Canonical bundles of Deligne-Lusztig varieties

Abstract

In this paper we consider Deligne–Lusztig varieties. We explicitly describe the canonical bundles of their smooth compactifications in terms of homogeneous line bundles pulled back from G/B. Using this description we show that the members (one member in each dimension) of a special family of Deligne–Lusztig varieties have ample canonical bundles. A consequence is that, unlike the closely related Schubert varieties, Deligne–Lusztig varieties are not in general Frobenius split. Several examples are given. Among these we exhibit (Example 4) an infinite family of counter-examples to the Miyaoka–Yau inequality (one for each prime power).