  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.

Algebraic Geometry;Complex Variables