Related papers: Motivic principal value integrals for hyperplane a…
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras,…
In this article we study motives corresponding to the moduli stacks of G-shtukas and their local models. In particular we deal with the question of describing their motivic fundamental invariants. As an application, we provide a criterion…
We improve some foundational connectivity results and the relative Hurewicz theorem in motivic homotopy theory, study functorial central series in motivic local group theory, establish the existence of functorial Moore--Postnikov…
A well-known conjecture of Gross and Zagier states that the values of the higher automorphic Green's function at pairs of points with complex multiplication in the upper half-plane are proportional to the logarithm of an algebraic number.…
This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta…
We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of…
We extend Ayoub's formalism of motivic nearby cycle functor to the $\infty$-categorical level, and prove some desired cohomological properties by relating the motivic nearby cycle functor to the notion of local acyclicity in motivic…
By associating a `motivic integral' to every complex projective variety X with at worst canonical, Gorenstein singularities, Kontsevich proved that, when there exists a crepant resolution of singularities Y of X, the Hodge numbers of Y do…
In this paper we make a systematic study of certain motivic cohomology classes ("Rankin-Eisenstein classes") attached to the Rankin--Selberg convolution of two modular forms of weight $\ge 2$. The main result is the computation of the…
We consider a family of generic weighted arrangements of $n$ hyperplanes in $\C^k$ and show that the Gauss-Manin connection for the associated hypergeometric integrals, the contravariant form on the space of singular vectors, and the…
Given a Nori motivic local system over a smooth, connected complex algebraic variety, we define its exceptional locus as a way to measure the variation in the motivic complexity of its stalks. The definition is given explicitly in terms of…
Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the \'etale cohomology with…
The main result of this paper is a computation of the motivic cohomology of varieties of n \times m-matrices of of rank m, including both the ring structure and the action of the reduced power operations. The argument proceeds by a…
A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which…
A mixed Weil cohomology with values in an abelian rigid tensor category is a cohomological functor on Voevodsky's category of motives which is satisfying K\"unneth formula and such that its restriction to Chow motives is a Weil cohomology.…
Grothendieck-Chow motives of quadric hypersurfaces have provided many insights into the theory of quadratic forms. Subsequently, the landscape of motives of more general projective homogeneous varieties has begun to emerge. In particular,…
The integral identity conjecture of Kontsevich and Soibelman plays an important role in proving the existence of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau manifolds. There are a number of different…
For any cohomology theory $H$ that can be factorized through (the Morel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ we establish the $SH^{S^1}(k)$-functoriality of coniveau spectral sequences for $H$. We also prove:…
We develop a motivic framework for Feynman integrals of one-loop graphs in momentum space. Its advantage compared to the already existing framework in Feynman representation is that it naturally includes graphs with cuts. To each such…
We give an explicit expression for the contact loci of hyperplane arrangements and show that their cohomology rings are combinatorial invariants. We also give an expression for the restricted contact loci in terms of Milnor fibers of…