Related papers: Height Pairing on Higher Cycles and Mixed Hodge St…
We attach a mixed Hodge structure and associate two versions of heights to a pair of Bloch higher cycles. Both these heights generalize the biextension height attached to a pair of classical algebraic cycles homologous to zero. We also…
In algebraic geometry there is the notion of a height pairing of algebraic cycles, which lies at the confluence of arithmetic, Hodge theory and topology. After explaining a motivating example situation, we introduce new directions in this…
We exhibit a precise connection between N\'eron--Tate heights on smooth curves and biextension heights of limit mixed Hodge structures associated to smoothing deformations of singular quotient curves. Our approach suggests a new way to…
We consider the problem of explicitly computing Beilinson--Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes,…
We compute the mixed Hodge structure on the cohomology ring of complements of complex coordinate subspace arrangements. The mixed Hodge structure can be described in terms of the special bigrading on the cohomology ring of complements of…
For the product $X=C\times S$ of a curve and a surface over a number field, we construct unconditionally a Beilinson--Bloch type height pairing for homologically trivial algebraic cycles on $X$. Then for an embedding $f: C\to S$, we define…
We calculate the mixed Hodge numbers of smooth 3-dimensional cluster varieties and show that they are of mixed Tate type. We also study the mixed Hodge structures of the cohomology and intersection cohomology groups of some singular cluster…
We consider the limiting behaviour of the archimedean height pairing for homologically trivial algebraic cycles in a degenerating one-parameter family of smooth projective complex varieties. We conjecture that the limit is controlled by the…
We prove that, as was conjectured by Spencer Bloch, the Hodge period of some limit Hodge structures equals the height pairing of algebraic cycles on the resolution of singularities of the singular fiber.
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 prove an analog of Schmid's $\text{\rm SL}_2$-orbit theorem for a class of variations of mixed Hodge structure which includes logarithmic deformations, degenerations of 1-motives and archimedean heights. In particular, as consequence…
It's well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall {\em explicitly} determine these structures related to multiple logarithms and some other multiple…
With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory and polarization, Deligne's Mixed Hodge Structure on cohomology of complex algebraic varieties is described.
We extend to the topological setting the classical constructions of the Abel-Jacobi mapping on homologically trivial algebraic cycles and the height pairing between two such cycles. We further interpret the height pairing between…
In this article, we extend Grothendieck's standard conjectures to cycles on degenerated fibers and use them to define some decompositions for the arithmetic Chow group of Gillet--Soul\'e. In the local setting, our decompositions provide…
The purpose of this paper is to introduce the notion of mixed twistor structure, a generalization of the notion of mixed Hodge structure. The utility of this notion is to make possible a theory of weights for various things surrounding…
In the first section of his seminal paper on height pairings, Beilinson constructed an $\ell$-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth projective varieties over the…
In this paper we introduce a certain space of higher order modular forms of weight 0 and show that it has a Hodge structure coming from the geometry of the fundamental group of a modular curve. This generalizes the usual structure on…
Above a Laurent polynomial f one makes grow a vector space of vanishing cycles (after the work of Sabbah, singularity setting), a graded Milnor ring (after the work of Kouchnirenko) and an orbifold cohomology ring (after the work of…
In this article we are interested in morphisms without slope for mixed Hodge modules. We first show the commutativity of iterated nearby cycles and vanishing cycles applied to a mixed Hodge module in the case of a morphism without slope.…