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Geometric pushforward in Hodge filtered complex cobordism and secondary invariants

Knut Bjarte Haus, Gereon Quick

Abstract

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.

Geometric pushforward in Hodge filtered complex cobordism and secondary invariants

Abstract

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.
Paper Structure (11 sections, 37 theorems, 292 equations)

This paper contains 11 sections, 37 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Currential geometric Hodge filtered cobordism
  3. Currents
  4. Geometric Hodge filtered cobordism groups
  5. Currential Hodge filtered complex cobordism
  6. Hodge filtered MU-orientations
  7. Pushforward along proper Hodge filtered MU-oriented maps
  8. A canonical Hodge filtered MU-orientation for holomorphic maps
  9. Fundamental classes and secondary cobordism invariants
  10. Hodge filtered Thom morphism
  11. Image and kernel for compact Kähler manifolds

Key Result

Theorem 1.1

Let $g \colon X\to Y$ be a proper holomorphic map between complex manifolds of complex codimension $d=\dim_{\mathbb{C}} Y-\dim_{\mathbb{C}} X$. Then there is a pushforward homomorphism of $MU^*(*)(Y)$-modules which is functorial for proper holomorphic maps and compatible with pullbacks.

Theorems & Definitions (113)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8
  • ...and 103 more