Let X be a projective manifold, ρ : X̃ → X its universal covering and ρ∗ : V ect(X) → V ect(X̃) the pullback map for the isomorphism classes of vector bundles. This article establishes a connection between the properties of the pullback map ρ∗ and the properties of the function theory on X̃. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holo...

We prove some closed mapping theorems on k-spaces with point-countable k-networks. One of them generalizes Lašnev’s theorem. We also construct an example of a Hausdorff space Ur with a countable base that admits a closed map onto metric space which is not compact-covering. Another our result says that a k-space X with a point-countable k-network admitting a closed surjection which is not compac...

In this paper we study when a bipartite graph is a covering of a non-bipartite graph. We give a characterization of all coverings in terms of factoring the covering map through the canonical double covering. We also consider regular bipartite coverings described in terms of voltage assignments. We give an algebraic characterization of such coverings involving the subgroup generated by voltages ...

Any covering Y ! X of a hyperbolic Riemann surface X of-nite area determines an inclusion of Teichm uller spaces Teich(X) ,! Teich(Y). We show this map is an isometry for the Teichm uller metric if the covering is amenable, and contracting otherwise. In particular , we establish jjjj < 1 for classical Poincar e series (Kra's `Theta conjecture'). The appendix develops the theory of geometric lim...

Let T : X → Y be a bounded linear map between Banach spaces X and Y . Let T ∗ : Y ∗ → X∗ be its adjoint. Let BX and BY ∗ be the closed unit balls of X and Y ∗ respectively. We obtain apparently new estimates for the covering numbers of the set T ∗ (BY ∗ ). These are expressed in terms of the covering numbers of T (BX), or, more generally, in terms of the covering numbers of a “significant” subs...

Every continuous map X → S defines, by composition, a homomorphism between the corresponding algebras of real-valued continuous functions C(S) → C(X). This paper deals with algebraic properties of the homomorphism C(S)→ C(X) in relation to topological properties of the map X → S. The main result of the paper states that a continuous map X → S between topological manifolds is a finite (branched)...

One of the cornerstones of complex geometry is the link between positivity of curvature and ampleness. Let X be a compact complex manifold and L ! X be a holomorphic line bundle over X. Suppose that L has a unitary connection whose curvature form is ?2i! where ! is a positive (1; 1)-form on X. Then for large k the line bundle L k has many holomorphic sections. More precisely, the holo-morphic s...

We obtain generalizations of the Fan’s matching theorem for an open (or closed) covering related to an admissible map. Each of these is restated as a KKM theorem. Finally, applications concerning coincidence theorems and section results are given.