The Geometry of Supermanifolds

Abstract

Soient X et Y deux surfaces hyperkählériennes et Q un instanton généralisé sur X ×Y ; il est possible d'introduire un foncteur de Fourier-Mukai, qui, sous des hypothèses convenables, transforme fibrés sur X en fibrés sur Y . Dans le cas où X et Y sont deux tores complexes duales l'un à l'autre, ce foncteur transforme instantons sur X en instantons sur Y . Après un bref aperçu de ces résultats, on définit une transformation de Fourier-Mukai lorsque X est une surface K3 et on étudie le comportement des instantons sous l'action de cette transformation. Fourier-Mukai transform on hyperkähler surfacesAbstract -Given two compact hyperkähler surfaces X and Y and a holomorphic vector bundle Q on X × Y , which is a generalized instanton, one can define a Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles on X to vector bundles on Y . If X and Y are dual complex tori, this transform maps instantons on X to instantons on Y . After a quick review of these results, we define a Fourier-Mukai transform in the case when X is a K3 surface, and study the behaviour of instantons on X under this transform.

Noninvertibility, Semisupermanifolds and Categories Regularization

“…In the supermanifold theory [17,18,19] the phenomenon of noninvertibility obviously arises from odd nilpotent elements and zero divisors of Grassmann algebras (also in the infinite dimensional case [20]). Despite the invertibility question is quite natural, the answer is not so simple and in some cases can be nontrivial, e.g.…”

Locally free sheaves on complex supermanifolds

“…It possesses also the Z 2 -grading given by (2). Our aim now is to classify locally free sheaves of O-modules on supermanifolds (M, O) assuming that (M, gr O) is fixed.…”

“…, we can construct a locally free sheaf of O-modules on (M, O(γ ij )), where (M, O(γ ij )) is the supermanifold corresponding to the cocycle (γ ij ) ∈ Z 1 (U, Aut (2) …”

The category of Z2n-supermanifolds

“…Of course, there is an extensive literature on Supergeometry and related topics and it is impossible to give complete references; let us at least mention [BBHR91], [Ber79], [Ber87], [DeW84], [Kos75], [Rog07], [Tuy04]. The sources that had the highest impact on the present text are: [Lei80], [Rog07], [Tuy04], [Lei11], [Var04], [Man02], [DM99], [CCF11], [DSB03], [Vor12], [BP12], [GKP09], [GKP10], [Sch97].…”