D. Hernández Ruipérez

Stable sheaves on elliptic fibrations

Abstract Let X → B be an elliptic surface and M (a,b) the moduli space of torsion-free sheaves on X which are stable of relative degree zero with respect to a polarization of type aH + bμ , H being the section and μ the elliptic fibre ( b ≫0). We characterize the open subscheme of M (a,b) which is isomorphic, via the relative Fourier–Mukai transform, with the relative compactified Simpson–Jacobian of the family of those curves D ↪ X which are flat over B . This generalizes and completes earlier constructions due to Friedman, Morgan and Witten. We also study the relative moduli scheme of torsion-free and semistable sheaves of rank n and degree zero on the fibres. The relative Fourier–Mukai transform induces an isomorphic between this relative moduli space and the relative n th symmetric product of the fibration. These results are relevant in the study of the conjectural duality between F-theory and the heterotic string.

Foundations of supermanifold theory: the axiomatic approach☆

Abstract We discuss an axiomatic approach to supermanifolds valid for arbitrary ground graded- commutative Banach algebras B. Rothsteins axiomatics is revisited and completed by a further requirement which calls for the completeness of the rings of sections of the structure sheaves, and allows one to dispose of some undesirable features of Rothstein supermanifolds. The ensuing system of axioms determines a category of supermanifolds which coincides with graded manifolds when B  R , and with G-supermanifolds when B is a finite-dimensional exterior algebra. This category is studied in detail. The case of holomorphic supermanifolds is also outlined.

Semistability vs. nefness for (Higgs) vector bundles

Generalizing a result of Miyaoka, we prove that the semistability of a vector bundle E on a smooth projective curve over a field of characteristic zero is equivalent to the nefness of any of certain divisorial classes θs, λs in the Grassmannians Grs(E) of locally-free quotients of E and in the projective bundles PQs, respectively (here 0<s<rkE and Qs is the universal quotient bundle on Grs(E)). The result is extended to Higgs bundles. In that case a necessary and sufficient condition for semistability is that all classes λs are nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the classes λs is equivalent to the semistability of the bundle E together with the vanishing of the characteristic class Δ(E)=c2(E)−r−12rc1(E)2.

A hyperkähler Fourier transform

Abstract Given two hyperkahler manifolds X and Y and a quaternionic instanton on X × Y , one can define a Fourier-Mukai transform, which maps quaternionic instantons on X to quaternionic instantons on Y . This construction encompasses the cases of two-dimensional algebraic tori and K3 surfaces already treated elsewhere. We also sketch some higher dimensional examples.

Fourier-Mukai transform and index theory

Given a submersive morphism of complex manifoldsf: X→Y, and a complex vector bundleE onX, there is a relationship between the higher direct images of ε (the sheaf of holomorphic sections ofE) and the index of the relative Dolbeault complex twisted byE. This relationship allows one to yield a global and simple proof of the equivalence between the Mukai transform of stable vector bundles on a torusT of complex dimension 2 and the Nahm transform of instantons. We also offer a proof of Mukai’s inversion theorem which circumvents the use of derived categories by introducing spectral sequences of sheaves onT (this is related to Donaldson and Kronheimer’s proof, but is automatically global and somehow simpler). The general framework developed in the first part of this paper may be applied to the study of the Mukai transform for more general varieties.

Global structures for the moduli of (punctured) super Riemann surfaces

Abstract A fine moduli superspace for algebraic super Riemann surfaces with a level- n structure is constructed as a quotient of the split superscheme of local spin-gravitivo fields by an etale equivalence relation. This object is not a superscheme, but still has an interesting structure: it is an algebraic superspace, that is, an analytic superspace with sufficiently many meromorphic functions. The moduli of super Riemann surfaces with punctures (fixed points in the supersurface) is also constructed as an algebraic superspace. Moreover, when one only considers ordinary punctures (fixed points in the underlying ordinary curve), it turns out that the moduli is a true superscheme. We prove furthermore that this moduli superscheme is split.

THE VARIETY OF POSITIVE SUPERDIVISORS OF A SUPERCURVE (SUPERVORTICES)

Abstract The supersymmetric product of a supercurve is constructed with the aid of a theorem of algebraic invariants and the notion of positive relative superdivisor (supervortex) is introduced. A supercurve of positive superdivisors of degree 1 (supervortices of vortex number 1) of the original supercurve is constructed as its supercurve of conjugate fermions, as well as the supervariety of relative positive superdivisors of degree p (supervortices of vortex number p ). A universal superdivisor is defined and it is proved that every positive relative superdivisor can be obtained in a unique way as a pull-back of the universal superdivisor. The case of SUSY-curves is discussed.

Supersymmetric products of SUSY-curves °

The supersymmetric product of a SUSY-curve over a field is constructed with the aid of a theorem of invariants, and the notion of relative superdivisor is introduced. A universal superdivisor is defined in the supersymmetric product by means of Manins superdiagonal, and it is proven that every superdivisor can be obtained in a unique way as a pullback of the universal superdivisor.

Some Results on Line Bundles over Susy-Curves

It is usually thought that any approach to superstrings a la Polyakov should involve a suitable generalization of the notion of Riemann surface.