We prove that the category of (rigidified) Breuil–Kisin–Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil–Kisin–Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a ‘$p$-adic Serre group’.

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.

In the joint work with Elmanto, Hoyois, Khan and Sosnilo, we computed infinite $\mathbb{P}^{1}$-loop spaces of motivic Thom spectra using the technique of framed correspondences. This result allows us to express non-negative $\mathbb{G}_{m}$-homotopy groups of motivic Thom spectra in terms of geometric generators and relations. Using this explicit description, we show that the unit map of the algebraic special linear cobordism spectrum induces an isomorphism on $\mathbb{G}_{m}$-homotopy sheaves.

This paper stems from the observation (arising from work of Delzant) that “most” Kähler groups $G$ virtually algebraically fiber, that is, admit a finite index subgroup that maps onto $\mathbb{Z}$ with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, that is, they have virtual Albanese dimension $va(G)\leqslant 1$. We show that the existence of algebraic fibrations has implications in the study of coherence and higher BNSR invariants of the fundamental group of aspherical Kähler surfaces. The class of Kähler groups with $va(G)=1$ includes virtual surface groups. Further examples exist; nonetheless, they exhibit a strong relation with surface groups. In fact, we show that the Green–Lazarsfeld sets of groups with $va(G)=1$ (virtually) coincide with those of surface groups, and furthermore that the only virtually RFRS groups with $va(G)=1$ are virtually surface groups.

We consider the space $X=\bigwedge ^{3}\mathbb{C}^{6}$ of alternating senary 3-tensors, equipped with the natural action of the group $\operatorname{GL}_{6}$ of invertible linear transformations of $\mathbb{C}^{6}$. We describe explicitly the category of $\operatorname{GL}_{6}$-equivariant coherent ${\mathcal{D}}_{X}$-modules as the category of representations of a quiver with relations, which has finite representation type. We give a construction of the six simple equivariant ${\mathcal{D}}_{X}$-modules and give formulas for the characters of their underlying $\operatorname{GL}_{6}$-structures. We describe the (iterated) local cohomology groups with supports given by orbit closures, determining, in particular, the Lyubeznik numbers associated to the orbit closures.

Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O’Grady’s filtration on the $\text{CH}_{0}$-group of the $K3$ surface. This solves a conjecture of O’Grady and improves on previous results of Huybrechts, O’Grady, and Voisin. Second, we propose a candidate for the Beauville–Voisin filtration on the $\text{CH}_{0}$-group of the moduli space of stable objects. We discuss its connection with Voisin’s recent proposal via constant cycle subvarieties, and prove a conjecture of hers on the existence of special algebraically coisotropic subvarieties for the moduli space.

The germ of the universal isomonodromic deformation of a logarithmic connection on a stable $n$-pointed genus $g$ curve always exists in the analytic category. The first part of this article investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this work studies the dynamics of this action in the particular case of reducible rank 2 representations and genus $g>0$, allowing to classify all finite orbits. Both of these results extend recent ones concerning the genus 0 case.

We introduce and study various categories of (equivariant) motives of (versal) flag varieties. We relate these categories with certain categories of parabolic (Demazure) modules. We show that the motivic decomposition type of a versal flag variety depends on the direct sum decomposition type of the parabolic module. To do this we use localization techniques of Kostant and Kumar in the context of generalized oriented cohomology as well as the Rost nilpotence principle for algebraic cobordism and its generic version. As an application, we obtain new proofs and examples of indecomposable Chow motives of versal flag varieties.

We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_{M}^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_{N}^{\ast }}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

Let $k$ be a perfect field of characteristic $p>0$ and let $\operatorname{W}$ be the ring of Witt vectors of $k$. In this article, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over $k$ relative to $\operatorname{W}$. This proof allows us to deduce an analogue of the de Rham complexes comparison theorem of Berthelot [$\mathscr{D}$-modules arithmétiques. II. Descente par Frobenius, Mém. Soc. Math. Fr. (N.S.) 81 (2000)] without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot’s conjecture on the preservation of convergent isocrystals under the higher direct image by a smooth proper morphism of $k$-varieties.

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$-finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly $F$-regular varieties over arbitrary fields.

In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that $H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$-linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with $\mathbb{Z}[\frac{1}{p}]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable ${\mathcal{O}}_{K}$-model ${\mathcal{X}}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same ${\mathcal{O}}_{K}$-lattice inside $H_{\text{dR}}^{i}(X/K)$ and, moreover, that this lattice is functorial in $X$. For this, we extend the results of Bhatt–Morrow–Scholze on the construction and the analysis of an $A_{\text{inf}}$-valued cohomology theory of $p$-adic formal, proper, smooth ${\mathcal{O}}_{\overline{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{\text{inf}}$-cohomology to the $p$-adic étale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine–Jannsen.

We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including $\overline{\mathbb{Q}}$ and $\overline{\mathbb{F}_{p}}$, we arrive at a complete description of the tensor triangular spectrum and a classification of the thick tensor ideals.

Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i.e. a line arrangement) its freeness depends solely on its combinatorics, but this has only been proved for sets of up to 12 lines. In looking for a counter-example to Terao’s conjecture, the nearly free curves introduced by Dimca and Sticlaru arise naturally. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non-zero section that vanishes on one single point, P say, called the jumping point, and that this characterises the bundle. We then give a precise description of the behaviour of P. Based on detailed examples we then show that the position of P relative to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics.

We will establish a nearby and vanishing cycle formalism for the arithmetic $\mathscr{D}$-module theory following Beilinson’s philosophy. As an application, we define smooth objects in the framework of arithmetic $\mathscr{D}$-modules whose category is equivalent to the category of overconvergent isocrystals.

The notion of Hochschild cochains induces an assignment from $\mathsf{Aff}$, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor $\mathbb{H}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$, where the latter denotes the category of monoidal DG categories and bimodules. Any functor $\mathbb{A}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ gives rise, by taking modules, to a theory of sheaves of categories $\mathsf{ShvCat}^{\mathbb{A}}$. In this paper, we study $\mathsf{ShvCat}^{\mathbb{H}}$. Loosely speaking, this theory categorifies the theory of $\mathfrak{D}$-modules, in the same way as Gaitsgory’s original $\mathsf{ShvCat}$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $\mathsf{ShvCat}^{\mathbb{H}}$, its descent properties and the notion of $\mathbb{H}$-affineness. We then prove the $\mathbb{H}$-affineness of algebraic stacks: for ${\mathcal{Y}}$ a stack satisfying some mild conditions, the $\infty$-category $\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ is equivalent to the $\infty$-category of modules for $\mathbb{H}({\mathcal{Y}})$, the monoidal DG category of higher differential operators. The main consequence, for ${\mathcal{Y}}$ quasi-smooth, is the following: if ${\mathcal{C}}$ is a DG category acted on by $\mathbb{H}({\mathcal{Y}})$, then ${\mathcal{C}}$ admits a theory of singular support in $\operatorname{Sing}({\mathcal{Y}})$, where $\operatorname{Sing}({\mathcal{Y}})$ is the space of singularities of ${\mathcal{Y}}$. As an application to the geometric Langlands programme, we indicate how derived Satake yields an action of $\mathbb{H}(\operatorname{LS}_{{\check{G}}})$ on $\mathfrak{D}(\operatorname{Bun}_{G})$, thereby equipping objects of $\mathfrak{D}(\operatorname{Bun}_{G})$ with singular support in $\operatorname{Sing}(\operatorname{LS}_{{\check{G}}})$.

Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Déglise and a version of the Röndigs and Østvær theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor–Witt correspondences in the sense of Calmès and Fasel.