Related papers: Motivic HyperK\"ahler Resolution Conjecture : I. G…

Hypertoric varieties are hyperk\"ahler analogues of toric varieties, and are constructed as abelian hyperk\"ahler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold…

A conjecture by Corvaja and Zannier predicts that smooth, projective, simply connected varieties over a number field with Zariski dense set of rational points have the Hilbert Property; this was proved by Demeio for Kummer surfaces which…

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant…

In this paper we study some new theories of characteristic homology classes for singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which…

We introduce the notion of algebraic cogroup over a subfield $k$ of the complex numbers, and use it to prove that every Nori motive over $k$ is isomorphic to a quotient of a motive of the form $H^n(X, Y)(i)$.

The Beilinson--Bloch conjecture is a generalization of the Birch and Swinnerton-Dyer conjecture, which relates the ranks of Chow groups of smooth projective varieties over global fields to the order of vanishing of $L$-functions. We prove…

For a projective $2n$-dimensional irreducible holomorphic symplectic manifold $Y$ of generalized Kummer deformation type and $j$ the smallest prime number dividing $n+1$, we prove the Lefschetz standard conjectures in degrees…

Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X) of every smooth proper Deligne-Mumford…

The purpose of this article is to construct a Hecke-equivariant Chow motive whose realizations equal interior (or intersection) cohomology of Hilbert-Blumenthal varieties with non-constant algebraic coefficients.

The Euler characteristic of all the Chow varieties, of a fixed projective variety, can be collected in a formal power series called the Euler-Chow series. This series coincides with the Hilbert series when the Picard group is a finite…

The goal of this paper is to study non-$\mathbb{A}^1$-invariant motivic cohomology, recently defined by Elmanto, Morrow, and the first-named author, for smooth schemes over possibly non-discrete valuation rings. We establish that the cycle…

For a smooth quasi-projective scheme $X$ over a field $k$ with an action of a reductive group, we establish a spectral sequence connecting the equivariant and the ordinary higher Chow groups of $X$. For $X$ smooth and projective, we show…

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new…

Let $f: X \rightarrow A$ be an abelian cover from a complex algebraic variety with quotient singularities to an abelian variety. We show that $f^*$ induces an isomorphism between the rational cohomology rings $H^\bullet(A, \mathbb{Q})$ and…

For an Abelian surface $A$ with a symplectic action by a finite group $G$, one can define the partition function for $G$-invariant Hilbert schemes \[Z_{A, G}(q) = \sum_{d=0}^{\infty} e(\text{Hilb}^{d}(A)^{G})q^{d}.\] We prove the reciprocal…

A generalized Kummer surface $X=Km_{3}(A,G_{A})$ is the minimal resolution of the quotient of a $2$-dimensional complex torus by an order 3 symplectic automorphism group $G_{A}$. A Kummer structure on $X$ is an isomorphism class of pairs…

Let $X$ be a compact Riemann surface of genus $g \geq 2$ and let $D\subset X$ be a fixed finite subset. We considered the moduli spaces of parabolic Higgs bundles and of parabolic connections over $X$ with the parabolic structure over $D$.…

To an arbitrary variety over a field of characteristic zero, we associate a complex of Chow motives, which is, up to homotopy, unique and bounded. We deduce that any variety has a natural Euler characteristic in the Grothendieck group of…

For a reductive group $G$, we prove that complex irreducible rigid $G$-local systems with quasi-unipotent monodromies and finite order abelianization on a smooth curve are motivic, generalizing a theorem of Katz for $GL_n$. We do so by…

Let $A^{[[n]]}$ denote the $2(n - 1)$-dimensional generalised Kummer variety constructed from the abelian surface $A$. Further, let $X$ be an arbitrary smooth projective surface with $\int_X c_1(X)^2 \neq 0$, and $X^{[k]}$ the Hilbert…