Torus-equivariantly embedded toric manifolds associated to affine subspaces

TL;DR

This paper investigates the conditions under which the closure of a complex subtorus in a toric manifold forms a smooth, Hamiltonian submanifold, and describes the geometric properties of such embeddings via moment maps and Delzant polytopes.

Contribution

It introduces the concept of torus-equivariantly embedded toric manifolds associated to affine subspaces and characterizes their moment map images.

Findings

The closure of a complex subtorus can be a smooth Hamiltonian submanifold under certain conditions.

The moment map image of the submanifold matches the pullback of the Delzant polytope.

The paper defines and studies torus-equivariantly embedded toric manifolds.

Abstract

We study the closure of a complex subtorus in a toric manifold. If the closure of the complex subtorus is a smooth complex submanifold in the toric manifold, then the subtorus action on such submanifold is Hamiltonian. In this case, we may think of the embedding of the submanifold as torus-equivariant. We show that the image of the moment map for the Hamiltonian subtorus action on our submanifold coincides with the image of the Delzant polytope of the ambient toric manifold under the pullback of the inclusion of the tori. The submanifolds constructed in the present paper are called torus-equivariantly embedded toric manifolds with respect to the subtorus action.