Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 4

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

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n4.a4

Authors

Alexander Caviedes Castro (Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia)

Milena Pabiniak (Department Mathematik/Informatik, Universität zu Köln, Germany)

Silvia Sabatini (Department Mathematik/Informatik, Universität zu Köln, Germany)

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.

Keywords

symplectic geometry, combinatorics

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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