Related papers: Height pairing and nearby cycles

In this paper we give a geometrical interpretation of an extension of mixed Hodge structures (MHS) obtained from the canonical MHS on the group ring of the fundamental group of a hyperelliptic curve modulo the fourth power of its…

We study the variation of heights of cycles in flat families over number fields or, more generally, globally valued fields. To a finite type scheme over a GVF we associate a locally compact Hausdorff space which we refer to as its GVF…

In this article, we revisit the construction of some algebraic cycles due to Chad Schoen on certain Prym Varieties. More precisely, we show that these cycles arise naturally from (unramified) geometric class field theory, and apply it to…

In one of the fundamental results of Arakelov's arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for arithmetic surfaces by relating the intersection pairing to the negative of the…

The Hodge Conjecture, posits a profound connection between the topology and algebraic geometry of complex algebraic varieties. It asserts that Hodge cycles, specific elements in the cohomology of a K\"ahler variety with rational properties,…

There are many instances such that deformation space of the homology class of an algebraic cycle as a Hodge cycle is larger than its deformation space as algebraic cycle. This phenomena can occur for algebraic cycles inside hypersurfaces,…

We apply the classical technique on cyclic objects of Alain Connes to various objects, in particular to the higher Chow complex of S. Bloch to prove a Connes periodicity long exact sequence involving motivic cohomology groups. The Cyclic…

This paper introduces and develops the "Spectral Fingerprint Philosophy" for detecting algebraic cycles on complex algebraic varieties, particularly K3 surfaces. This framework proposes that algebraic cycles can be revealed through…

Let $f:\mathbb{C}^{n+1} \to \mathbb{C}$ be a germ of hypersurface with isolated singularity. One can associate to $f$ a polarized variation of mixed Hodge structure $\mathcal{H}$ over the punctured disc, where the Hodge filtration is the…

The Griffiths height of a variation of Hodge structures over a projective curve is defined as the degree of its canonical line bundle, as defined by Griffiths and generalized by Peters to allow bad reduction points. It may be seen as a…

A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families…

In this text we apply the methods of Hodge theory for isolated hypersurface singularities to define a signature for the Grothendieck residue pairing of these singularities.

We prove compatibility relations between mixed Hodge numbers of $k$-Du Bois fibers in flat projective families and versal deformations of isolated $k$-Du Bois singularities. These extend the notion of polarized relations in asymptotic Hodge…

We prove that the wreath product orbifolds studied earlier by the first author provide a large class of higher dimensional examples of orbifolds whose orbifold Hodge numbers coincide with the ordinary ones of suitable resolutions of…

Let $P$ be the image of a period map. We discuss progress towards a conjectural Hodge theoretic completion $\overline{P}$, an analogue of the Satake-Baily-Borel compactification in the classical case. The set $\overline{P}$ is defined and…

We provide a Hodge theoretical characterization of the set of algebraic numbers which arises from the complete list, due to A. Beauville, of semistable families of elliptic curves over $\mathbb{P}^1$ with four singular fibers. Our technical…

We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are…

We define a period pairing for flat, irregular singular, rank one connections, satisfying a technical condition regarding its stationary set, on complex surfaces between de Rham cohomology of the connection and a modified singular homology,…

We investigate self-similar scalar field solutions to the Einstein equations in whole cylinder symmetry. Imposing self-similarity on the spacetime gives rise to a set of single variable functions describing the metric. Furthermore, it is…

In this article we prove that locally Griffiths' horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system. As a consequence of this description we obtain strong local…