Vector fields on ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}(\mathbb{C})$-flag supermanifolds

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 ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}(\mathbb{C})$.