Finite Type Properties of (Tame) Fundamental Groups

• Jakob Stix (Goethe-Universität Frankfurt)

Room: Centre de conférences Marilyn et James Simons We are interested in finite generation or finite presentation of fundamental groups as topological profinite groups. Our knowledge of group theoretic properties of étale fundamental groups relies traditionally on Riemann's existence theorem (in char 0) and Grothendieck's specialization map (for the transition to char $p$). But not all varieties lift to characteristic 0. Building on recent results by Esnault, Shusterman and Srinivas for smooth projective varieties in char $p$, we are going to explain in the talk how to generalize finite presentation to arbitrary proper varieties (joint work with Lara and Srinivas). Furthermore, we introduce an adic tameness condition and discuss finite generation/presentation of tame fundamental groups for rigid analytic spaces. The second part is joint work with Achinger, Lara and Hübner.

Pure Local Systems Over Local Fields

• Moritz Kerz (Universität Regensburg)

Room: Centre de conférences Marilyn et James Simons In joint work with Hélène we study certain pure $l$-adic local systems on varieties over $p$-adic local fields which are analogs of variations of pure Hodge structures. In the talk I will explain an approach via tilting to the most basic open problems in this setting: analogs of limiting mixed Hodge structures and purity of cohomology for a curve.

Various Remarks on the Donagi-Pantev Program for Construction of Hecke Eigensheaves

• Carlos Simpson (Université Nice-Sophia Antipolis)

Room: Centre de conférences Marilyn et James Simons Donagi and Pantev set out a program for the construction of the parabolic logarithmic Higgs sheaves associated to Hecke eigensheaves in the geometric Langlands correspondence. One of the main features is that their spectral varieties over $\mathrm{Bun}_G$ are Hitchin fibers viewed birationally as subvarieties of $T^*(\mathrm{Bun}_G)$. We'll discuss various aspects of this construction: cases where it is known, the difficulties that can arise, and relationships with the geometry of the Hitchin moduli space.

Different Notions of Tameness Revisited

• Katharina Hübner (Goethe-Universität Frankfurt)

Room: Centre de conférences Marilyn et James Simons For an étale morphism $f:Y \to X$ of schemes over a base $S$ there are different approaches to define what it means that $f$ is tame. Behind all of them lies the intuition that the induced morphism of compactifications $\bar{f} : \bar{Y} \to \bar{X}$ is tamely ramified along the boundary $\bar{Y} \setminus Y$ (in an appropriate sense). Many of the tameness definitions work with valuations without relying on the choice of a compactification. Kerz and Schmidt compare these different notions of tameness in their article “On different notions of tameness” mainly working with compactifications. The disadvantage of this approach is that they need to assume resolution of singularities in order to obtain nice compactifications. In my talk I want to present work in progress with Michael Temkin that approaches the problem purely valuation theoretic by using nonachimedean geometry. As a consequence we can drop the assumption on resolution of singularities. The heart of the project lies in a careful study of the geometry of adic curves over an arbitrary affinoid field (of higher rank) and of the wild locus of an étale morphism of such curves.

The Bloch-Esnault-Kerz Fiber Square

• Lars Hesselholt (Nagoya University & University of Copenhagen)

Room: Centre de conférences Marilyn et James Simons A theorem of Bloch-Esnault-Kerz published in 2014 states that the formal part of the Fontaine-Messing $p$-adic variational Hodge conjecture holds for schemes smooth and proper over an unramified local number ring. The theorem states that a class in the rational $p$-adic Grothendieck group of the special fiber admits a lifting to the rational $p$-adic continuous Grothendieck group of the formal completion along the special fiber if and only if the image of its crystalline Chern class under the de Rham-crystalline comparison isomorphism lies in the appropriate part of the Hodge filtration. In a paper also published in 2014, Beilinson generalized the equivalence of the relative rational $p$-adic $K$-theory and cyclic homology, implicit in the Bloch-Esnault-Kerz paper. As much else, this work, was greatly clarified by the Bhatt-Morrow-Scholze unification of $p$-adic Hodge theory and topological cyclic. Indeed, Antieau-Mathew-Morrow-Nikolaus showed that Beilinson's equivalence is given by the map of horizontal fibers in a square in which the map of vertical fibers is an equivalence by the Nikolaus-Scholze Tate-Orbit-Lemma. In this talk, I will recall how said cartesian square appears from the Nikolaus-Scholze Frobenius of $\mathbb{Z}$ and explain a proposal by Clausen for how it may lead to a definition of the Hodge-Tate period map that does not require any calculational input.

On Secondary Invariants and Arithmetic Rigidity

• Dustin Clausen (IHES)

Room: Centre de conférences Marilyn et James Simons A complex local system on a space $S$ gives rise to "secondary" Chern classes in $H^{2p-1}(S; \mathbb{C}/\mathbb{Z}(p))$, refining the usual "primary" Chern classes in $H^{2p}(S;\mathbb{Z}(p))$. In fact, Esnault in a survey article describes four methods of defining such classes, of which 3 are proved to be equivalent by means of her "modified splitting principle". I will explain how to show that the remaining 1 out of 4 definitions, that of Cheeger-Simons, agrees with the others. Then, changing gears, I will describe some arithmetic analogs of the phenomenon of rigidity of secondary Chern classes. This has bearing on another question from Esnault's article, and leads us to some motivic speculations.

Finiteness Questions for Étale Coverings with Bounded Wild Ramification at the Boundary

• Vasudevan Srinivas (SUNY, Buffalo)

Room: Centre de conférences Marilyn et James Simons We will consider étale coverings $ f : Y {\rightarrow} X$ of varieties over an algebraically closed field in characteristic $p$ > 0 (with some further restrictions on the boundary ramification, in the non-proper case). This talk will give an overview of some work done with Hélène Esnault and others over the last few years, as well as open problems, related to this theme.

Donaldson-Thomas Invariants: Classical, Motivic, Quadratic and Real

• Marc Levine (Universität Duisburg-Essen)

Room: Centre de conférences Marilyn et James Simons Let $X$ be a smooth projective 3-fold over the complex numbers. Following work of Thomas, Behrend-Fantechi, and others, one has a virtual fundamental class in the Chow group of 0-cycles on the Hilbert scheme of dimension 0, length $n$ subschemes of $X$, the degree of which is the $n$th Donaldson-Thomas invariant of $X$. Now take $X$ over an arbitrary field $k$. We have developed a construction of virtual fundamental classes with values in an arbitrary motivic cohomology theory. An example of such, a "quadratic" analog of the Chow groups, is the cohomology of the sheaf of Witt rings, which leads to a refinement of the classical DT-invariants to quadratic DT-invariants with values in the Witt ring of quadratic forms over $k$. We will discuss some developments and conjectures for these refined DT invariants, including some computations of the signature of these invariants due to Anneloes Viergever.

Characteristic Cycle and Pushforward

• Tomoyuki Abe (IPMU - University of Tokyo)

Room: Centre de conférences Marilyn et James Simons Characteristic cycle for $l$-adic sheaf was introduced by T. Saito after the existence of singular support by Beilinson. This measures the ramification of the sheaf, and can be viewed as a vast generalization of Swan conductor. Various compatibility with cohomological operation had been verified by Saito and Beilinson, but the compatibility of pushforward along proper morphism has been left open. In this talk, I wish to discuss this compatibility.

Vanishing Theorems for the Irregular Hodge Filtration

• Claude Sabbah (École polytechnique)

Room: Centre de conférences Marilyn et James Simons I will give an overview of recent advances concerning the irregular Hodge filtration (introduced by Deligne 40 years ago) and I will focus on Kodaira vanishing theorems similar to those of Saito for mixed Hodge modules.

The Non-Abelian $p$-Curvature Conjecture

• Daniel Litt (University of Toronto)

Room: Centre de conférences Marilyn et James Simons The classical Grothendieck-Katz $p$-curvature conjecture gives an arithmetic criterion for the solutions to an algebraic linear ODE to be algebraic functions. We formulate a version of the $p$-curvature conjecture for certain non-linear ODEs arising from algebraic geometry (for example, the Painlevé VI equation or the Schlesinger system), which implies the classical conjecture, and prove it for "Picard-Fuchs initial conditions." The proof is inspired in part by Katz's resolution of the classical p-curvature conjecture for Picard-Fuchs equations, and in part by Esnault-Groechenig's recent resolution of the classical conjecture for rigid $\mathbb{Z}$-local systems. This is joint work with Josh Lam.

Motives and (Super-)Representation Theory: Principles and Case Studies

• Yves André (IMJ-PRG)

Room: Centre de conférences Marilyn et James Simons I shall outline how existence and shape of motives can sometimes be (not only predicted but) established using abstract motivic Galois theory, bypassing concrete constructions of algebraic cycles.

Characteristic Classes of Étale Local Systems

• Alexander Petrov (Harvard University)

Room: Centre de conférences Marilyn et James Simons Given an étale $\mathbb{Z}_p$-local system of rank $n$ on an algebraic variety $X$, continuous cohomology classes of the group $\rm{GL}_n(\mathbb{Z}_p)$ give rise to classes in (absolute) étale cohomology of the variety with coefficients in $\mathbb{Z}_p$. These characteristic classes can be thought of as $p$-adic analogs of Chern-Simons characteristic classes of vector bundles with a flat connection. For a smooth projective variety over complex numbers, Reznikov proved that the usual Chern-Simons classes in degrees $>1$ of all $\mathbb{C}$-local systems are torsion. It turns out that characteristic classes of étale $\mathbb{Z}_p$-local systems on algebraic varieties over non-closed fields are often non-zero even rationally. In particular, if $X$ is a smooth variety over a $p$-adic field, and the local system is de Rham, then its characteristic classes are related to Chern classes of the graded quotients of the Hodge filtration on the associated vector bundle with connection. This relation can be established through considering an analog of Chern classes for vector bundles on the pro-étale site of $X$. This is a joint work with Lue Pan.

Towards an Eichler-Shimura Decomposition for Ordinary $p$-adic Siegel Modular Forms

• Ana Caraiani (Imperial College London)

Room: Centre de conférences Marilyn et James Simons There are two different ways to construct families of ordinary $p$-adic Siegel modular forms. One is by $p$-adically interpolating classes in Betti cohomology, first introduced by Hida and then given a more representation-theoretic interpretation by Emerton. The other is by $p$-adically interpolating classes in coherent cohomology, once again pioneered by Hida and generalised in recent years by Boxer and Pilloni. I will explain these two constructions and then discuss joint work with James Newton and Juan Esteban Rodríguez Camargo, very much in progress, that aims to compare them.

The Frobenius Action on the De Rham Moduli Space

• Michael Gröchenig (University of Toronto)

Room: Centre de conférences Marilyn et James Simons In a suitable mixed characteristic setting, the moduli stack of flat vector connections can be endowed with a Frobenius-pullback operation. This talk is devoted to the properties of this map, which yields amongst other things a new construction of the $F$-isocrystal structure for rigid flat connections. This is joint work with Hélène Esnault.

Analytic Prismatization

• Peter Scholze (MPIM - University of Bonn)

Room: Centre de conférences Marilyn et James Simons Prismatic cohomology is a unifying $p$-adic cohomology of $p$-adic formal schemes. Motivated by questions on locally analytic representations of $p$-adic groups and the $p$-adic Simpson correspondence, an extension of prismatic cohomology to rigid-analytic spaces (over $\mathbb{Q}_p$ or over $\mathbb{F}_p((t))$ has been sought. We will explain what form this should take, and our progress on realizing this picture. This includes a degeneration from the analytic Hodge-Tate stack underlying the $p$-adic Simpson correspondence to a similar (analytic) stack related to the Ogus-Vologodsky correspondence in characteristic $p$. This is joint work in progress with Johannes Anschütz, Arthur-César le Bras and Juan Esteban Rodriguez Camargo.

Local Systems and Higgs Bundles in $p$-adic Geometry

• Bhargav Bhatt (IAS - Princeton University & University of Michigan)

Room: Centre de conférences Marilyn et James Simons The classical Corlette--Simpson (CS) correspondence relates local systems on complex varieties to Higgs bundles; it is highly transcendental in nature. Its characteristic $p$ counterpart surprisingly turns out to be purely algebraic: Bezrukavnikov identified de Rham local systems on a smooth variety $X$ over $\mathbb{F}_p$ with Higgs bundles twisted by a natural $\mathbb{G}_m$-gerbe on the cotangent bundle $T^*X$. By trivializing the gerbe over suitable loci in $T^*X$ using additional choices, Ogus--Vologodsky then recovered an honest CS correspondence (i.e., with untwisted Higgs bundles). In this talk, I'll explain that this story has an exact analog for a smooth rigid space $X$ over a perfectoid $p$-adic field: (generalized) local systems identify with Higgs bundles twisted by a natural $\mathbb{G}_m$-gerbe on $T^*X$, and honest CS correspondes (as studied by many authors in the last 2 decades) can be recovered by trivializing the gerbe over suitable loci in $T^*X$. This is joint work in progress with Mingjia Zhang, and is inspired by recent work of Heuer.

Vanishing Theorems in Positive Characteristic

• Emelie Arvidsson (University of Utah)

Room: Centre de conférences Marilyn et James Simons Starting from the seminal book of Hélène Esnault and Eckart Viehweg on vanishing theorems my talk will be centered around vanishing theorems in positive characteristics. The Kodaira and Kawamata—Viehweg vanishing theorems are incredibly useful in Complex geometry but fail in general to be true over fields of positive characteristics. It was long expected that this failure would be pathological and that these theorems still would be true for some important classes of varieties, such as log Fano varieties. It turns out that starting from dimension two there are log Fano varieties which contradict Kodaira vanishing. However, the known constructions have the dimension of the Fano variety increasing with the characteristic of the base field. One could therefore ask if in any given dimension log Fano's satisfy this vanishing theorem in large enough characteristic depending on the dimension? In this direction, joint with Fabio Bernasconi and Justin Lacini we proved that the Kawamata—Viehweg vanishing theorem holds on log del Pezzo surfaces over a perfect field of characteristic $p$>5.

Weight Filtration on Log Crystalline Site

• Atsushi Shiho (University of Tokyo)

Room: Centre de conférences Marilyn et James Simons Let $p$ be a prime. For a family of simple normal crossing log varieties on which $p$ is nilpotent, we construct a filtered complex on certain log crystalline site which gives rise to the weight filtered $p$-adic Steenbrink complex defined by Mokrane and Nakkajima when we project it to the Zariski site.

Integrality of the Betti Moduli Space

• Johan de Jong (Columbia University)

Room: Centre de conférences Marilyn et James Simons This is a report on joint work with Hélène Esnault. Let $X$ be a smooth projective variety over the complex numbers $\mathbb{C}$. Let $M$ be the moduli space of irreducible representations of the topological fundamental group of $X$ of a fixed rank $r$. Then $M$ is a finite type scheme over the spectrum of the integers $\mathbb{Z}$. We may ask whether $M$ is pure over $\mathbb{Z}$ in the sense of Raynaud-Gruson, for example we can ask if the irreducible components of $M$ which dominate ${\rm Spec}(\mathbb{Z})$ actually surject onto ${\rm Spec}(\mathbb{Z})$. We will explain what this means, present a weak answer to this question, apply this to exclude some abstract groups as the fundamental groups of smooth projective varieties over $\mathbb{C}$, and we discuss what other phenomena can be studied using the method of proof.