On the stable principal Higgs sheaves

Let G be a connected reductive linear algebraic group defined over C with Lie algebra g. Let be a stable principal Higgs G-sheaf on a compact connected Kähler manifold. We consider all holomorphic sections of the adjoint vector bundle ad(EG) of EG that commute with the Higgs field φ. These correspond to the infinitesimal automorphisms of the principal Higgs G-sheaf. Any element of the center of g gives such a section. We prove that all the sections are given by the center of g.     

Equivalence of certain principal vectors of polynomial operator sheaves

The equivalence of derived chains constructed from the principal vectors of polynomial sheafs of operators acting in Hilbert space is studied. These derived chains correspond to different boundary-value problems on the semi-axis for operator-differential equations whose symbol is these operator sheafs. On the basis of equivalence tests assertions are deduced concerning the minimality of derived chains corresponding to a boundary-value problem on the semi-axis in the case in which the initial conditions of the vector solution at zero are known, and the solution itself obeys radiation-type conditions at infinity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 948–970, July, 1992.     

Moduli spaces of stable sheaves on abelian surfaces

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. Received: 28 February 2000 / Revised version: 15 September 2000 / Published online: 24 September 2001     

The derived category of quasi-coherent sheaves and axiomatic stable homotopy

We prove in this paper that for a quasi-compact and semi-separated (nonnecessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category Dqc(X) (which is equivalent to D(Aqc(X))) in the case of a usual scheme.     

Simplicity of principal eigenvalue for p-Laplace operator with singular indefinite weight

Given a connected open set and a function w ∈L^N/p(Ω) if 1 < p < N and w ∈L^r (Ω) for some r ∈(1, ∞) if p ≧ N, with we prove that the positive principal eigenvalue of the problem

Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine

According to Mukai and Iliev, a smooth prime Fano threefold $X$ of genus $9$ is associated with a surface $\mathbb{P }(\mathcal{V })$ , ruled over a smooth plane quartic $\varGamma $ , and the derived category of $\varGamma $ embeds into that of $X$ by a theorem of Kuznetsov. We use this setup to study the moduli spaces of rank- $2$ stable sheaves on $X$ with odd determinant. For each $c_2 \ge 7$ , we prove that a component of their moduli space $\mathsf{M}_X(2,1,c_2)$ is birational to a Brill–Noether locus of vector bundles with fixed rank and degree on $\varGamma $ , having enough sections when twisted by $\mathcal{V }$ . For $c_2=7$ , we prove that $\mathsf{M}_X(2,1,7)$ is isomorphic to the blow-up of the Picard variety $\text{ Pic}^{2}({\varGamma })$ along the curve parametrizing lines contained in $X$ .     

Moduli of stable sheaves on a smooth quadric and a Brill-Noether locus

We prove that the moduli space of stable sheaves of rank 2 with the Chern classes c₁=O_Q(1,1) and c₂=2 on a smooth quadric Q in P₃ is isomorphic to P₃. Using this identification, we give a new proof that a Brill-Noether locus, defined as the closure of the stable bundles with at least three linearly independent sections, on a non-hyperelliptic curve of genus 4, is isomorphic to the Donagi-Izadi cubic threefold.     

Symmetric polynomials constructed from moduli of stable sheaves with oddc 1 on ruled surfaces

We shall describe a universal sheaf for the moduli of stable sheaves with odd first chern class over ruled surfacesX. Then, we define and compute a set of symmetric polynomials in the symmetric algebra Sym^* (H ²(X;Z)), which are algebro-geometric analogues of Donaldson's polynomials (see [7] and [8]).     

Birational properties of moduli spaces of stable locally free rank-2 sheaves on algebraic surfaces

LetX be a smooth algebraic surface over the complex number field. Fix a polarizationL, an invertible sheafc ₁ and an integerc ₂ such that (4c ₂-c ₁ ² ) is positive. letM _L(c ₁,c ₂) be the moduli space ofL-stable locally free rank-2 sheaves onX with chern classesc ₁ andc ₂ respectively, and let ξ be a numerical equivalence class defining a nonempty wall of type (c ₁,c ₂). We study the properties ofE _ξ(c ₁,c ₂) and obtain estimations for its dimension. Then, we discuss the existence of trivial polarizations, and determine the birational structures of moduli spacesM _L(c ₁,c ₂) whenX is a minimal surface of general type and (4c ₂-c ₁ ² ) is sufficiently large.     

A complex Frobenius theorem, multiplier ideal sheaves and Hermitian-Einstein metrics on stable bundles

A complex Frobenius theorem is proved for subsheaves of a holomorphic vector bundle satisfying a finite generation condition and a differential inclusion relation. A notion of `multiplier ideal sheaf' for a sequence of Hermitian metrics is defined. The complex Frobenius theorem is applied to the multiplier ideal sheaf of a sequence of metrics along Donaldson's heat flow to give a construction of the destabilizing subsheaf appearing in the Donaldson-Uhlenbeck-Yau theorem, in the case of algebraic surfaces.

     

Cat-valued sheaves

Let \(\mathcal{\widetilde{O}}\)(B) be the category of (open) subcategories of a topological groupoid B: In this paper we study Cat-valued sheaves over category \(\mathcal{\widetilde{O}}\)(B): The paper introduces a notion of categorical union, such that the categorical union of subcategories is a subcategory. We use this definition of categorical unions to define a categorical cover of a topological category. Instead of assuming a Grothendieck topology, we define Cat-valued sheaves in terms of the categorical cover defined in this paper. The main result is the following. For a fixed category C, the categories of local functorial sections from B to C define a Catvalued sheaf on \(\mathcal{\widetilde{O}}\)(B): Replacing C with a categorical group G; we find a CatGrp-valued sheaf on \(\mathcal{\widetilde{O}}\)(B): We also relate and distinguish our construction with the notion of stacks.