In this article, we study a Gysin triangle in the category of motives with modulus (Theorem 1.2). We can understand this Gysin triangle as a motivic lift of the Gysin triangle of log-crystalline cohomology due to Nakkajima and Shiho. After that we compare motives with modulus and Voevodsky motives (Corollary 1.6). The corollary implies that an object in $\operatorname {\mathbf {MDM}^{\operatorname {eff}}}$ decomposes into a p-torsion part and a Voevodsky motive part. We can understand the corollary as a motivic analogue of the relationship between rigid cohomology and log-crystalline cohomology.

Let X be a smooth proper variety over a field k and suppose that the degree map ${\mathrm {CH}}_0(X \otimes _k K) \to \mathbb {Z}$ is isomorphic for any field extension $K/k$. We show that $G(\operatorname {Spec} k) \to G(X)$ is an isomorphism for any $\mathbb {P}^1$-invariant Nisnevich sheaf with transfers G. This generalises a result of Binda, Rülling and Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge–Witt cohomology is a $\mathbb {P}^1$-invariant Nisnevich sheaf with transfers.

We study the Chow ring of the moduli stack $\mathfrak {M}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. This paper provides foundations for the paper [BS21].

In the appendix (jointly with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of $\mathfrak {M}_{g,n}$ and is important in its own right. We also develop operational Chow groups for algebraic stacks.

In this article, we improve our main results from [LL21] in two directions: First, we allow ramified places in the CM extension $E/F$ at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla–Rapoport conjecture for exotic smooth Rapoport–Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomology of integral models of unitary Shimura varieties with Drinfeld level structures.

In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$ . In this process, we establish the decomposition of Chow groups for the cases of the Cayley trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds.

We give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the compactification of the stacks of hyperelliptic curves given by admissible double coverings.

This article is about Lehn–Lehn–Sorger–van Straten eightfolds $Z$ and their anti-symplectic involution $\iota$. When $Z$ is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of $\iota$ on the Chow group of $0$-cycles of $Z$. The formula is in agreement with the Bloch–Beilinson conjectures and has some non-trivial consequences for the Chow ring of the quotient.

Let Y be a smooth complete intersection of three quadrics, and assume the dimension of Y is even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow–Künneth decomposition for double planes.

We consider Calabi–Yau n-folds X arising from certain hyperplane arrangements. Using Fu–Vial’s theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of X generated by divisors, Chern classes and intersections of subvarieties of positive codimension injects into cohomology. We also prove Voisin’s conjecture for X, and Voevodsky’s smash-nilpotence conjecture for odd-dimensional X.

Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the Levine–Morel sense. We provide a uniform approach to the $A^*$-motives of geometrically cellular smooth projective $G$-varieties based on the Hopf algebra structure of $A^*(G)$. Using this approach, we provide various applications to the structure of motives of twisted flag varieties.

We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

In this article we introduce the local versions of the Voevodsky category of motives with $\mathbb{F} _p$-coefficients over a field k, parametrized by finitely generated extensions of k. We introduce the so-called flexible fields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructed local motivic categories are much simpler than the global one and more reminiscent of a topological counterpart. This provides handy ‘local’ invariants from which one can read motivic information. We compute the local motivic cohomology of a point for $p=2$ and study the local Chow motivic category. We introduce local Chow groups and conjecture that over flexible fields these should coincide with Chow groups modulo numerical equivalence with$\mathbb{F} _p$-coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this conjecture in various cases.

Generalising the classical work of Atiyah and Hirzebruch on non-algebraic classes, recently Quick proved the existence of torsion non-algebraic elements in the Brown–Peterson tower. We construct non-torsion non-algebraic elements in the Brown–Peterson tower for the prime number 2.

The Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of $X$. We prove a weak version of this conjecture when $X$ is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn’s formula and the Li–Qin–Wang $W_{1+\infty }$ algebra action from cohomology to Chow groups for the Hilbert scheme of an arbitrary smooth projective surface $S$.

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.

In this paper we prove the Rigidity Theorem for motives of rigid analytic varieties over a non-Archimedean valued field $K$. We prove this theorem both for motives with transfers and without transfers in a relative setting. Applications include the construction of étale realization functors, an upgrade of the known comparison between motives with and without transfers and an upgrade of the rigid analytic motivic tilting equivalence, extending them to $\mathbb{Z}[1/p]$-coefficients.

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.

This note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.

In this paper we study the subgroup of the Picard group of Voevodsky’s category of geometric motives $\operatorname{DM}_{\text{gm}}(k;\mathbb{Z}/2)$ generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [On the invertibility of motives of affine quadrics, Doc. Math. 22 (2017), 363–395], but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics over $k$. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the criterion of motivic equivalence of projective quadrics.

According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti–Tate group. We extend this result to 1-motives.