Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

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The geometry of moduli spaces of sheaves

We briefly review the formal picture in which a Calabi-Yau $n$-fold is the complex analogue of an oriented real $n$-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of \cite{LT}, \cite{BF} in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in $\Pee^3$, and Donaldson-- and Gromov-Witten-- like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold $X$, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general $K3$ fibration $X$, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the ``Mukai-dual'' 3-fold for others. As an example the invariant is shown to distinguish Gross' diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of $X$.

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A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations

In this paper, we consider basic problems on moduli spaces of stable sheaves on abelian surfaces. Our main assumption is the primitivity of the associated Mukai vector. We determine the deformation types, albanese maps, Bogomolov factors and their weight 2 Hodge structures. We also discuss the deformation types of moduli spaces of stable sheaves on K3 surfaces.

https://link.springer.com/content/pdf/10.1007/s002080100255.pdf

Moduli spaces of stable sheaves on abelian surfaces

where XKL π X X¬YMN π Y Y are the projection maps. Functors of this type which are equivalences of categories are called Fourier–Mukai transforms, and have proved to be powerful tools for studying moduli spaces of vector bundles [4, 5, 11]. A vector bundle 0 on X¬Y is called strongly simple over Y if, for each point y `Y, the bundle 0 y on X is simple, and if, for any two distinct points y " , y # of Y, and any integer i, one has Exti X (0 y " ,0 y # ) 0. One might think of the family 20 y :y `Y as an ‘orthonormal’ set of bundles on X. The following basic result allows one to construct many examples of Fourier– Mukai transforms.

http://www.tom-bridgeland.staff.shef.ac.uk/publications/pub2.pdf

Equivalences of triangulated categories and Fourier-Mukai transforms

We compute a large number of moduli spaces of stable bun- dles on a general algebraic elliptic surface using a new class of relative Fourier-Mukai transforms.

http://www.tom-bridgeland.staff.shef.ac.uk/publications/pub1.pdf

Fourier-Mukai transforms for elliptic surfaces

A Foruier-Mukai transform for stable bundles on K3 surfaces.

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3 surfaces $\hat X$ can be chosen to be isomorphic to $X$; then the Fourier-Mukai transform is invertible, and the image of a zero-degree stable bundle $F$ is stable and has the same Euler characteristic as $F$.

https://arxiv.org/pdf/alg-geom/9405006.pdf

A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

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.

Semistability vs. nefness for (Higgs) vector bundles

Stable sheaves on elliptic fibrations

We discuss an axiomatic approach to supermanifolds valid for arbitrary ground graded- commutative Banach algebras B. Rothstein's 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.

D. Hernández Ruipérez

Papers

A Foruier-Mukai transform for stable bundles on K3 surfaces.

A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

Semistability vs. nefness for (Higgs) vector bundles

Stable sheaves on elliptic fibrations

Foundations of supermanifold theory: the axiomatic approach