme for threefolds admitting a conic bundle structure, and we give applications to rationality for conic bundle threefolds over non-closed fields. In particular, we construct new examples of geometrically rational, irrational conic bundles over P^2 where various obstructions to rationality vanish. This work is joint with S. Frei, S. Sankar, B. Viray, and I. Vogt.

Alicia Lamarche, Derived Categories and Rational Points for a class of toric Fano varieties

Abstract: I will discuss forthcoming work with Matthew Ballard as well as joint work with Matthew Ballard, Alexander Duncan, and Patrick McFaddin on using the derived category of coherent sheaves to detect the existence of rational points on a particular family of arithmetic toric Fano varieties. More precisely, I will explain how we show that a member of this family of varieties is rational if and only if its bounded derived category of coherent sheaves admits a full étale exceptional collection.

Joaquín Moraga, Coregularity of Fano varieties.

Abstract: In this talk, we will introduce the coregularity of Fano varieties. This invariant measures how large of a dual complex can we find among log Calabi-Yau structures on a Fano variety. The coregularity relates to log canonical thresholds, existence of complements, and the index of log Calabi-Yau pairs. In this talk, we will discuss some recent results about this invariant and future directions. The results of this talk are joint work with Fernando Figueroa, Stefano Filipazzi, Mirko Mauri and Junyao Peng.

Karl Schwede, Perfectoid signature and an application to étale fundamental groups

Abstract: In characteristic p > 0 commutative algebra, the F-signature measures how close a strongly F-regular ring is from being non-singular. Here F-regular singularities are a characteristic p > 0 analog of klt singularities. In this talk, using the perfectoidization of Bhatt-Scholze, we will introduce a mixed characteristic analog of F-signature. As an application, we show it can be used to provide an explicit upper bound on the size of the étale fundamental group of the regular locus of BCM-regular singularities (related to results of Xu, Braun, Carvajal-Rojas, Tucker and others in characteristic zero and characteristic p). This is joint work with Hanlin Cai, Seungsu Lee, Linquan Ma and Kevin Tucker.

Burt Totaro, The failure of vanishing theorems in characteristic p

Abstract: The Kodaira vanishing theorem is a central result of algebraic geometry that fails in characteristic p. We construct counterexamples even in "nice" situations such as Fano varieties or terminal singularities. It is an open question how far the minimal model program will work in characteristic p, despite these obstacles. There are also some related counterexamples in mixed characteristic.