arXiv subjects

G. Pearlstein

Publications and source records attributed to G. Pearlstein.

3 recordsLinked to original sources

arXiv2024

Height Pairing on Higher Cycles and Mixed Hodge Structures II

To a pair of Bloch higher cycles that intersect properly and have complementary codimensions, we attach a mixed Hodge structure with some extra data (a framed mixed Hodge structure). Using this mixed Hodge structure, we define two different archimedean local height pairings. Both constructions generalize the biextension archimedean height attached to a pair of classical algebraic cycles homologous to zero. When applied to the polylogarithm variation of mixed Hodge structures, we recover both the single-valued polylogarithm of Bloch, Wigner et al. and the one defined by Brown. We also prove several salient properties of these heights, including various vanishing results.

math.AG↗

arXiv2022

Height Pairing on Higher Cycles and Mixed Hodge Structures

For a smooth, projective complex variety, we introduce several mixed Hodge structures associated to higher algebraic cycles. Most notably, we introduce a mixed Hodge structure for a pair of higher cycles which are in the refined normalized complex and intersect properly. In a special case, this mixed Hodge structure is an oriented biextension, and its height agrees with the higher archimedean height pairing introduced in a previous paper by the first two authors. We also compute a non-trivial example of this height given by Bloch-Wigner dilogarithm function. Finally we study the variation of mixed Hodge structures of Hodge-Tate type, and show that the height extends continuously to degenerate situations.

math.AG↗

arXiv2016

Nilpotent cones and their representation theory

We describe two approaches to classifying the possible monodromy cones C arising from nilpotent orbits in Hodge theory. The first is based upon the observation that C is contained in the open orbit of any interior point N in C under an associated Levi subgroup determined by the limit mixed Hodge structure. The possible relations between the interior of C its faces are described in terms of signed Young diagrams. The second approach is to understand the Tannakian category of nilpotent orbits via a category D introduced by Deligne in a letter to Cattani and Kaplan. In analogy with Hodge theory, there is a functor from D to a subcategory of SL(2)-orbits. We prove that these fibers are, roughly speaking, algebraic. We also give a correction to a result of K. Kato.

math.AG↗