On the cohomology of the Bigolin complex

Abstract: Given a compact complex manifold, we develop Hodge theory for the elliptic complex of differential forms defined by Bigolin in 1969 and recently referred as the Schweitzer complex. We exhibit several $L^2$ orthogonal decompositions of spaces of forms and prove a Hodge decomposition for harmonic forms on compact K\"ahler manifolds. Then we compute the cohomology of this Bigolin complex on the small deformations of the complex structure of the Iwasawa manifold, showing that, in this example, this Bigolin cohomology is as powerful as Aeppli and Bott-Chern cohomology, in order to distinguish complex structures. 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. We prove that, on any compact complex manifold of 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. This result is sharp, meaning that if we remove Hodge or Bigolin numbers from the previous statement then it becomes false. We then apply this last result to fully describe the double complexes of the small deformations of the Iwasawa manifold. Finally, we partially extend the definition of this complex on almost complex manifolds, providing a cohomological invariant on $1$-forms which is finite dimensional when the manifold is compact.