Hodge decomposition for Kato manifolds

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 Kähler 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.

Figures (2)

• Figure 1: Construction on $\widehat{\mathbb{B}}$
• Figure 2: Hodge diamond of $X = X(\pi, \sigma)$, where $h^{p,q} := h^{p,q}(\widehat{\mathbb{B}})$.

Theorems & Definitions (50)

• Conjecture 1.1: Ornea--Verbitsky
• Theorem 1.2
• Definition 2.1
• Theorem 2.2: Voisin2002
• Theorem 2.3: Meng20
• Corollary 2.4: rao2018dolbeaultcohomologiesblowingcomplex
• Theorem 3.1: Cartan's Theorem B
• Lemma 3.2
• proof
• Lemma 3.3