Balanced Superprojective Varieties

TL;DR

This paper explores the geometry of superprojective spaces, extending the concept of balanced manifolds to supermanifolds, and discusses their relation to supercosets and superpoints.

Contribution

It introduces a new framework for balanced supermanifolds based on superprojective space geometry, extending Donaldson's classical definition.

Findings

Derived relations between superprojective spaces and supercosets

Extended Donaldson's balanced manifold concept to supermanifolds

Applied the theory to superpoints as submanifolds

Abstract

We first review the definition of superprojective spaces from the functor-of-points perspective. We derive the relation between superprojective spaces and supercosets in the framework of the theory of sheaves. As an application of the geometry of superprojective spaces, we extend Donaldson's definition of balanced manifolds to supermanifolds and we derive the new conditions of a balanced supermanifold. We apply the construction to superpoints viewed as submanifolds of superprojective spaces. We conclude with a list of open issues and interesting problems that can be addressed in the present context.