Variation of Hodge structures for non-K\"ahler manifolds

TL;DR

This paper investigates how Hodge structures vary in non-Kähler manifolds, showing that under certain conditions, the period map remains holomorphic and transversal, extending classical results beyond Kähler geometry.

Contribution

It demonstrates the holomorphicity and transversality of the period map for a class of non-Kähler manifolds with specific cohomological properties.

Findings

Period map is holomorphic for the considered class of non-Kähler manifolds.

The period map exhibits transversality under the given conditions.

Extension of Hodge theory concepts to non-Kähler settings.

Abstract

In this note, we discuss unpolarized, complex variation of Hodge structures for non-K\"ahler manifolds. In particular, given a holomorphic family of compact complex manifolds whose central fiber satisfies: the inclusions $F^{p}A^{p+q+1}(X)\hookrightarrow A^{p+q+1}(X), F^{p}A^{p+q}(X)\hookrightarrow A^{p+q}(X)$ are injective in cohomology, it is shown that the period map is holomorphic and transversal.