| For an algebraic surface X , Schwarzanberger [Sw61] has shown that every topological vector bundle whose first Chern class lies in the Neron-Severi group of X, admits a holomorphic structure. In the non-algebraic case, the analogous condition c1 (E) NS(X) (which is always necessary) is no longer sufficient, as the works of Elencwajg-Forster [EF82] and Banica-Le Potier [BP87] showed. We recall that a holomorphic vector bundle of rank r over a complex manifold is irreducible if it admits no coherent analytic subsheaf of rank k with 0 < k < r. Hence, irreducibility implies non-filtrability, but non-filtrability does not necessarily mean irreducibility. However, if E is a rank- 2 bundle, then E is irreducible if and only if E is non-filtrable. Irreducible vector bundle do not exist over algebraic manifold , but there are plenty of them over compact complex non-algebraic surfaces.In [EF82], Elencwajg-Forster showed the existence of irreducible rank-2 vector bundles over 2-tori with trivial Neron-Severi group. This was done by comparing the versal deformation of a filtrable rank-2 vector bundle with the space parametrising extensions producing filtrable vector bundles. In this way they proved that in general the versal deformation is richer, hence it contains also irreducible vector bundles. They showed that on a 2-torus X with NS(X) = 0, There exist non-filtrable holomorpbic vector bundles E of rank 2 with c2(E) = 2.In [BP87], Banica-Le Potier, using the relative Douady space of quotients associated to the versal deformation of a filtrable vector bundle, showed the existence of irreducible vector bundle E in any rank over surfaces with algebraic dimension zero, for a large range of Chern classes c1(E), c2(E). They also showed the existence of irreducible rank 2 vector bundles over surfaces with algebraic dimension one and trivial canonical line bundle. As it was remarked by M. Toma (cf. [To92]), their proof can be extended to any surface X with a(X) = 1. And the Chern classes of all these examples of irreducible vector bundles are in the same range as for the filtrable vector bundles.In the non-algebraic case, the following are in general open problems, i.e. on a given topological vector bundle of rank 2 whether a holomorphic structure exists or not and in what circumstances a known holomorphic vector bundle is filtrable. Hence holomorphic vector bundles on non-algebraic surface are deserved to receive increasing attention . In the present paper, we obtained some remarks on holomorphic vector bundles on some non-algebraic compact complex surfaces X, that is, certain non-algebraic compact complex surfaces without divisors and certain non-algebraic compact complex surfaces with odd first betti number. Especially, we got a description of IS2(X, 0) of exceptional Hopf surface here.In the classification theory of compact complex manifold, the Kodaira dimension and the algebraic dimension play important role. It is well-known that for a compact complex surface X, X is algebraic iff its algebraic dimension a(X) = dim X = 2, and X is Kahlerian iff the first betti number of X is even. Furthermore, for a non-algebraic surface X, a(X) = 1 iff X is elliptic, and a(X) = 0 iff X is non-elliptic.After reading the paper of Elencwajg and Forster [EF82J. We can not help wondering what kind of surfaces having the property of having no divisors except for non-elliptic 2-tori. Later on , we came across the paper of Inoue[In74] and learned that Inoue's surfaces do not admit divisors. But we were a little unsatisfied.Lemma 1 Suppose X be a compact complex surface without divisors. Then for an arbitrary line bundle L Pic(X), we have2 Doctoral thesis: Vector Bundles on Certain Non-algebraic SurfacesIn particular,where m 6 Z, which implies that the Kodaira dimension of X must be either - or 0.Enriques ( cf.[Enr49]) classified algebraic surfaces. Kodaira (cf. [Ko68], part IV, p. 1062) extended the Enriques classification to the Enriques-Kodaira classification of all compact complex surfaces.Therefore, according to the E...
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