An Abel-Jacobi invariant for cobordant cycles

TL;DR

This paper introduces a new Abel-Jacobi invariant for algebraic cobordism cycles that vanish in topological cobordism, providing a concrete description for smooth complex varieties.

Contribution

It constructs a concrete Abel-Jacobi map and Hodge filtered cohomology groups for algebraic cobordism cycles on smooth projective complex varieties.

Findings

Defines a new Abel-Jacobi invariant for cobordant cycles

Provides explicit descriptions of Hodge filtered cohomology groups

Connects algebraic and topological cobordism through the invariant

Abstract

We discuss an Abel-Jacobi invariant for algebraic cobordism cycles whose image in topological cobordism vanishes. The existence of this invariant follows by abstract arguments from the construction of Hodge filtered cohomology theories in joint work of Michael J. Hopkins and the author. In this paper, we give a concrete description of the new Abel-Jacobi map and Hodge filtered cohomology groups for projective smooth complex varieties.