Real Seifert Forms and Polarizing Forms of Steenbrink Mixed Hodge Structures

Abstract Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

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Chapter Three. Mixed Hodge Structures

Hodge Theory ◽

10.1515/9781400851478.123 ◽

2014 ◽

pp. 123-216

Author(s):

Fouad El Zein ◽

Lê Dũng Tráng

Keyword(s):

Hodge Structures ◽

Mixed Hodge Structures

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Iterated integrals and mixed hodge structures on homotopy groups

Lecture Notes in Mathematics - Hodge Theory ◽

10.1007/bfb0077530 ◽

1987 ◽

pp. 75-83 ◽

Cited By ~ 1

Author(s):

Richard M. Hain

Keyword(s):

Homotopy Groups ◽

Hodge Structures ◽

Iterated Integrals ◽

Mixed Hodge Structures

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THE KUGA–SATAKE CONSTRUCTION UNDER DEGENERATION

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748019000239 ◽

2019 ◽

Vol 19 (6) ◽

pp. 2165-2182

Author(s):

Stefan Schreieder ◽

Andrey Soldatenkov

Keyword(s):

Abelian Varieties ◽

Hodge Theory ◽

Hodge Structures ◽

Mixed Hodge Structures

We extend the Kuga–Satake construction to the case of limit mixed Hodge structures of K3 type. We use this to study the geometry and Hodge theory of degenerations of Kuga–Satake abelian varieties associated with polarized variations of K3 type Hodge structures over the punctured disc.

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