Homomorphisms of infinitely generated analytic sheaves

We prove that every homomorphism \(\mathcal{O}^{E}_{\zeta}\rightarrow\mathcal{O}^{F}_{\zeta}\), with E and F Banach spaces and ζ∈? ^m , is induced by a \(\mathop{\mathrm{Hom}}(E,F)\)-valued holomorphic germ, provided that 1≤m<∞. A similar structure theorem is obtained for the homomorphisms of type \(\mathcal{O}^{E}_{\zeta}\rightarrow\mathcal{S}_{\zeta}\), where \(\mathcal{S}_{\zeta}\) is a stalk of a coherent sheaf of positive depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheaves which guarantees the sheaf to be equipped with a unique analytic structure in the sense of Lempert–Patyi.     

Criteria for the injectivity of analytic sheaves

It is shown that a moduleL over the sheafO of germs of holomorphic functions on a domain G of Cⁿ is injective if and only if the following conditions are satisfied; a)L is flabby; b) for every closed set S ?G and every point z λ G, the stalk se _z of the sheafS ^L;U1→Γ S ^(U:L) is an injectiveO _z -module. It follows in particular that the sheaf of germs of hyperfunctions is injective over the sheaf of germs of analytic functions.     

Chern classes in Deligne cohomology for coherent analytic sheaves

In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes satisfy the functoriality property under pullbacks, the Whitney formula and the Grothendieck–Riemann–Roch theorem for projective morphisms between smooth complex compact manifolds.     

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.     

Quotients of Bing spaces

A Bing space is a compact Hausdorff space whose every component is a hereditarily indecomposable continuum. We investigate spaces which are quotients of a Bing space by means of a map which is injective on components. We show that the class of such spaces does not include every compact space, but does properly include the class of compact metric spaces.     

Quotients of gravitational instantons

A classification result for Ricci-flat anti-self-dual asymptotically locally Euclidean 4-manifolds is obtained: they are either hyperk?hler (one of the gravitational instantons classified by Kronheimer), or they are a cyclic quotient of a Gibbons–Hawking space. The possible quotients are described in terms of the monopole set in \mathbbR³{\mathbb{R}^3} , and it is proved that every such quotient is actually K?hler. The fact that the Gibbons–Hawking spaces are the only gravitational instantons to admit isometric quotients is proved by examining the possible fundamental groups at infinity: most can be ruled out by the classification of three-dimensional spherical space form groups, and the rest are excluded by a computation of the Rohklin invariant (in one case) or the eta invariant (in the remaining family of cases) of the corresponding space forms.     

Quotients of incidence geometries

We develop a theory for quotients of geometries and obtain sufficient conditions for the quotient of a geometry to be a geometry. These conditions are compared with earlier work on quotients, in particular by Pasini and Tits. We also explore geometric properties such as connectivity, firmness and transitivity conditions to determine when they are preserved under the quotienting operation. We show that the class of coset pregeometries, which contains all flag-transitive geometries, is closed under an appropriate quotienting operation.     

Quotients of rigid graphs

A consequence of the main theorem presented here is a companion result to that of |2| (see also |1|): a finite undirected graph is a quotient of a rigid one if and only if it has no 2-colourable components. In fact, any graph of finitely many components none of which is 2-colourable is a common quotient of graphs that form a universal (or binding) category.     

Affine Quotients of Supergroups

In this paper we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If G is an affine supergroup and H is its normal supersubgroup, then we prove that a dur K-sheaf is again an affine supergroup. Additionally, if G is algebraic, then a K-sheaf is also an algebraic supergroup and it coincides with . In particular, any normal supersubgroup of an affine supergroup is faithfully exact. Supported by RFFI 07-01-00392 and by FAPESP (proc. 07/54834-9).     

Quotients of uniform spaces

This paper develops the basic theory of quotients of uniform spaces via sufficiently nice group actions. We generalize and unify two fundamental constructions: quotients of topological groups via closed normal subgroups and quotients of metric spaces via actions by isometries. Basic results about inverse limits of topological groups are extended to inverse limits of group actions on uniform spaces, and notions of prodiscrete action and generalized covering map are introduced.