Related papers: Arithmetic Hodge structure and higher Abel-Jacobi …

This paper develops a cohomology theory for Hom-Jacobi-Jordan algebras using and applies it to classify non-abelian extensions. The main result establishes that equivalence classes of split extensions of a Hom-Jacobi-Jordan algebra $J$ by…

We introduce a theory of finite polynomial cohomology with coefficients in this paper. We prove several basic properties and introduce an Abel-Jacobi map with coefficients. As applications, we use such a cohomology theory to study…

We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.

Jannsen asked whether the rational cycle class map in continuous $\ell$-adic cohomology is injective, in every codimension for all smooth projective varieties over a field of finite type over the prime field. As recently pointed out by…

We present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the N\'eron-Severi group and provide a proof under additional assumptions on an…

We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomolgy with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural…

In this article we prove a result comparing rationality of integral algebraic cycles over the function field of a quadric and over the base field. This is an integral version of the result known for coefficients modulo 2. Those results have…

Given a smooth and separated K(pi,1) variety X over a field k, we associate a "cycle class" in etale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute…

We are showing that the Deligne--Beilinson cohomology sheaves ${\cal H}^{q+1}({\bf Z}(q)_{\cal D})$ are torsion free by assuming Kato's conjectures hold true for function fields. This result is `effective' for $q=2$; in this case, by…

This paper studies the Hilbert scheme of a curve on a complete-intersection K-trivial threefold, in the case in which the curve is unobstructed in the ambient variety in which the threefold lives. The basic result is that the obstruction…

This paper presents a novel symbolic analytic framework to address the Hodge Conjecture, utilizing a refined invariant called the Hermitian spectral fingerprint. We modify the fingerprint functional to specifically exclude $(k,k)$…

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…

Motivated by the Bloch-Beilinson conjectures, Voisin has made a conjecture concerning zero-cycles on self-products of Calabi-Yau varieties. We reformulate Voisin's conjecture in the setting of hyperk\"ahler varieties, and we prove this…

We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use…

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.

On a threefold with trivial canonical bundle, Kuranishi theory gives an algebro-geometry construction of the (local analytic) Hilbert scheme of curves at a smooth holomorphic curve as a gradient scheme, that is, the zero-scheme of the…

Let X be a complex surface with no nontrivial 2-forms. Then we show that Bloch's conjecture is true (i.e. the Albanese map in this case is injective) if and only if any homologically trivial idempotent in the ring of correspondences…

We give explicit formulas for maps in a long exact sequence connecting bialgebra cohomology to Hochschild cohomology. We give a sufficient condition for the connecting homomorphism to be surjective. We apply these results to compute all…

Suppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D, and that U is the complement of the ramification locus in Y. The first theorem implies that the Beilinson-Hodge conjecture holds for U if certain…

We demonstrate that the two (1 + 1)-dimensional (2D) free 1-form Abelian gauge theory provides an interesting field theoretical model for the Hodge theory. The physical symmetries of the theory correspond to all the basic mathematical…