The generalized Mukai conjecture for symmetric varieties
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- by Giuliano Gagliardi and Johannes Hofscheier PDF
- Trans. Amer. Math. Soc. 369 (2017), 2615-2649
Abstract:
We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp ({X})$, which we conjecture to satisfy the inequality $\wp ({X}) \le \dim X - \mathrm {rank} X$ with equality holding if and only if $X$ is isomorphic to a toric variety. We show that, for spherical varieties, our conjecture implies the generalized Mukai conjecture on the pseudo-index of smooth Fano varieties due to Bonavero, Casagrande, Debarre, and Druel. We also deduce from our conjecture a smoothness criterion for spherical varieties. It follows from the work of Pasquier that our conjecture holds for horospherical varieties. We are able to prove our conjecture for symmetric varieties.References
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Additional Information
- Giuliano Gagliardi
- Affiliation: Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
- Address at time of publication: Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
- MR Author ID: 1040639
- Email: gagliardi@math.uni-hannover.de
- Johannes Hofscheier
- Affiliation: Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
- Address at time of publication: Institut für Algebra und Geometrie, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
- Email: johannes.hofscheier@ovgu.de
- Received by editor(s): December 30, 2014
- Received by editor(s) in revised form: April 15, 2015
- Published electronically: May 2, 2016
- © Copyright 2016 by the authors
- Journal: Trans. Amer. Math. Soc. 369 (2017), 2615-2649
- MSC (2010): Primary 14M27; Secondary 14J45, 14L30, 52B20
- DOI: https://doi.org/10.1090/tran/6738
- MathSciNet review: 3592522