The generalized Mukai conjecture for spherical varieties with a reductive general isotropy group

TL;DR

This work proves the Gagliardi–Hofscheier conjecture for spherical varieties with reductive general isotropy, thereby establishing the generalized Mukai conjecture in this setting. The authors reduce the problem to combinatorial data via the spherical skeleton $\mathcal R=(\Delta,S^p,\Sigma,\Gamma)$ and its associated polytope $Q_{\mathcal R}$, then connect equality in the numerical bound $\mathscr P(\mathcal R) \le |R^+|-|R^+_{S^p}|$ to skeletons that arise from spherical modules. A key conceptual result is that, for spherical modules, $\mathscr P(X)=\dim(X)-\operatorname{rank}(X)$, proved using the weight monoid and a constructive completion, which implies skeleton completeness in this case. The main theorem follows from a case-by-case, LP-based verification over the non-symmetric indecomposable spherically closed reductive subgroups, showing equality occurs precisely for skeletons isomorphic to those of spherical modules. Consequently, the generalized Mukai conjecture holds for the targeted class, with the module-skeleton cases exactly characterizing the extremal equality instances. The approach bridges polyhedral combinatorics, representation theory, and the geometry of spherical varieties, providing a tractable path to broader classifications via the spherical skeleton framework.

Abstract

In the case of spherical varieties with reductive general isotropy group we prove a conjecture of G. Gagliardi and J. Hofscheier, which implies the generalized Mukai conjecture of L. Bonavero, C. Casagrande, O. Debarre and S. Druel for these varieties.

Theorems & Definitions (30)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.1
• Definition 3.1
• Conjecture 3.1
• Proposition 4.1
• proof