Serre duality, Abel’s theorem, and Jacobi inversion for supercurves over a thick superpoint

The principal aim of this paper is to extend Abel’s theorem to the setting of complex supermanifolds of dimension 1|q1|q over a finite-dimensional local supercommutative CC-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincaré duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic groups and class fields  , combined with an invariance result for the topology on the dual of the space of répartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov’s characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf DD of differential operators. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if nn is an integer sufficiently large that the generic divisor of degree nn is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree nn.