The character sheaves on the group compactification

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in [G. Lusztig, Parabolic character sheaves, II, Mosc. Math. J. 4 (4) (2004) 869-896]. We also prove some properties of the character sheaves on the group compactification.     

Semi-stable Higgs sheaves and Bogomolov type inequality

In this paper, we study semi-stable Higgs sheaves over compact Kähler manifolds. We prove that there is an admissible approximate Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently, the Bogomolov type inequality holds on a semi-stable reflexive Higgs sheaf.     

Complexes pondérés sur les compactifications de Baily-Borel: Le cas des variétés de Siegel

In this work, we calculate the trace of a Hecke correspondance composed with a power of the Frobenius endomorphism on the fibre of the intersection complexes of the Baily-Borel compactification of a Siegel modular variety.

Our main tool is Pink's theorem about the restriction to the strata of the Baily-Borel compactification of the direct image of a local system on the Shimura variety. To use this theorem, we give a new construction of the intermediate extension of a pure perverse sheaf as a weight truncation of the full direct image.

More generally, we are able to define analogs in positive characteristic of the weighted cohomology complexes introduced by Goresky, Harder and MacPherson.

     

Orbifolds, Sheaves and Groupoids

We characterize orbifolds in terms of their sheaves, and show that orbifolds correspond exactly to a specific class of smooth groupoids. As an application, we construct fibered products of orbifolds and prove a change-of-base formula for sheaf cohomology.     

Localisation de faisceaux caractères

We obtain a formula for the values of the characteristic function of a character sheaf, in terms of the representation theory of certain finite groups related to the Weyl group. This formula, a generalization of previous results due to Mœglin and Waldspurger, depends on knowledge of certain reductive subgroups that admit cuspidal character sheaves. For quasi-simple groups, we make the formula truly explicit by determining all such subgroups upto conjugation.     

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 .     

A Construction of the Rational Function Sheaves on Elliptic Curves

Abstract The authors introduce an effective method to construct the rational function sheaf κ on an elliptic curve E, and further study the relationship between κ and any coherent sheaf on E. Finally, it is shown that the category of all coherent sheaves of finite length on E is completely characterized by κ.     

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U⊂X be an open set whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor.Under suitable hypotheses, we introduce a construction (called “S₂-extension”) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical “S₂-ification” of appropriate X. The construction also has applications to the “Macaulayfication” problem, and it is particularly well-behaved when X is Gorenstein.Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S₂-extension to give a uniform construction of the desired variety.     

Twisted stability and Fourier-Mukai transform II

 In this note, we define the twisted stability for a purely 1-dimensional sheaf and study the problem of the preservation of the stability condition under the relative Fourier-Mukai transform on an elliptic surface. As an application, we compute the Hodge polynomials of some moduli spaces of sheaves on an elliptic surface. We also construct the moduli space of twisted semi-stable sheaves. Received: 29 January 2002 / Revised version: 16 October 2002 Published online: 24 January 2003 Mathematics Subject Classification (2000): 14D20     

Duality and intersection theory in complex manifolds. I

We introduce the concept of a twisting cochain and a twisted complex associated to a coherent sheaf. For sheaves of submanifolds these twisted complexes are used to construct on cochain level the Grothendieck theory of dual class and Gysin map. These explicit constructions give, for instance, a local formula for dual class of higher codimensional submanifolds. We prove a refined version of the Hirzebruch Riemann Roch using such local formulas. We also prove a theorem on when global analytic intersection classes can be computed from first order geometric data. This theory will be used to prove the Holomorphic Lefschetz formula (in Part II) and the Hirzebruch Riemann Roch for analytic coherent sheaves.The first author is supported in part by NSF grants GP-36418X1 and MCS 76-08478. The second by MCS 75-07986 and Sonderforschungsbereich Theoretische Mathematik at Bonn University     

Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles II: Higgs sheaves and admissible structures

We study the basic properties of Higgs sheaves over compact Kähler manifolds and establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we prove some properties of semistable Higgs sheaves.     

L-fuzzy集理论的范畴基础与层表示

鉴于L-fuzzy集在理论上的重要性和应用上的广泛性,旨在建立L-fuzzy集理论的范畴基础与它的层表示,提出完备范畴中对象上的格值结构概念,这一概念是L-fuzzy结构在范畴层面上的提升,进一步提出完备范畴上格值结构提升范畴概念,证明了在集合范畴中L-fuzzy结构与格值结构是同构的.以集层、群层、环层和左R-模层以及Grothendieck层等概念为基础,提出完备范畴中对象上的层结构以及完备范畴上层结构提升范畴概念,证明了在集合范畴中L-fuzzy结构与层结构也是同构的.     

Coherent Sheaves on P1: their Families and their Deformations Related to a Behavioral Approach to Singular Systems

Recently, Lomadze, Ravi, Rosenthal and Schumacher gave a natural and geometrically meaningful one-to-one correspondence between abstract linar behaviors and coherent sheaves on P ¹. Motivated by their result, here we give a complete picture of the deformation theory of coherent sheaves on P ¹. We use our results to study deformations over the field R obtaining a connectedness result for the real locus of the algebraic parameter spaces.     

Calabi–Yau varieties with semi-stable fibre structures

Motivated by the Strominger–Yau–Zaslow conjecture, we study Calabi–Yau varieties with semi-stable fibre structures. We use Hodge theory to study the higher direct images of wedge products of relative cotangent sheaves of certain semi-stable families over higher dimensional quasi-projective bases, and obtain some results on positivity. We then apply these results to study non-isotrivial Calabi–Yau varieties fibred by semi-stable Abelian varieties (or hyperkähler varieties).     

A Microlocal Riemann–Hilbert Correspondence

We present a microlocal version of the Riemann–Hilbert correspondence for regular holonomic D-modules. We show that a regular holonomic system of microdifferential equations is associated to a perverse sheaf concentrated in degree 0. Moreover, we show that this perverse sheaf can be recovered from the local system it determines on the complementary of its singular locus. We characterize the classes of perverse sheaves and local systems associated to regular holonomic systems of microdifferential equations.     

Rigidity of generalized laplacians and some geometric applications

Summary Every generalized laplacianL defined on a manifoldM determines a sheaf of L-harmonic sections namely the sheaf of local solutions ofLu = 0. We study the converse problem: to what extent this sheaf determines the operator. Our main result states that the sheaf ofL-harmonic sections determines the operator up to a conformal factor. Moreover, when the operator is a covariant laplacian and the dimension ofM is greater than 2, the sheaf determinesL up to a multiplicative constant. An interesting consequence is the following: if two Riemann metrics on a smooth manifold of dimension greater than 2 have the same sheaves of harmonic functions then they are homothetic.     

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     

On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics

We decompose each moduli space of semi-stable sheaves on the complex projective plane with support of dimension one and degree four into locally closed subvarieties, each subvariety being the good or geometric quotient of a set of morphisms of locally free sheaves modulo a reductive or a non-reductive group. We find locally free resolutions of length one for all these sheaves and describe them.     

The characteristic class and ramification of an ℓ-adic étale sheaf

We introduce the characteristic class of an ℓ-adic étale sheaf using a cohomological pairing due to Verdier (SGA5). As a consequence of the Lefschetz–Verdier trace formula, its trace computes the Euler–Poincaré characteristic of the sheaf. We compare the characteristic class to two other invariants arising from ramification theory. One is the Swan class of Kato-Saito [17] and the other is the 0-cycle class defined by Kato for rank 1 sheaves in [16]. Dedicated to Luc Illusie, with admiration