Generic Vanishing Fails for Surfaces in Positive Characteristic

TL;DR

This paper demonstrates that the Generic Vanishing theorem does not hold for smooth surfaces in positive characteristic, providing counterexamples and extending the failure to higher dimensions.

Contribution

It constructs explicit counterexamples showing Generic Vanishing fails for smooth surfaces in any characteristic p ≥ 3, extending known results to higher dimensions.

Findings

Counterexamples to Generic Vanishing in characteristic p ≥ 3

Failure of Generic Vanishing for surfaces in positive characteristic

Extension of counterexamples to dimension 3 and higher

Abstract

We show that there exist smooth surfaces violating Generic Vanishing in any characteristic $p \geq 3$. As a corollary, we recover a result of Hacon and Kov\'acs, producing counterexamples to Generic Vanishing in dimension 3 and higher.