Higher singularities for hypersurfaces

TL;DR

This paper investigates conditions under which certain differential forms on complex hypersurfaces with higher singularities are isomorphic, extending results to positive characteristic cases.

Contribution

It establishes a new criterion linking the isomorphism of differential forms of order m to all lower orders on hypersurfaces with high codimension singularities.

Findings

If ^m_D \u2261 _D^m, then ^i_D \u2261 _D^i for all i  to m.

Results extend to hypersurfaces in positive characteristic.

Provides insights into the structure of differential forms on singular hypersurfaces.

Abstract

With an assumption on the codimension of the singular locus of a complex hypersurface $D$ in smooth variety $X$, we show that if $\underline{\Omega}^m_D \cong \Omega^m_D$, then $\underline{\Omega}^i_D \cong \Omega^i_D$ for all $0 \leq i \leq m$. We also discuss an analogue of this statement in positive characteristic.