Projective superspaces in practice

This paper is devoted to the study of supergeometry of complex projective superspaces Pn|m. First, we provide formulas for the cohomology of invertible sheaves of the form OPn|m(ℓ), that are pullbacks of ordinary invertible sheaves on the reduced variety Pn. Next, by studying the even Picard group Pic0(Pn|m), classifying invertible sheaves of rank 1|0, we show that the sheaves OPn|m(ℓ) are not the only invertible sheaves on Pn|m, but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the Π-Picard group PicΠ(Pn|m), classifying Π-invertible sheaves of rank 1|1, proving that there are also non-split Π-invertible sheaves on supercurves P1|m. Further, we investigate infinitesimal automorphisms and first order deformations of Pn|m, by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special attention is paid to the meaningful case of supercurves P1|m and of Calabi-Yau's Pn|n+1. Last, with an eye to applications to physics, we show in full detail how to endow P1|2 with the structure of N=2 super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra osp(2|2). A particular effort has been devoted to keep the exposition as concrete and explicit as possible.