On the Poincar\'e Lemma for reflexive differential forms

TL;DR

This paper investigates the exactness properties of the de Rham complex of reflexive differential forms on normal complex spaces, revealing conditions under which it is exact and relating these results to broader conjectures and vanishing theorems.

Contribution

It establishes new conditions for the exactness of the reflexive de Rham complex and explores its implications for the Lipman-Zariski conjecture and vanishing theorems on singular spaces.

Findings

Exactness in degree one under certain topological conditions.

High-degree exactness for holomorphically contractible spaces with mild singularities.

Connections between de Rham complex exactness, Lipman-Zariski conjecture, and vanishing theorems.

Abstract

In this paper we study the cohomology of the de Rham complex of sheaves of reflexive differential forms on a normal complex space. First, we prove that the complex is exact in degree one under suitable conditions on the underlying topological space, but that exactness in general depends on the complex structure. Second, we show exactness in high degrees for holomorphically contractible spaces under mild assumptions on the nature of singularities, e.g. klt singularities. Subsequently, the exactness of the de Rham complex of reflexive differential forms is related to the Lipman-Zariski conjecture and the failure of vanishing theorems of Kodaira-Akizuki-Nakano type on singular spaces.