Pure and Applied Mathematics Quarterly
Volume 19 (2023)
Special Issue in honor of Victor Guillemin
Guest Editors: Yael Karshon, Richard Melrose, Gunther Uhlmann, and Alejandro Uribe
Generalizing the Mukai Conjecture to the symplectic category and the Kostant game
Pages: 1803 – 1837
In this paper we pose the question of whether the (generalized) Mukai inequalities hold for compact, positive monotone symplectic manifolds. We first provide a method that enables one to check whether the (generalized) Mukai inequalities hold true. This only makes use of the almost complex structure of the manifold and the analysis of the zeros of the so-called generalized Hilbert polynomial, which takes into account the Atiyah-Singer indices of all possible line bundles.
We apply this method to generalized flag varieties. In order to find the zeros of the corresponding generalized Hilbert polynomial we introduce a modified version of the Kostant game and study its combinatorial properties.
symplectic geometry, combinatorics
The authors were partially supported by SFB-TRR 191 grant Symplectic Structures in Geometry, Algebra and Dynamics, funded by the Deutsche Forschungsgemeinschaft.
Received 30 November 2021
Received revised 1 June 2022
Accepted 11 August 2022
Published 20 November 2023