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Computation of the multiplicities of zigzags

Victor Chen

Abstract

In this note, we explore various cohomological invariants on double complexes with the aim of finding their decomposition into irreducible parts, which are of square and zigzag shape. By studying the growth rate of the number of invariants given by the multiplicities of zigzags in the double complex of an n-dimensional complex manifold, we show that the De Rham, Dolbeault, Bott-Chern, Aeppli, and Varouchas cohomologies do not suffice to distinguish non-isomorphic double complexes. We also describe the zigzags counted by the Bigolin cohomology, and show how their dimensions are related to the multiplicities of odd zigzags. A special class of complex manifolds is given by the nilmanifolds. For a nilmanifold, the double complex of left-invariant forms is quasi-isomorphic to the double complex of differential forms. In dimension 6, we compute the double complex of forms of two nilmanifolds having the same Betti, Hodge and Bott-Chern numbers, but whose double complexes are non-isomorphic. We also compute the double complexes of a subclass of almost abelian nilmanifolds, which exist in any dimension.

Computation of the multiplicities of zigzags

Abstract

In this note, we explore various cohomological invariants on double complexes with the aim of finding their decomposition into irreducible parts, which are of square and zigzag shape. By studying the growth rate of the number of invariants given by the multiplicities of zigzags in the double complex of an n-dimensional complex manifold, we show that the De Rham, Dolbeault, Bott-Chern, Aeppli, and Varouchas cohomologies do not suffice to distinguish non-isomorphic double complexes. We also describe the zigzags counted by the Bigolin cohomology, and show how their dimensions are related to the multiplicities of odd zigzags. A special class of complex manifolds is given by the nilmanifolds. For a nilmanifold, the double complex of left-invariant forms is quasi-isomorphic to the double complex of differential forms. In dimension 6, we compute the double complex of forms of two nilmanifolds having the same Betti, Hodge and Bott-Chern numbers, but whose double complexes are non-isomorphic. We also compute the double complexes of a subclass of almost abelian nilmanifolds, which exist in any dimension.
Paper Structure (17 sections, 19 theorems, 142 equations)

This paper contains 17 sections, 19 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. General theory about double complexes
  3. Zigzags and Squares
  4. The cohomological toolbox
  5. The Frölicher spectral sequence
  6. Computing the multiplicities of zigzags
  7. Locally similar double complexes
  8. Varouchas cohomologies
  9. Construction of locally similar double complexes
  10. The Bigolin complex
  11. Lie algebras
  12. The Chevalley-Eilenberg complex
  13. Lie algebras of dimension $6$
  14. Almost abelian Lie algebras
  15. Some decompositions of double complexes
  16. ...and 2 more sections

Key Result

Theorem 1.2

Starting from formal dimension 4, there are locally similar double complexes with different zigzag multiplicities.

Theorems & Definitions (58)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.7
  • ...and 48 more