Related papers: A construction of the polylogarithm motive

Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact K\"ahler manifold or on the…

We prove a motivic version of the Lefschetz hyperplane theorem for a smooth ample divisor $\Theta$ on an Abelian variety. We use this to construct a motive $P$ that realizes the primitive cohomology of $\Theta$.

Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be…

For each fs log scheme $(X,\mathcal M_X)$ over a field $k$ we construct a geometrical Voevodsky motive $[X]^{log}\in DM_{gm}(k,\mathbb Q)$. We prove that, for $k=\mathbb C$, the Betti realization of $[X]^{log}$ is the log Betti cohomology…

Given a mixed Hodge module E on a scheme X over the complex numbers, and a quasi-projective morphism f:X->S, we construct in this paper a natural resolution of the nth exterior tensor power of E restricted to the nth configuration space of…

The integral $t$-motivic cohomology and the class module of a (rigid analytically trivial) Anderson $t$-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor…

We generalize the motivic incarnation morphism from the theory of arithmetic integration to the relative case, where we work over a base variety S over a field k of characteristic zero. We develop a theory of constructible effective Chow…

We construct motivic power operations on the mod-$p$ motivic cohomology of $\Fb_p$-schemes using a motivic refinement of Nizio{\l}'s theorem. The key input is a purity theorem for motivic cohomology established by Levine. Our operations…

Given a polynomial map $f:\Bbb C^{n+1}\to\Bbb C$, one can attach to it a geometrical variation of mixed Hodge structures (MHS) which gives rise to a limit MHS. The equivariant Hodge numbers of this MHS are analytical invariants of the…

We begin with modular form periods, a focal point of several Yuri Manin's works. The similarity is discussed between the corresponding zeta-polynomials and superpolynomials of algebraic links, closely related to Khovanov-Rozansky…

We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules…

Many combinatorial proofs rely on induction. When these proofs are formulated in traditional language, they can be bulky and unmanageable. Coalgebras provide a language which can reduce reduce many inductive proofs in graded poset theory to…

The Hodge correlators ${\rm Cor}_{\mathcal H}(z_0,z_1,\dots,z_n)$ are functions of several complex variables, defined by Goncharov (arXiv:0803.0297) by an explicit integral formula. They satisfy some linear relations: dihedral symmetry…

For any subfield K of the complex numbers which is not contained in an imaginary quadratic number field, we construct conjugate varieties whose algebras of K-rational (p,p)-classes are not isomorphic. This compares to the Hodge conjecture…

We construct algebraic cycles in Bloch's cubical cycle group which correspond to multiple polylogarithms with generic arguments. Moreover, we construct out of them a Hopf subalgebra in the Bloch-Kriz cycle Hopf algebra. In the process, we…

The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.

We review some recent results and conjectures saying that, roughly speaking, periodic cyclic homology of a smooth non-commutative algebraic variety should carry all the additional "motivic" structures possessed by the usual de Rham…

We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative…

A. Huber and B. Kahn construct a relative slice filtration on the motive M(X) associated to a principal T-bundle X over a smooth scheme Y. As a consequence of their result, one can observe that the mixed Tateness of the motive M(Y) implies…

In this article we make several contributions of independent interest. First, we introduce the notion of stressed hyperplane of a matroid, essentially a type of cyclic flat that permits to transition from a given matroid into another with…