Complexes of differential forms and singularities: The injectivity theorem

TL;DR

This paper proves an injectivity theorem for varieties with specific Du Bois singularities, confirming a conjecture and advancing understanding of the relationship between Du Bois complexes and their duals.

Contribution

It establishes the injectivity of a natural morphism for varieties with (m-1)-Du Bois singularities, confirming a key conjecture in the field.

Findings

Confirmed Conjecture G of Popa, Shen, and Vo [PSV24]

Proved injectivity of a morphism between duals of Du Bois complexes

Enhanced understanding of singularities in algebraic geometry

Abstract

In this paper, it is proved, that for varieties with (m-1)-Du Bois singularities, the natural morphism from the Grothendieck dual of the m-th graded Du Bois complex to the Grothendieck dual of its zero-th cohomology sheaf is injective on cohomology. This confirms Conjecture G of Popa, Shen, and Vo [PSV24].