Generalizing the Mukai Conjecture to the symplectic category and the Kostant game

TL;DR

This paper explores whether generalized Mukai inequalities apply to compact, positive monotone symplectic manifolds by developing a new method based on the analysis of the generalized Hilbert polynomial and a modified Kostant game.

Contribution

It introduces a novel approach using the almost complex structure and a modified Kostant game to verify Mukai inequalities on symplectic manifolds, especially generalized flag varieties.

Findings

Method to check Mukai inequalities using generalized Hilbert polynomial

Application of the method to generalized flag varieties

Introduction of a modified Kostant game for combinatorial analysis

Abstract

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.