Albanese varieties with modulus and Hodge theory

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).