Localization formulas on complex supermanifolds

TL;DR

This paper develops localization formulas and residue theorems for odd holomorphic super vector fields on compact complex supermanifolds, extending classical results to the supergeometric setting.

Contribution

It introduces new localization formulas and residue calculations specifically for supermanifolds with equal fermionic and bosonic dimensions, a novel extension of classical complex geometry.

Findings

Established residue theorem for super vector fields with isolated singularities

Derived explicit residue formulas under generic local conditions

Extended classical localization techniques to supergeometry

Abstract

In this work we provide a localization formulae for odd holomorphic super vector fields on compact complex supermanifolds with fermionic dimension equal to the bosonic dimension. We prove a residue theorem for holomorphic super vector fields with non-degenerated and isolated singularities and we determine the residues under certain generic local conditions.