On the cohomology of the Bigolin complex

TL;DR

This paper investigates the cohomology of the Bigolin (Schweitzer) complex on compact complex manifolds, especially in dimension 3, linking it to Betti, Hodge, Aeppli, and Bigolin numbers, and explores its behavior under deformations and on Kähler manifolds.

Contribution

It characterizes zigzag multiplicities in the cohomology decomposition for complex dimension 3 using various invariants and extends the theory to almost complex manifolds.

Findings

Zigzag multiplicities are determined by Betti, Hodge, Aeppli, and Bigolin numbers in dimension 3.

Computed Bigolin cohomology for small deformations of the Iwasawa manifold.

Established a Hodge decomposition for Bigolin harmonic forms on Kähler manifolds.

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 K\"ahler 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.