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arXiv · 1412.6084

The generalized Mukai conjecture for symmetric varieties

Giuliano Gagliardi·Johannes Hofscheier
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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 $\wp(X) \le \operatorname{dim} X - \operatorname{rank} X$ 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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Giuliano Gagliardi, Johannes Hofscheier. 2016-05-09. The generalized Mukai conjecture for symmetric varieties. https://doi.org/10.1090/tran%2F6738

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