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On the cohomology of the Bigolin complex

Riccardo Piovani

Abstract

Given a compact complex manifold, we study the cohomology and the Hodge theory for the elliptic complex of differential forms defined by Bigolin in 1969 and recently referred to as the Schweitzer complex. Recall that the double complex of a compact complex manifold decomposes into a direct sum of so-called squares and zigzags, and the zigzags are the only components contributing to cohomology. The main result of this paper states that in complex dimension 3, the multiplicities of zigzags in this decomposition are completely characterised by Betti, Hodge, Aeppli numbers plus Bigolin numbers, namely the dimensions of the Bigolin cohomology. The result is sharp, meaning that if we remove Hodge or Bigolin numbers from the previous statement then it becomes false. In addition, we compute the Bigolin cohomology on the small deformations of the complex structure of the Iwasawa manifold, and then apply the main theorem to fully describe the double complexes of all the small deformations. We also prove a Hodge decomposition for Bigolin harmonic forms on compact Kähler manifolds of any dimension. Finally, we partially extend the definition of this complex on almost complex manifolds, providing a cohomological invariant on $1$-forms which is finite when the manifold is compact.

On the cohomology of the Bigolin complex

Abstract

Given a compact complex manifold, we study the cohomology and the Hodge theory for the elliptic complex of differential forms defined by Bigolin in 1969 and recently referred to as the Schweitzer complex. Recall that the double complex of a compact complex manifold decomposes into a direct sum of so-called squares and zigzags, and the zigzags are the only components contributing to cohomology. The main result of this paper states that in complex dimension 3, the multiplicities of zigzags in this decomposition are completely characterised by Betti, Hodge, Aeppli numbers plus Bigolin numbers, namely the dimensions of the Bigolin cohomology. The result is sharp, meaning that if we remove Hodge or Bigolin numbers from the previous statement then it becomes false. In addition, we compute the Bigolin cohomology on the small deformations of the complex structure of the Iwasawa manifold, and then apply the main theorem to fully describe the double complexes of all the small deformations. We also prove a Hodge decomposition for Bigolin harmonic forms on compact Kähler manifolds of any dimension. Finally, we partially extend the definition of this complex on almost complex manifolds, providing a cohomological invariant on -forms which is finite when the manifold is compact.
Paper Structure (13 sections, 16 theorems, 159 equations, 3 tables)

This paper contains 13 sections, 16 theorems, 159 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The complex
  3. Hodge Theory
  4. Cohomological relations on compact complex manifolds
  5. The double complex of a complex 3-manifold and cohomological invariants
  6. An explicit computation: the small deformations of the Iwasawa manifold
  7. $h^1_{1,1}$
  8. $h^1_{1,2}$
  9. $h^2_{1,2}$
  10. $h^3_{1,3}$
  11. Remaining cases
  12. Conclusions
  13. Almost complex manifolds

Key Result

Theorem 1.1

Given a compact Hermitian manifold $(M,g)$, there are $L^2$ orthogonal decompositions

Theorems & Definitions (57)

  • Theorem 1.1: see Theorem \ref{['theorem hodge decomposition']}
  • Theorem 1.2: see Theorem \ref{['thm_kahler']}
  • Theorem 1.3: see Theorem \ref{['thm zigzags 3manifolds']}
  • Definition 2.1: The Bigolin complex
  • Remark 2.2: The range of $k$
  • Remark 2.3: Graphic representation of the complex
  • Remark 2.4: Explicit formulation of $\delta^k_{p,q}$ for $k<p+q$
  • Definition 2.5: Bigolin cohomology
  • Remark 2.6: Conjugation
  • Remark 2.7: Aeppli and Bott-Chern cohomologies
  • ...and 47 more