Tag - Hodge theory

Matthew Morrow: Algebraic K-theory and p-adic arithmetic geometry

22nd April, 2024

To any unital, associative ring R one may associate a family of invariants known as its algebraic K-groups. Although they are essentially constructed out of simple linear algebra data over the ring, they see an extraordinary range of information: depending on the ring, its K-groups can be related to zeta functions, corbordisms, algebraic cycles and the Hodge conjecture, elliptic operators, Grothendieck's theory of motives, and so on.

Our understanding of algebraic K-groups, at least as far as they appear in algebraic and arithmetic geometry, has rapidly improved in the past few years. This talk will present some of the fundamentals of the subject and explain why K-groups are related to the ongoing special year in p-adic arithmetic geometry. The intended audience is non-specialists.

Sally Gilles: The v-Picard Group of Stein Spaces

3rd April, 2024

In this talk, I will present a computation of the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties. When the variety is the affine space, Heuer has proved that this image is equal to the group of closed differential forms. In general, we will see that the image always contains such forms but the quotient can be non-trivial: it contains a ℤp-module that maps, via the Bloch-Kato exponential map, to integral classes in the proétale cohomology.

Peter Scholze: Real local Langlands as geometric Langlands on the twistor-ℙ1

4th March, 2024

In 2014, Fargues realized that one can formulate the local Langlands correspondence over p-adic fields as a geometric Langlands correspondence on the Fargues-Fontaine curve. This raises the question of a similar realization of the local Langlands correspondence over the real numbers. The goal of these lectures is to explain a possible formulation. As part of this, we will give a new perspective on the theory of variations of twistor structure, a generalization of the theory of variations of Hodge structure. This uses the theory of analytic stacks developed in our joint work with Clausen, of which we will give a brief overview.

Alexander Petrov: On Local Systems of Geometric Origin

1st November, 2023

I will discuss the following conjecture: an irreducible ℚ̅ℓ-local system L on a smooth complex algebraic variety S arises in cohomology of a family of varieties over S if and only if L can be extended to an etale local system over some descent of S to a finitely generated subfield of complex numbers. I will describe the motivation for this conjecture coming from relative p-adic Hodge theory, known partial results, and possible approaches (not very successful so far) to formulating a purely p-adic (and thus hopefully more tractable) version of this conjecture. A large part of the talk will be expository, including material based on the ideas of Hélène Esnault, Raju Krishnamoorthy, and Josh Lam.

Kalyani Kansal: Irregular Loci in the Emerton-Gee Stack for GL2

9th October, 2023

Let K be a finite extension of ℚp. The Emerton-Gee stack for GL2 is a stack of etale (φ, Γ)-modules of rank two. Its reduced part, X, is an algebraic stack of finite type over a finite field, and can be viewed as a moduli stack of two-dimensional mod p representations of the absolute Galois group of K. By the work of Caraiani, Emerton, Gee and Savitt, it is known that in most cases, the locus of mod p representations admitting crystalline lifts with specified regular Hodge-Tate weights is an irreducible component of X. Their work relied on a detailed study of a closely related stack of etale phi-modules which admits a map from a stack of Breuil-Kisin modules with descent data. In our work, we assume K is unramfied and further study this map with a view to studying the loci of mod p representations admitting crystalline lifts with small, irregular Hodge-Tate weights. We identify these loci as images of certain irreducible components of the stack of Breuil-Kisin modules and obtain several inclusions of the non-regular loci into the irreducible components of X.

Christian Lehn: Lagrangian Fibrations of Holomorphic Symplectic Varieties

11th September, 2023

By Matsushita's fundamental results, Lagrangian fibrations are essentially the only morphisms on irreducible holomorphic symplectic varieties with positive fibre dimension. We will start by reviewing these results and discuss their validity also for singular symplectic varieties. We will study singular fibres and some of the fundamental conjectures. Towards the end of the course, we will turn to some of the fascinating recent developments in the Hodge theory of Lagrangian fibrations.

Arthur-Cesar Le Bras: A stacky perspective on p-adic non-abelian Hodge theory

24th April, 2023

p-adic non-abelian Hodge theory, also known as the p-adic Simpson correspondence, aims at describing p-adic local systems on a smooth rigid analytic variety in terms of Higgs bundles. I will explain in this talk why the 'Hodge-Tate stacks' recently introduced by Bhatt-Lurie and Drinfeld in their work on prismatic cohomology can be useful to study this kind of questions.

Alan Thompson: The mirror Clemens-Schmid sequence

20th September, 2021

I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a 'mirror P=W' conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting.

Marcello Bernardara: Fano of K3 Type: Isomorphisms and classification of Hodge structures and K3 categories

13th May, 2021

Fano varieties of (derived) K3 type are Fano varieties whose Hodge structure (derived category) contains a K3-type sub-Hodge structure (subcategory). Many examples of such varieties are known, arising as zeroes of homogeneous bundles on Grassmannians, in dimensions that grow up to 19. In this talk, I will first present joint work with Fatighenti and Manivel showing that many of these examples can be related by geometric correspondences and have actually the same K3-type Hodge structure. I will also present an ongoing project with Fatighenti, Manivel and Tanturri, whose aim is to show that in the case of Fano fourfolds, the only possible K3-type structures which are not actual K3 can arise from Gushel-Mukai, cubics and Küchle c5 fourfolds.