Pushforwards of Tilting Sheaves

TL;DR

This paper studies how tilting sheaves behave under pushforward by finite Galois morphisms, identifying conditions for preservation and providing examples and counterexamples involving Severi Brauer and toric varieties.

Contribution

It establishes criteria for when pushforward of a tilting sheaf remains tilting and offers new examples and counterexamples in algebraic geometry.

Findings

Conditions for pushforward of tilting sheaves to be tilting

Examples of Severi Brauer and toric varieties with tilting sheaves

Counterexamples showing pushforward may not be tilting

Abstract

We investigate the behaviour of tilting sheaves under pushforward by a finite Galois morphism. We determine conditions under which such a pushforward of a tilting sheaf is a tilting sheaf. We then produce some examples of Severi Brauer flag varieties and arithmetic toric varieties in which our method produces a tilting sheaf, adding to the list of positive results in the literature. We also produce some counterexamples to show that such a pushfoward need not be a tilting sheaf.