Vector fields on $\mathfrak{gl}_{m|n}(\mathbb C)$-flag supermanifolds

TL;DR

This paper computes the Lie superalgebras of holomorphic vector fields on complex flag supermanifolds, showing most are fundamental with respect to the natural $rak{gl}_{m|n}(b C)$ action, advancing understanding of supermanifold symmetries.

Contribution

It provides a detailed computation of the holomorphic vector fields on flag supermanifolds, revealing their structure and relation to the Lie superalgebra $rak{gl}_{m|n}(b C)$, with specific exceptions.

Findings

Most holomorphic vector fields are fundamental under the $rak{gl}_{m|n}(b C)$ action

Explicit description of the Lie superalgebra of vector fields on flag supermanifolds

Identification of several exceptions to the fundamental vector fields

Abstract

The main result of this paper is the computation of the Lie superalgebras of holomorphic vector fields on complex flag supermanifolds, introduced by Yu.I.Manin. We prove that with several exceptions any holomorphic vector field is fundamental with respect to the natural action of the Lie superalgebra $\mathfrak {gl}_{m|n}(\mathbb C)$.