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.

Let ${\mathcal{V}}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $u:{\mathcal{Z}}{\hookrightarrow}\mathfrak{X}$ be a closed immersion of smooth, quasi-compact, separated formal schemes over ${\mathcal{V}}$, $T$ be a divisor of $X$ such that $U:=T\cap Z$ is a divisor of $Z$, and $\mathfrak{D}$ a strict normal crossing divisor of $\mathfrak{X}$ such that $u^{-1}(\mathfrak{D})$ is a strict normal crossing divisor of ${\mathcal{Z}}$. We pose $\mathfrak{X}^{\sharp }:=(\mathfrak{X},\mathfrak{D})$, ${\mathcal{Z}}^{\sharp }:=({\mathcal{Z}},u^{-1}\mathfrak{D})$ and $u^{\sharp }:{\mathcal{Z}}^{\sharp }{\hookrightarrow}\mathfrak{X}^{\sharp }$ the exact closed immersion of smooth logarithmic formal schemes over ${\mathcal{V}}$. In Berthelot’s theory of arithmetic ${\mathcal{D}}$-modules, we work with the inductive system of sheaves of rings $\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T):=(\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}(T))_{m\in \mathbb{N}}$, where $\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}$ is the $p$-adic completion of the ring of differential operators of level $m$ over $\mathfrak{X}^{\sharp }$ and where $T$ means that we add overconvergent singularities along the divisor $T$. Moreover, Berthelot introduced the sheaf ${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}:=\underset{\underset{m}{\longrightarrow }}{\lim }\,\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}(T)\otimes _{\mathbb{Z}}\mathbb{Q}$ of differential operators over $\mathfrak{X}^{\sharp }$ of finite level with overconvergent singularities along $T$. Let ${\mathcal{E}}^{(\bullet )}\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{Q},\text{coh}}^{\text{b}}(\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T))$ and ${\mathcal{E}}:=\varinjlim ~({\mathcal{E}}^{(\bullet )})$ be the corresponding object of $D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}})$. In this paper, we study sufficient conditions on ${\mathcal{E}}$ so that if $u^{\sharp !}({\mathcal{E}})\in D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{{\mathcal{Z}}^{\sharp }}^{\dagger }(\text{}^{\dagger }U)_{\mathbb{Q}})$ then $u^{\sharp (\bullet )!}({\mathcal{E}}^{(\bullet )})\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{Q},\text{coh}}^{\text{b}}(\widehat{{\mathcal{D}}}_{{\mathcal{Z}}^{\sharp }}^{(\bullet )}(U))$. For instance, we check that this is the case when ${\mathcal{E}}$ is a coherent ${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}$-module such that the cohomological spaces of $u^{\sharp !}({\mathcal{E}})$ are isocrystals on ${\mathcal{Z}}^{\sharp }$ overconvergent along $U$.

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.

We study the singularity at the origin of $\mathbb{C}^{n+1}$ of an arbitrary homogeneous polynomial in $n+1$ variables with complex coefficients, by investigating the monodromy characteristic polynomials $\unicode[STIX]{x1D6E5}_{l}(t)$ as well as the relation between the monodromy zeta function and the Hodge spectrum of the singularity. In the case $n=2$, we give a description of $\unicode[STIX]{x1D6E5}_{C}(t)=\unicode[STIX]{x1D6E5}_{1}(t)$ in terms of the multiplier ideal.

We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric ${\mathcal{D}}$-modules. We obtain, in particular, a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss–Manin systems of Laurent polynomials via Fourier–Laplace and Radon transformations.

We prove constancy of Newton polygons of all convergent $F$-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of proper smooth families of curves over Abelian varieties. More generally, we prove the isotriviality over projective smooth varieties on which any convergent $F$-isocrystal has constant Newton polygons.

We show that a $\mathbb{P}$-object and simple configurations of $\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.

In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for $k$ a perfect field of characteristic $p$, a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with $\mathbb{Q}_{p}$-coefficients.

Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index $1$ or $2$ in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

Soient $S$ un schéma nœthérien et $f:X\rightarrow S$ un morphisme propre. D’après SGA 4 XIV, pour tout faisceau constructible $\mathscr{F}$ de $\mathbb{Z}/n\mathbb{Z}$-modules sur $X$, les faisceaux de $\mathbb{Z}/n\mathbb{Z}$-modules $\mathtt{R}^{i}f_{\star }\mathscr{F}$, obtenus par image directe (pour la topologie étale), sont également constructibles : il existe une stratification $\mathfrak{S}$ de $S$ telle que ces faisceaux soient localement constants constructibles sur les strates. À la suite de travaux de N. Katz et G. Laumon, ou L. Illusie, dans le cas particulier où $S$ est génériquement de caractéristique nulle ou bien les faisceaux $\mathscr{F}$ sont constants (de torsion inversible sur $S$), on étudie ici la dépendance de $\mathfrak{S}$ en $\mathscr{F}$. On montre qu’une condition naturelle de constructibilité et modération « uniforme » satisfaite par les faisceaux constants, introduite par O. Gabber, est stable par les foncteurs $\mathtt{R}^{i}f_{\star }$. Si $f$ n’est pas supposé propre, ce résultat subsiste sous réserve de modération à l’infini, relativement à $S$. On démontre aussi l’existence de bornes uniformes sur les nombres de Betti, qui s’appliquent notamment pour les fibres des faisceaux $\mathtt{R}^{i}f_{\star }\mathbb{F}_{\ell }$, où $\ell$ parcourt les nombres premiers inversibles sur $S$.

We define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural λ-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized (‘motivic’) Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher-order Euler characteristics of wreath products.

We study the equivariant oriented cohomology ring $\mathtt{h}_{T}(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in $\mathtt{h}_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory $\mathfrak{h}$ corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in $\mathfrak{h}_{T}(G/P)$. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.

We compare two cohomological Hall algebras (CoHA). The first one is the preprojective CoHA introduced in [19] associated with each quiver Q, and each algebraic oriented cohomology theory A. It is defined as the A-homology of the moduli of representations of the preprojective algebra of Q, generalizing the K-theoretic Hall algebra of commuting varieties of Schiffmann-Vasserot [15]. The other one is the critical CoHA defined by Kontsevich-Soibelman associated with each quiver with potentials. It is defined using the equivariant cohomology with compact support with coefficients in the sheaf of vanishing cycles. In the present paper, we show that the critical CoHA, for the quiver with potential of Ginzburg, is isomorphic to the preprojective CoHA as algebras. As applications, we obtain an algebra homomorphism from the positive part of the Yangian to the critical CoHA.

In this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

We discuss the kernel of the localization map from étale motivic cohomology of a variety over a number field to étale motivic cohomology of the base change to its completions. This generalizes the Hasse principle for the Brauer group, and is related to Tate–Shafarevich groups of abelian varieties.

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over $\mathbb{Q}$ whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.