Injective resolutions of BG and derived moduli spaces of local systems

It was suggested on several occasions by Deligne, Drinfeld and Kontsevich that all the moduli spaces arising in the classical problems of deformation theory should be extended to natural “derived” moduli spaces which are always smooth in an appropriate sense and whose tangent spaces involve the entire cohomology of the sheaf of infinitesimal automorphisms, not just H¹. In this note we give an algebraic construction of such an extension for the simplest class of moduli spaces, namely for moduli of local systems (representations of the fundamental group).     

Image multiple point spaces and rectified homotopical depth

We investigate the local topology of multiple point spaces in the image of a finite complex analytic map. In particular, we find examples of maps for which the constant sheaf on an image multiple point space of the map is perverse. These results are proved by the use of a spectral sequence which calculates the homology of the image of a continuous map.

     

An additive formula for Samuel multiplicities on Hilbert spaces of analytic functions

We establish a short exact sequence to relate the germ model of invariant subspaces of a Hilbert space of vector-valued analytic functions and the sheaf model of the corresponding coinvariant subspaces. As a consequence we obtain an additive formula for Samuel multiplicities. As an application, we give a different proof for a formula relating the fibre dimension and the Samuel multiplicity which is first proved in Fang (2005) [11]. The feature of the new proof is that the analytic arguments in Fang (2005) [11] are now subsumed by algebraic machinery.     

Toric sheaves,stability and fibrations

For an equivariant reflexive sheaf over a polarised toric variety, we study slope stability of its reflexive pullback along a toric fibration. Examples of such fibrations include equivariant blow-ups and toric locally trivial fibrations. We show that stability (resp. unstability) is preserved under such pullbacks for so-called adiabatic polarisations. In the strictly semistable situation, under locally freeness assumptions, we provide a necessary and sufficient condition on the graded object to ensure stability of the pulled back sheaf. As applications, we provide various stable perturbations of semistable tangent sheaves, either by changing the polarisation, or by blowing-up a subvariety. Finally, our results apply uniformly in specific flat families and induce injective maps between the associated moduli spaces.     

L’application cotangente des surfaces de type général

We study surfaces of general type S whose cotangent sheaf is generated by its global sections. We define a map called the cotangent map of S that enables us to understand the obstructions to the ampleness of the cotangent sheaf of S. These obstructions are curves on S that we call “non-ample”. We classify the surfaces with an infinite number of non-ample curves and we partly classify the non-ample curves.     

The deformation theory of sheaves of commutative rings

We define a sheaf of abelian groups whose cohomology is represented by the cotangent complex, permitting a rapid introduction to the theory of the cotangent complex in the same generality as it was defined by Illusie, but avoiding simplicial methods. We show how obstructions to some standard deformation problems arise as the classes of torsors under and gerbes banded by this sheaf. This generalizes results of Quillen, Rim, and Gaitsgory.     

Invariants de Von Neumann des faisceaux analytiques cohérents

In order to study the group of holomorphic sections of the pull-back to the universal covering space of an holomorphic vector bundle on a compact complex manifold, it would be convenient to have a cohomological formalism, generalizing Atiyah's index theorem. In [Eys99], such a formalism is proposed in a restricted context. To each coherent analytic sheaf on a n-dimensionnal smooth projective variety and each Galois infinite unramified covering , whose Galois group is denoted by , cohomology groups denoted by are attached, such that: 1. The underly a cohomological functor on the abelian category of coherent analytic sheaves on X. 2. If is locally free, is the group of holomorphic sections of the pull-back to of the holomorphic vector bundle underlying . 3. belongs to a category of -modules on which a dimension function with real values is defined. 4. Atiyah's index theorem holds [Ati76]: The present work constructs such a formalism in the natural context of complex analytic spaces. Here is a sketch of the main ideas of this construction, which is a Cartan-Serre version of [Ati76]. A major ingredient will be the construction [Farb96] of an abelian category containing every closed -submodule of the left regular representation. In topology, this device enables one to use standard sheaf theoretic methods to study Betti numbers [Ati76] and Novikov-Shubin invariants [NovShu87]. It will play a similar r?le here. We first construct a -cohomology theory () for coherent analytic sheaves on a complex space endowed with a proper action of a group such that conditions 1-2 are fulfilled. The -cohomology on the Galois covering of a coherent analytic sheaf onX is the ordinary cohomology of a sheaf on X obtained by an adequate completion of the tensor product of by the locally constant sheaf on X associated to the left regular representation of the discrete group in the space of functions on . Then, we introduce an homological algebra device, montelian modules, which can be used to calculate the derived category of and are a good model of the Čech complex calculating -cohomology. Using this we prove that , if X is compact. This is stronger than condition 3, since this also yields Novikov-Shubin type invariants. To explain the title of the article, Betti numbers and Novikov-Shubin invariants of are the Von Neumann invariants of the coherent analytic sheaf . We also make the connection with Atiyah's -index theorem [Ati76] thanks to a Leray-Serre spectral sequence. From this, condition 4 is easily deduced.

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Received: 30 October 1998 / Published online: 8 May 2000     

Periodic Cyclic Homology as Sheaf Cohomology Dedicated to Daniel Quillen on his 60th Birthday

We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example, we show that infinitesimal hypercohomology with coefficients in K-theory gives the fiber of the Jones–Goodwillie character which goes from K-theory to negative cyclic homology.     

Grothendieck topologies and deformation Theory II

Starting from a sheaf of associative algebras over a scheme we show thatits deformation theory is described by cohomologies of a canonical object,called the cotangent complex, in the derived category of sheaves ofbi-modules over this sheaf of algebras. The passage from deformations tocohomology is based on considering a site which is naturally constructed outof our sheaf of algebras. It turns out that on the one hand, cohomology ofcertain sheaves on this site control deformations, and on the other hand,they can be rewritten in terms of the category of sheaves of bi-modules.     

Direct image of parabolic line bundles

Given a vector bundle E, on an irreducible projective variety X, we give a necessary and sufficient criterion for E to be a direct image of a line bundle under a surjective étale morphism. The criterion in question is the existence of a Cartan subalgebra bundle of the endomorphism bundle $\text{End}(E)$. As a corollary, a criterion is obtained for E to be the direct image of the structure sheaf under an étale morphism. The direct image of a parabolic line bundle under any ramified covering map has a natural parabolic structure. Given a parabolic vector bundle, we give a similar criterion for it to be the direct image of a parabolic line bundle under a ramified covering map.     

Hochschild cohomology of abelian categories and ringed spaces

This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in complete generality that Hochschild cohomology satisfies a Mayer-Vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf.     

Vanishing cotangent cohomology for Plücker algebras

We use representation theory and Bott’s theorem to show vanishing of higher cotangent cohomology modules for the homogeneous coordinate ring of Grassmannians in the Plücker embedding. As a by-product, we answer a question of Wahl about the cohomology of the square of the ideal sheaf for the case of Plücker relations. We obtain slightly weaker vanishing results for the cotangent cohomology of the coordinate rings of isotropic Grassmannians.     

Hitchin System on Singular Curves

We study the Hitchin system on singular curves. We consider curves obtainable from the projective line by matching at several points or by inserting cusp singularities. It appears that on such singular curves, all basic ingredients of Hitchin integrable systems (moduli space of vector bundles, dualizing sheaf, Higgs field, etc.) can be explicitly described, which can be interesting in itself. Our main result is explicit formulas for the Hitchin Hamiltonians. We also show how to obtain the Hitchin integrable system on such curves by Hamiltonian reduction from a much simpler system on a finite-dimensional space. We pay special attention to a degenerate curve of genus two for which we find an analogue of the Narasimhan–Ramanan parameterization of the moduli space of SL(2) bundles as well as the explicit expressions for the symplectic structure and Hitchin-system Hamiltonians in these coordinates. We demonstrate the efficiency of our approach by rederiving the rational and trigonometric Calogero–Moser systems, which are obtained from Hitchin systems on curves with a marked point and with the respective cusp and node.     

On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Fix a smooth very ample curve C on a K3 or abelian surface X. Let $ \mathcal{M} $ denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on $ \mathcal{M} $. On the other hand, $ \mathcal{M} $ has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on $ \mathcal{M} $ coincides with the pullback of the symplectic form on the Hilbert scheme.     

Moduli of reflexive sheaves on smooth projective 3-folds

We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf, and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi-Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.     

An approximation theorem for sections of reflexive sheaves

 We show that, if X is a Stein manifold and D ? X an open set (not necessarily Stein) such that the restriction map has dense image, then, for any reflexive coherent analytic sheaf ℱ on X, the map has dense image, too. We also characterize the reflexivity of a torsion-free coherent sheaf on complex manifolds in terms of absolute gap sheaves or Kontinuit?tssatz. Received: 14 September 2001 / Revised version: 29 January 2002     

Intersection Poincaré Polynomial for Nagaraj–Seshadri moduli space

We compute Betti numbers of both the components of the moduli space of rank 2 semi‐stable torsion‐free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.