Extensions of mixed Hodge modules and Picard-Fuchs equations

TL;DR

This paper demonstrates that the extensions of mixed Hodge modules, local systems, and D-modules associated with algebraic cycles and their Picard-Fuchs equations are equivalent, deepening understanding of their interrelations.

Contribution

It establishes the equivalence of different extensions related to normal functions, algebraic cycles, and Picard-Fuchs equations within the framework of mixed Hodge modules.

Findings

Extensions of D-modules and local systems coincide with the mixed Hodge module extension.

Normal functions determine a unique extension of mixed Hodge modules.

The results unify various perspectives on algebraic cycles and differential equations.

Abstract

The normal function associated to algebraic cycles in higher Chow groups defines a differential equation. This Picard-Fuchs equation defines an extension of $D$-modules as well as an extension of local systems. In this paper, we show that both extensions define the same extension of mixed Hodge modules determined by the normal function.