Albanese varieties with modulus and Hodge theory

TL;DR

This paper explores the generalized Albanese variety with modulus for complex smooth varieties, providing Hodge theoretic descriptions and extending classical concepts to higher dimensions with multiplicities.

Contribution

It introduces Hodge theoretic presentations for Albanese varieties with modulus, generalizing classical Jacobian concepts to higher dimensions and multiplicities.

Findings

Hodge theoretic descriptions of Alb(X,Y) are established.

The algebraic group Alb(X,Y) can include an additive part due to multiplicities.

Extension of classical Jacobian theory to higher dimensions with modulus.

Abstract

Let X be a proper smooth variety over the complex numbers. We consider the generalized Albanese variety Alb(X,Y) of X of modulus Y, which is a higher dimensional analogue of the generalized Jacobian variety with modulus of Rosenlicht-Serre. Note that the divisor Y can have multiplicity, so the algebraic group Alb(X,Y) can have an additive part. The purpose of this paper is to give Hodge theoretic presentations of Alb(X,Y).