Duality Construction of Moduli Spaces

The duality of moduli spaces of coherent sheaves on curves and surfaces gives a GIT free construction of these spaces. It is a suitable generalization for topics as: strange duality, jumping lines of vector bundles on projective space, and Faltings' construction of the moduli space of semistable vector bundles on a complex curve given in his article Stable G-bundles and projective connections. Using this duality we propose a construction for the moduli space of semistable sheaves on a surface.     

Counterexamples of Kodaira vanishing for smooth surfaces of general type in positive characteristic

We generalize the construction of Raynaud [14] of smooth projective surfaces of general type in positive characteristic that violate the Kodaira vanishing theorem. This corrects an earlier paper [19] of the same purpose. These examples are smooth surfaces fibered over a smooth curve whose direct images of the relative dualizing sheaves are not nef, and they violate Kollár's vanishing theorem. Further pathologies on these examples include the existence of non-trivial vector fields and that of non-closed global differential 1-forms.     

Self-dual Sheaves on Reductive Borel—Serre Compactifications of Hilbert Modular Surfaces

The main theorem of this article asserts that the category of self-dual sheaves compatible with the intersection chain sheaves (for upper/lower middle perversity) on the reductive Borel—Serre compactification of a Hilbert modular surface is nonempty. Also we prove that the direct image of such a sheaf under the canonical map to the Baily—Borel compactification is isomorphic (in the derived category) to the intersection chain sheaf for upper and lower middle perversity. As a consequence of the main theorem, there exist characteristic L-classes of these sheaves in the rational homology of . In fact, these classes do not depend on the choice of a self-dual sheaf and hence are invariants of the compactification .     

Inflectional loci of scrolls

Let be a scroll over a smooth curve C and let denote the hyperplane bundle. The special geometry of X implies that certain sheaves related to the principal part bundles of are locally free. The inflectional loci of X can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.      

Déformations de fibrés vectoriels sur les variétés de dimension 3

In the first part, we give some necessary conditions for a torsion free sheaf on a smooth threefold to be a reduced limit of vector bundles.In a second part of the article, we illustrate these results by reinterpreting a condition described by G. Ellingsrud and S.A. Strømme as a condition obtained in the first part. This enables us to identify, in a well known too big familly, the torsion free sheaves that are limit of vector bundles.     

On dual-projectively flat affine connections

Given a pseudo Riemannian metrich and a torsion-free affine connection on a smoothn-manifold M,a dual geodesic curve of is defined as a curve whose tangent 1-form is parallel along the curve. The corresponding dual-projective group is defined as a group of transformations of connections preserving dual-geodesic curves. The class of connections semi-compatible with the metrich and pairs of semi-conjugate connections are defined using the relations between their geodesics and dual-geodesics. The dual-projective curvature tensor for a connection semi-compatible withh is determined as an invariant of the dual projective group. Dual-projectively flat connections semi-compatible withh are characterized as connections with vanishing dual-projective curvature tensor. As an application we recover the fundamental theorem for non-degenerate hypersurface immersions.Research partialy supported by Contract MM 18/1991 with the Ministry of Science and Education of Bulgaria and by Contract with the University of Sofia.     

On dual holomorphically projectively flat affine connections

Given a complex Riemannian metrich and a torsion-free complex affine connection on a complex manifold, a dual holomorphically-planar curve of is defined as a curve whose tangent complex plane, generated by its tangent 1-form, is parallel along the curve. The corresponding dual holomorphically projective group is defined as a group of transformations of connections preserving dual holomorphically-planar curves. The class of connections complex semi-compatible with the metrich and pairs of complex semi-conjugate connections are defined using the relations between their holomorphically-planar curves and their dual holomorphically-planar curves. The dual holomorphically-projective curvature tensor for a connection complex semi-compatible withh is determined as an invariant of the dual holomorphically-projective group. Dual holomorphically-projectively flat connections complex semi-compatible withh are characterized as connections with vanishing dual holomorphically-projective curvature tensor.Research partialy supported by Contract MM 423/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridski.I would like to thank the referee for indicating omissions in the text and for the helpful advices during the preparation of the final form of the paper.     

A Structure Theorem for \displaystyle\mathbb{P}^{{1}} — Spec k-bimodules

Let k be an algebraically closed field. Using the Eilenberg–Watts theorem over schemes (Nyman, J Pure Appl Algebra 214:1922–1954, 2010), we determine the structure of k-linear right exact direct limit and coherence preserving functors from the category of quasi-coherent sheaves on $\mathbb{P}^{1}_{k}$ to the category of vector spaces over k. As a consequence, we characterize those functors which are integral transforms.     

Sheaf cohomology and free resolutions over exterior algebras

We derive an explicit version of the Bernstein-Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its ``Koszul dual' exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the ``linear part' of a resolution over the exterior algebra.

We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their ``linear parts" in the sense that erasing all terms of degree 1$"> in the complex yields a new complex which is eventually exact.

As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to prove two conjectures about the morphisms in the monad, and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data.

     

Topologien auf Schnittmoduln kohärenter komplex-und reell-analytischer Garben

In this paper we consider coherent complex-analytic sheaves F on a complex-analytic space X, and study two canonical topologies, inductive resp. projective locally convex, on F(A,F) for subsets ax. We are interested in conditions on A for which these topologies coincide, and get as a main result that this is the case for real analytic spaces which can be imbedded in some ^l and have the original X as a complexification. By complexification we apply our results to coherent real-analytic sheaves.     

\boldsymbol{\mathcal{Q}}-*-Categories

We consider the theory of categories enriched in an involutive quantaloid : the -*-categories. After giving an introduction to involutive quantaloids and nuclei, we use matrices with entries in to define -*-categories. Then we examine the relations between two kinds of morphisms between them, the functors and the *-maps, to provide a basis to study completeness properties. These results are used to provide a definition of pseudo-presheaves, presheaves and sheaves on involutive quantaloids in order to get a generalization of presheaves and sheaves on sites. Finally a characterization of these sheaves in terms of covers and compatible families is presented.      

Hyperholomorphic connections on coherent sheaves and stability

Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ? with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessary L ²-integrable. We show that such sheaves are polystable.     

Koszul duality for toric varieties

We show that certain categories of perverse sheaves on affine toric varieties and defined by dual cones are Koszul dual in the sense of Beilinson, Ginzburg and Soergel (1996). The functor expressing this duality is constructed explicitly by using a combinatorial model for mixed sheaves on toric varieties.

     

EXOTIC SYMMETRIC SPACE OVER A FINITE FIELD,III

Let \( \mathcal{X}=G/H\times V \) , where V is a symplectic space such that G = GL(V) and H = Sp(V). In previous papers, the authors constructed character sheaves on \( \mathcal{X} \) , based on the explicit data. On the other hand, there exists a conceptual definition of character sheaves on \( \mathcal{X} \) based on the idea of Ginzburg in the case of symmetric spaces. Our character sheaves form a subset of Ginzburg type character sheaves. In this paper we show that these two definitions actually coincide, which implies a classification of Ginzburg type character sheaves on \( \mathcal{X} \) .     

Perverse Sheaves on Image Multiple Point Spaces

Using multiple point spaces some new examples of perverse sheaves on images of maps are described. Furthermore, suppose is a finite and proper map of complex analytic manifolds of dimension n and n+1 such that every multiple point space is nonsingular and has the dimension expected of a generic map. Then we can describe the composition series for the constant sheaf on the image in the category of perverse sheaves.     

Formal structure of direct image of holonomic $${\mathcal{D}}$$ -modules of exponential type

We compute formal invariants associated with the cohomology sheaves of the direct image of holonomic -modules of exponential type. We also prove that every formal -modules is isomorphic, after a ramification, to a germ of formalized direct image of analytic -module of exponential type.     

Local connection forms revisited

Local connection forms provide a very useful tool for handling connections on principal bundles, because, essentially, they involve only the adjoint representation and the left (logarithmic) differential of the structure group, thus overcoming any complexities of the total space. The main results here characterize connections related together by bundle morphisms. A few applications refer to connections on (Banach) associated bundles and connections on projective limit bundles (in the Fréchet framework). The role of local connection forms is further illustrated by their sheaf-theoretic globalization, resulting in a sheaf-theoretic approach to principal connections. The latter point of view is naturally leading to a theory of connections on abstract principal sheaves.     

3-Mannigfaltigkeiten im ℙ5 und ihre zugehörigen Stabilen Garben

In this paper we begin to study 3-folds in a projective space of dimension 5. Using results from [9] we give a classification of all 3-folds in ⁵ , up to degree 6. There are only 3 different types of 3-folds in ⁵ of degree 6 which are not complete intersections. These manifolds can be represented as zero schemes of sections in certain (extremal) semistable reflexive sheaves of rank 2 on ⁵ . Finally we obtain examples of stable reflexive sheaves on ⁵ with homologieal dimension 1, which do not belong to the extremal sheaves [10].

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Dies ist ein Teil meiner Habilitationsschrift