manuscripta mathematica

, Volume 98, Issue 3, pp 363–375

Canonical bundles of Deligne–Lusztig varieties

  • Søren Have Hansen

DOI: 10.1007/s002290050146

Cite this article as:
Hansen, S. manuscripta math. (1999) 98: 363. doi:10.1007/s002290050146


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).

Mathematics Subject Classification (1991):14M15 

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Søren Have Hansen
    • 1
  1. 1.Department of Mathematical Sciences, University of Aarhus,¶DK-8000 Aarhus C, Denmark. e-mail:

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