The generalized Mukai conjecture for symmetric varieties

Gagliardi, Giuliano; Hofscheier, Johannes

Abstract

©2016 American Mathematical Society. We associate to any complete spherical variety X a certain nonnegative rational number ℘(X), which we conjecture to satisfy the inequality ℘(X) ≤ dimX − rankX 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.

Citation

Gagliardi, G., & Hofscheier, J. (2017). The generalized Mukai conjecture for symmetric varieties. Transactions of the American Mathematical Society, 369(4), 2615-2649. https://doi.org/10.1090/tran/6738

• Journal Article Type: Article
• Acceptance Date: Apr 15, 2015
• Online Publication Date: May 2, 2016
• Publication Date: Jan 1, 2017
• Deposit Date: Nov 12, 2019
• Journal: Transactions of the American Mathematical Society
• Print ISSN: 0002-9947
• Electronic ISSN: 1088-6850
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 369
• Issue: 4
• Pages: 2615-2649
• DOI: https://doi.org/10.1090/tran/6738
• Public URL: https://nottingham-repository.worktribe.com/output/3235100
• Publisher URL: https://www.ams.org/journals/tran/2017-369-04/S0002-9947-2016-06738-0/
• Additional Information: Date of acceptance not given. The date used is specified as "Received by editor(s) in revised form" date.