A generalization of Barannikov-Kontsevich theorem

TL;DR

This paper proves the $E_1$-degeneration of the Hodge-to-de Rham spectral sequence for a twisted de Rham complex on a Kähler manifold, extending the Barannikov-Kontsevich theorem to a broader setting.

Contribution

It generalizes the Barannikov-Kontsevich theorem by establishing $E_1$-degeneration for twisted de Rham complexes with compact critical sets.

Findings

Proves $E_1$-degeneration of the spectral sequence.

Extends the theorem to more general Kähler manifolds.

Provides new insights into the structure of twisted de Rham complexes.

Abstract

We study the twisted de Rham complex associated with a holomorphic function on a K\"ahler manifold whose critical point set is compact. We prove the $E_1$-degeneration of the Hodge-to-de Rham spectral sequence. It is a generalization of Barannikov-Kontsevich Theorem.