The generalized Mukai conjecture for symmetric varieties

TL;DR

This paper introduces a new invariant for complete spherical varieties, conjectures an inequality relating it to the variety's dimension and rank, and proves this conjecture for symmetric varieties, linking it to broader conjectures in algebraic geometry.

Contribution

The paper formulates a conjecture for spherical varieties, proves it for symmetric varieties, and connects it to the generalized Mukai conjecture and smoothness criteria.

Findings

Conjecture holds for horospherical and symmetric varieties.

Proves the conjecture for symmetric varieties.

Links the conjecture to the generalized Mukai conjecture.

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.