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This paper is devoted to the study of supergeometry of complex projective superspaces P n | m . First, we provide formulas for the cohomology of invertible sheaves of the form O P n | m ( ℓ ) , that are pullbacks of ordinary invertible sheaves on the reduced variety P n . Next, by studying the even Picard group Pic 0 ( P n | m ) , classifying invertible sheaves of rank 1 | 0 , we show that the sheaves O P n | m ( ℓ ) are not the only invertible sheaves on P n | 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 Pic Π ( P n | m ) , classifying Π -invertible sheaves of rank 1 | 1 , proving that there are also non-split Π -invertible sheaves on supercurves P 1 | m . Further, we investigate infinitesimal automorphisms and first order deformations of P n | 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 P 1 | m and of Calabi–Yau’s P n | n + 1 . Last, with an eye to applications to physics, we show in full detail how to endow P 1 | 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 o s p ( 2 | 2 ) . A particular effort has been devoted to keep the exposition as concrete and explicit as possible. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we estimate the eigenvalues of the twisted Dirac operator on Kähler submanifolds of the complex projective space and we discuss the sharpness of this estimate for the embedding . [Copyright &y& Elsevier]

Abstract: We study Kustin–Miller unprojections between Calabi–Yau threefolds, or more precisely the geometric transitions they induce. We use them to connect many families of Calabi–Yau threefolds with Picard number one to the web of Calabi–Yau complete intersections. This result enables us to find explicit description of a few known families of Calabi–Yau threefolds in terms of equations. Moreover, we find two new examples of Calabi–Yau threefolds with Picard group of rank one, which are described by Pfaffian equations in weighted projective spaces. [Copyright &y& Elsevier]