Vanishing theorems on wonderful varieties

TL;DR

This paper investigates vanishing theorems for tautological bundles on wonderful varieties, providing new results and applications including a characteristic-independent cohomology analogue and connections to White's basis conjecture.

Contribution

It introduces new vanishing theorems for tautological bundles on wonderful varieties and applies them to cohomology comparisons and conjecture reductions.

Findings

Proved a characteristic-independent analogue of Brieskorn's cohomology result.

Established a comparison between Orlik--Solomon algebra and logarithmic de Rham cohomology.

Extended Borel--Weil--Bott type vanishing theorems for tautological bundles.

Abstract

We study vanishing theorems of tautological bundles in the sense of Berget--Eur--Spink--Tseng restricted to wonderful varieties. As an application, we prove a characteristic-independent analogue of Brieskorn's result on cohomology of arrangement complements, in addition to a comparison theorem between Orlik--Solomon algebra and the logarithmic de Rham cohomology of wonderful varieties. In a different direction, we extend a vanishing theorem of Borel--Weil--Bott type for tautological bundles. Finally, we reduce the weak version of White's basis conjecture to a problem about cohomology vanishing of tautological bundles.