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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 

 

The generalized Mukai conjecture for symmetric varieties


Authors: Giuliano Gagliardi and Johannes Hofscheier
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
Published electronically: May 2, 2016
MathSciNet review: 3592522
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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.


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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
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

DOI: https://doi.org/10.1090/tran/6738
Received by editor(s): December 30, 2014
Received by editor(s) in revised form: April 15, 2015
Published electronically: May 2, 2016
Article copyright: © Copyright 2016 by the authors