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

Abstract

a b s t r a c tThe principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1|q over a finite-dimensional local supercommutative C-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 D of differential operators. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n.

Regularity of the superstring supermeasure and the superperiod map

“…There is an analogue of Grothendieck-Serre duality for families of supercurves in which ω X S plays a role of a relative dualizing sheaf, see [19,Sec. 2] (see also [16], where the case of Serre duality on supercurves is worked out in detail). This duality gives an isomorphism in the derived category of S, Rπ * (V ) ∨ ≃ Rπ * (V ∨ ⊗ ω X S ) [1], for any perfect complex V over X.…”

Pluri-Canonical Models of Supersymmetric Curves

“…General references about super geometry are [Man97], [DM99], [CCF11], [Wit12] and the first section of [DW13b]; let us mention also the slightly more technical [LPW90]. Two important references about supersymmetric curves are [Man91] and [Wit13]; other sources are [BR99], [RR14], [FK14] and [Kwo14]. Reference about moduli of susy curves are [LR88] and [DPHRSDS97], two recent important contributions are [DW13b] and [DW13a].…”