A convexity theorem for the real part of a Borel invariant subvariety

TL;DR

This paper extends convexity theorems for moment map images to the fixed sets of involutions on Borel-invariant subvarieties, generalizing previous results by Brion, Guillemin, and Sjamaar.

Contribution

It introduces a new convexity theorem for the moment map image of involution fixed sets in Borel-invariant subvarieties, broadening the scope of prior convexity results.

Findings

Proves convexity of the moment map image for involution fixed sets

Generalizes previous convexity theorems to new settings

Provides a unified framework for various convexity results

Abstract

M. Brion proved a convexity result for the moment map image of an irreducible subvariety of a compact integral Kaehler manifold preserved by the complexification of the Hamiltonian group action. V. Guillemin and R. Sjamaar generalized this result to irreducible subvarieties preserved only by a Borel subgroup. In another direction, L. O'Shea and R. Sjamaar proved a convexity result for the moment map image of the submanifold fixed by an antisymplectic involution. Analogous to Guillemin and Sjamaar's generalization of Brion's theorem, in this paper we generalize O'Shea and Sjamaar's result, proving a convexity theorem for the moment map image of the involution fixed set of an irreducible subvariety preserved by a Borel subgroup.