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Differential forms and invariants of complex manifolds

Jonas Stelzig

TL;DR

The work surveys how differential-form–based invariants—cohomology, Chern classes, rational homotopy, and higher holomorphic operations—govern the structure of compact complex manifolds beyond the Kähler realm. It develops a unifying algebraic framework via the bicomplex of forms $(A_X,\partial,\bar{\partial})$ and advances in rational homotopy theory, including Sullivan's minimal models and $C_\infty$-structures, as well as pluripotential analogues and model-categorical approaches. It critically analyzes cohomology theories (Dolbeault, Bott–Chern, Aeppli, Frölicher) and their universal relations, along with bimeromorphic and topological invariants, and introduces ABC Massey products as higher holomorphic operations that detect non-formality. The results illuminate realizability conditions and invariants for complex structures, offering a cohesive toolkit for deformation, topological classification, and invariants in non-Kähler geometry.

Abstract

A survey of some results and open questions related to the following algebraic invariants of compact complex manifolds, that can be obtained from differential forms: cohomology groups, Chern classes, rational homotopy groups, and higher operations.

Differential forms and invariants of complex manifolds

TL;DR

The work surveys how differential-form–based invariants—cohomology, Chern classes, rational homotopy, and higher holomorphic operations—govern the structure of compact complex manifolds beyond the Kähler realm. It develops a unifying algebraic framework via the bicomplex of forms and advances in rational homotopy theory, including Sullivan's minimal models and -structures, as well as pluripotential analogues and model-categorical approaches. It critically analyzes cohomology theories (Dolbeault, Bott–Chern, Aeppli, Frölicher) and their universal relations, along with bimeromorphic and topological invariants, and introduces ABC Massey products as higher holomorphic operations that detect non-formality. The results illuminate realizability conditions and invariants for complex structures, offering a cohesive toolkit for deformation, topological classification, and invariants in non-Kähler geometry.

Abstract

A survey of some results and open questions related to the following algebraic invariants of compact complex manifolds, that can be obtained from differential forms: cohomology groups, Chern classes, rational homotopy groups, and higher operations.

Paper Structure

This paper contains 14 sections, 19 theorems, 35 equations.

Table of Contents

  1. Spaces, manifolds, and almost complex structures
  2. Spaces
  3. Compact manifolds
  4. Almost complex manifolds
  5. Complex manifolds
  6. A zoo of cohomology theories
  7. The compact Kähler case
  8. Hirzebruch's question
  9. Common motives
  10. Topological invariants
  11. Bimeromorphism invariants
  12. Multiplicative matters
  13. A pluripotential version of rational homotopy theory
  14. Holomorphic higher operations and formality

Key Result

Theorem 1.1

For any degree-wise finite-dimensional simply connected rational cga $H^\ast$, there is a simply connected space $X$ with $H_{sing}^\ast(X;\mathbb{Q})\cong H^\ast$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: KoSchr_HodgeKaehl
  • Theorem 1.5: Sullivan-Barge realization theorem Sullivan, Barge
  • Theorem 1.6: Stong congruences, AtiyahHirzebruch_CohOpsChaChla, Stong_CN_I, Stong_CN_II, Hattori_CN
  • Theorem 1.7: Almost complex realization, Mili
  • Theorem 2.2: AnTo, PoSteUb, SteWi
  • Proposition 2.3: The $\partial\bar{\partial}$-Lemma, DGMS
  • Theorem 2.4: KoSchr_HodgeKaehl
  • ...and 12 more