Variation of Hodge structures for non-Kähler manifolds

TL;DR

This work extends Griffiths' variation of Hodge structures to non-Kähler manifolds by developing a deformation theory for filtered forms and showing that a period map on the associated filtered cohomology is holomorphic and Griffiths-transversal under specific cohomological injectivity assumptions. A central tool is the exponential operator $e^{i_{oldsymbol{ ho}(t)}}$, which preserves the Hodge filtration and furnishes an isomorphism between the filtrations on the central fiber and its deformations. The authors derive a jumping formula for the dimensions of $F^pH^{k}(X_t)$, identify canonical deformations and obstructions, and connect these to the Frölicher spectral sequence, enabling criteria for invariance of filtrations and semi-continuity statements. Although polarized VHS cannot be guaranteed in general without a Kähler structure, the work clarifies when a unpolarized VHS with holomorphic, transversal period maps can be defined in the non-Kähler setting, and highlights the special role of $E_1$-degeneration in stabilizing invariants.

Abstract

In this note, we discuss unpolarized, complex variation of Hodge structures for non-Kähler 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.

Theorems & Definitions (23)

• Corollary 1.1
• Theorem 1.2: =Theorem \ref{['thm-period-map']}
• Proposition 3.1
• Corollary 3.2
• proof
• Proposition 3.3
• proof
• Definition 3.4
• Definition 3.5
• Proposition 3.6