  We construct a functorial pushforward homomorphism in geometric Hodge
filtered complex cobordism along proper holomorphic maps between arbitrary
complex manifolds. This significantly improves previous results on such
transfer maps and is a much stronger result than the ones known for
differential cobordism of smooth manifolds. This enables us to define and
provide a concrete geometric description of Hodge filtered fundamental classes
for all proper holomorphic maps. Moreover, we give a geometric description of a
cobordism analog of the Abel-Jacobi invariant for nullbordant maps which is
mapped to the classical invariant under the Hodge filtered Thom morphism. For
the latter we provide a new construction in terms of geometric cycles.
