The Du Bois complex of a hypersurface and the minimal exponent

Mircea Mustata, Sebastian Olano, Mihnea Popa, Jakub Witaszek

Published 2021-05-04Version 1

We study the Du Bois complex of a hypersurface $Z$ in a smooth complex algebraic variety in terms of the minimal exponent $\widetilde{\alpha}(Z)$ and give various applications. We show that if $\widetilde{\alpha}(Z)\geq p+1$, then the canonical morphism $\Omega_Z^p\to \underline{\Omega}_Z^p$ is an isomorphism. On the other hand, if $Z$ is singular and $\widetilde{\alpha}(Z)>p\geq 2$, then ${\mathcal H}^{p-1}(\underline{\Omega}_Z^{n-p})\neq 0$.