Hodge decomposition for Kato manifolds

TL;DR

This paper proves that Kato manifolds satisfy the Hodge decomposition and explores their Bott--Chern and Aeppli cohomology, providing evidence for a broader conjecture about locally conformally Kähler manifolds.

Contribution

It establishes the Hodge decomposition for Kato manifolds and relates their cohomology to modification data, supporting a conjecture on locally conformally Kähler manifolds.

Findings

Kato manifolds satisfy the Hodge decomposition.

Bott--Chern and Aeppli cohomology coincide with Dolbeault cohomology in certain degrees.

Provides evidence for the conjecture that all compact locally conformally Kähler manifolds satisfy Hodge decomposition.

Abstract

We prove that any Kato manifold satisfies the Hodge decomposition, in the sense that $b_k=\sum_{p+q=k}h^{p, q}$, by relating its cohomology to the corresponding cohomology of its modification data. We give, therefore, more evidence supporting a conjecture of Ornea--Verbitsky stating that compact locally conformally K\"ahler manifolds satisfy the Hodge decomposition. We further study Bott--Chern and Aeppli cohomology of Kato manifolds, showing that in certain degrees they coincide with Dolbeault cohomology.