Higher F-injective singularities

TL;DR

This paper introduces higher F-injective singularities, linking them to Du Bois singularities in characteristic zero and exploring their properties and implications in algebraic geometry.

Contribution

It generalizes F-injectivity to higher levels and connects it to Du Bois singularities, providing new insights into singularity theory in algebraic geometry.

Findings

Isolated singularities are k-Du Bois if k-F-injective after reduction mod p.

Under the ordinarity conjecture, the converse holds.

Application to Frobenius liftable hypersurfaces.

Abstract

We introduce the concept of higher $F$-injectivity, a generalisation of $F$-injectivity. We prove that an isolated singularity over a field of characteristic zero is $k$-Du Bois if it is $k$-$F$-injective after reductions modulo infinitely many primes $p$. Under the ordinarity conjecture, we also establish the converse. As an application, we study Frobenius liftable hypersurfaces.