We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based on generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed.

We introduce a new method for expanding an abelian category and study it using recollements. In particular, we give a criterion for the existence of cotilting objects. We show, using techniques from noncommutative algebraic geometry, that our construction encompasses the category of coherent sheaves on Geigle–Lenzing weighted projective lines. We apply our construction to some concrete examples and obtain new weighted projective varieties, and analyze the endomorphism algebras of their tilting bundles.

Our object of study is a rational map  defined by homogeneous forms $g_{1},\ldots ,g_{n}$ , of the same degree $d$ , in the homogeneous coordinate ring $R=k[x_{1},\ldots ,x_{s}]$ of $\mathbb{P}_{k}^{s-1}$ . Our goal is to relate properties of $\unicode[STIX]{x1D6F9}$ , of the homogeneous coordinate ring $A=k[g_{1},\ldots ,g_{n}]$ of the variety parameterized by $\unicode[STIX]{x1D6F9}$ , and of the Rees algebra ${\mathcal{R}}(I)$ , the bihomogeneous coordinate ring of the graph of $\unicode[STIX]{x1D6F9}$ . For a regular map $\unicode[STIX]{x1D6F9}$ , for instance, we prove that ${\mathcal{R}}(I)$ satisfies Serre’s condition $R_{i}$ , for some $i>0$ , if and only if $A$ satisfies $R_{i-1}$ and $\unicode[STIX]{x1D6F9}$ is birational onto its image. Thus, in particular, $\unicode[STIX]{x1D6F9}$ is birational onto its image if and only if ${\mathcal{R}}(I)$ satisfies $R_{1}$ . Either condition has implications for the shape of the core, namely, $\text{core}(I)$ is the multiplier ideal of $I^{s}$ and $\text{core}(I)=(x_{1},\ldots ,x_{s})^{sd-s+1}.$ Conversely, for $s=2$ , either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of $g_{1},\ldots ,g_{n}$ , we give an explicit method to reduce the nonbirational case to the birational one when $s=2$ .

Using twisted nearby cycles, we define a new notion of slopes for complex holonomic ${\mathcal{D}}$ -modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections ${\mathcal{E}}_{t}$ parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the ${\mathcal{E}}_{t}$ . This generalizes the regularity of the Gauss–Manin connection proved by Griffiths, Katz and Deligne.

We start developing a notion of reciprocity sheaves, generalizing Voevodsky’s homotopy invariant presheaves with transfers which were used in the construction of his triangulated categories of motives. We hope that reciprocity sheaves will eventually lead to the definition of larger triangulated categories of motivic nature, encompassing non-homotopy invariant phenomena.

In this article, we present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$ -adic fields in terms of the Néron–Severi group and provide a proof under additional assumptions on an integral model of $X$ . The proof depends on a non-degeneracy result of Brauer–Manin pairing due to Saito–Sato and on Gabber–de Jong’s comparison result of cohomological and Azumaya–Brauer groups. We will also mention the local–global problem for the Albanese cokernel; the abelian group on the ‘local side’ turns out to be a finite group.

Given a field $k$ of characteristic zero and $n\geqslant 0$, we prove that $H_{0}(\mathbb{Z}F(\unicode[STIX]{x1D6E5}_{k}^{\bullet },\mathbb{G}_{m}^{\wedge n}))=K_{n}^{MW}(k)$, where $\mathbb{Z}F_{\ast }(k)$ is the category of linear framed correspondences of algebraic $k$-varieties, introduced by Garkusha and Panin [The triangulated category of linear framed motives $DM_{fr}^{eff}(k)$, in preparation] (see [Garkusha and Panin, Framed motives of algebraic varieties (after V. Voevodsky), Preprint, 2014, arXiv:1409.4372], § 7 as well), and $K_{\ast }^{MW}(k)$ is the Milnor–Witt $K$-theory of the base field $k$.

We compute the characters of the simple $\text{GL}$ -equivariant holonomic ${\mathcal{D}}$ -modules on the vector spaces of general, symmetric, and skew-symmetric matrices. We realize some of these ${\mathcal{D}}$ -modules explicitly as subquotients in the pole order filtration associated to the $\text{determinant}/\text{Pfaffian}$ of a generic matrix, and others as local cohomology modules. We give a direct proof of a conjecture of Levasseur in the case of general and skew-symmetric matrices, and provide counterexamples in the case of symmetric matrices. The character calculations are used in subsequent work with Weyman to describe the ${\mathcal{D}}$ -module composition factors of local cohomology modules with determinantal and Pfaffian support.

For a prime number $p$ , we show that differentials $d_{n}$ in the motivic cohomology spectral sequence with $p$ -local coefficients vanish unless $p-1$ divides $n-1$ . We obtain an explicit formula for the first non-trivial differential $d_{p}$ , expressing it in terms of motivic Steenrod $p$ -power operations and Bockstein maps. To this end, we compute the algebra of operations of weight $p-1$ with $p$ -local coefficients. Finally, we construct examples of varieties having non-trivial differentials $d_{p}$ in their motivic cohomology spectral sequences.

In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category $\text{}\underline{C}$ by a set $S$ of morphisms in the heart $\text{}\underline{Hw}$ of a weight structure $w$ on it one obtains a triangulated category endowed with a weight structure $w^{\prime }$. The heart of $w^{\prime }$ is a certain version of the Karoubi envelope of the non-commutative localization $\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ (of $\text{}\underline{Hw}$ by $S$). The functor $\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of $S$ invertible. For any additive category $\text{}\underline{A}$, taking $\text{}\underline{C}=K^{b}(\text{}\underline{A})$ we obtain a very efficient tool for computing $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$ coincides with the ‘abstract’ localization $\text{}\underline{A}[S^{-1}]$ (as constructed by Gabriel and Zisman) if $S$ contains all identity morphisms of $\text{}\underline{A}$ and is closed with respect to direct sums. We apply our results to certain categories of birational motives $DM_{gm}^{o}(U)$ (generalizing those defined by Kahn and Sujatha). We define $DM_{gm}^{o}(U)$ for an arbitrary $U$ as a certain localization of $K^{b}(Cor(U))$ and obtain a weight structure for it. When $U$ is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general $U$ the result is completely new. The existence of the corresponding adjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over $U$.

We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: there exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin–Matlis duality à la Dwyer–Greenless–Iyengar in the theory of ring spectra, and of Brown–Comenetz duality à la Neeman in stable homotopy theory.

Given a family of varieties $X\rightarrow \mathbb{P}^{n}$ over a number field, we determine conditions under which there is a Brauer–Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums $\text{Kl}_{p}(a)$ , as $a$ varies over $\mathbf{F}_{p}^{\times }$ and as $p$ tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

We define truncated displays over rings in which a prime $p$ is nilpotent, we associate crystals to truncated displays, and we define functors from truncated displays to truncated Barsotti–Tate groups.

We show that the subcategory of mixed Tate motives in Voevodsky’s derived category of motives is not closed under infinite products. In fact, the infinite product $\prod _{n=1}^{\infty }\mathbf{Q}(0)$ is not mixed Tate. More generally, the inclusions of several subcategories of motives do not have left or right adjoints. The proofs use the failure of finite generation for Chow groups in various contexts. In the positive direction, we show that for any scheme of finite type over a field whose motive is mixed Tate, the Chow groups are finitely generated.

Let $k=\mathbb{C}$ be the field of complex numbers. In this article we construct Hodge realization functors defined on the triangulated categories of étale motives with rational coefficients. Our construction extends to every smooth quasi-projective $k$-scheme, the construction done by Nori over a field, and relies on the original version of the basic lemma proved by Beĭlinson. As in the case considered by Nori, the realization functor factors through the bounded derived category of a perverse version of the Abelian category of Nori motives.

We show that the torsion in the group of indecomposable $(2,1)$-cycles on a smooth projective variety over an algebraically closed field is isomorphic to a twist of its Brauer group, away from the characteristic. In particular, this group is infinite as soon as $b_{2}-{\it\rho}>0$. We derive a new insight into Roǐtman’s theorem on torsion $0$-cycles over a surface.

We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$ , and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$ , and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$ . We also classify all morphisms in the category character sheaves on $G$ . As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of $p$ -adic tori.

We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over an algebraic closure of a finite field if the variety admits a normal projective compactification with boundary locus of codimension greater than or equal to $2$.