An Abel-Jacobi invariant for cobordant cycles

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.Comment: v2: 21 pages; revised section 4; final version to appear in Doc. Mat

Hodge filtered complex bordism

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and \A^1-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.Comment: minor revision; final version accepted for publication by the Journal of Topolog

On the cokernel of the Thom morphism for compact Lie groups

We give a complete description of the potential failure of the surjectivity of the Thom morphism from complex cobordism to integral cohomology for compact Lie groups via a detailed study of the Atiyah-Hirzebruch spectral sequence and the action of the Steenrod algebra. We show how the failure of the surjectivity of the topological Thom morphism can be used to find examples of non-trivial elements in the kernel of the induced differential Thom morphism from differential cobordism of Hopkins and Singer to differential cohomology. These arguments are based on the particular algebraic structure and interplay of the torsion and non-torsion parts of the cohomology and cobordism rings of a given compact Lie group. We then use the geometry of special orthogonal groups to construct concrete cobordism classes in the non-trivial part of the kernel of the differential Thom morphism.Comment: 36 pages, comments welcom

Geometric pushforward in Hodge filtered complex cobordism and secondary invariants

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.Comment: 39 pages, comments very welcom