Related papers: Motivic Steenrod problem away from the characteris…

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold $X$ in the family has an involution such that the induced involution on the Fano variety…

In the past two decades, extensive research has been conducted on the (co)homology of various models of random simplicial complexes. So far, it has always been examined merely as a list of groups. This paper expands upon this by describing…

We define the logarithmic tautological rings of the moduli spaces of Deligne-Mumford stable curves (together with a set of additive generators lifting the decorated strata classes of the standard tautological rings). While these algebras…

Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G$. Let $w$ be an element of the Weyl group $W$ and let $X(w)$ be the…

We present a semicontinuity result, proven in recent joint work with Morrow and Scholze, relating the mod $p$ singular cohomology of a smooth projective complex algebraic variety X to the de Rham cohomology of a smooth characteristic $p$…

For a simple normal crossing variety $X$, we introduce the concepts of prelog Chow ring, saturated prelog Chow group, as well as their counterparts for numerical equivalence. Thinking of $X$ as the central fibre in a (strictly) semistable…

Chow stability is one notion of Mumford's Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its…

We provide a framework for the study of structured manifolds with singularities and their locally determined invariants. This generalizes factorization homology, or topological chiral homology, to the setting of singular manifolds equipped…

Let $k$ be a perfect field and $X$ be a smooth projective surface over $k$ with a rational point, we discuss the condition of splitting off the top cell for the motivic stable homotopy type of $X$. We also study some outlying examples, such…

We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface and study its fundamental properties. Using Voevodsky's category of effective geometrical motives, we then study the motive of the…

Let $X$ be a complex smooth projective variety of dimension $d$. Under some assumption on the cohomology of $X$, we construct mutually orthogonal idempotents in $CH_d(X \times X) \otimes \Q$ whose action on algebraically trivial cycles…

We define real topological Hochschild homology of separated log schemes with involutions. We show that real topological Hochschild homology is $(\mathbb{P}^n,\mathbb{P}^{n-1})$-invariant, which leads to the definition of the motivic real…

We study a double solid X branched along a nodal sextic surface in a projective space and the 2-torsion subgroup in the third integer cohomology group of a resolution of singularities of X. This group can be considered as an obstruction to…

We construct an analog of the intrinsic normal cone of Behrend-Fantechi in the equivariant motivic stable homotopy category over a base-scheme B and construct a fundament class in E-cohomology for any cohomology theory E in SH(B). For…

Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of…

A smooth projective scheme $X$ over a field $k$ is said to satisfy the Rost nilpotence principle if any endomorphism of $X$ in the category of Chow motives that vanishes on an extension of the base field $k$ is nilpotent. We show that an…

The motivic version of the $c_1$-spherical cobordism spectrum is constructed. A connection of this spectrum with other motivic Thom spectra is established. Using this connection, we compute the $\mathbb{P}^1$-diagonal of the homotopy groups…

We study Chow groups and \'etale motivic cohomology groups of smooth complete intersections with Hodge structures of level one, classified by Deligne and Rapoport, with particular attention to fivefolds. We extend these results to an…

We show that the classical homology theory of Steenrod may be enriched with descriptive set-theoretic information. We prove that the resulting definable homology theory provides a strictly finer invariant than Steenrod homology for compact…

Let $X$ be a cubic fourfold in $P^5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable…