ON SURFACES OF CLASS VII WITH NUMERICALLY ANTICANONICAL DIVISOR

Georges Dloussky (1)

1 I2M - Institut de Mathématiques de Marseille

We consider minimal compact complex surfaces S with Betti numbers b\ = 1 and n = bi > 0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m > 1 and a flat line bundle F such that H®(S,-mK 0; these surfaces admit no nonconstant mero-morphic functions. The major problem in classification of non-kahlerian surfaces is to achieve the classification of surfaces S of class VIIJ. All known surfaces of this class contain Global Spherical Shells (GSS), i.e., admit a biholomorphic map ip\ U-> V from a neighbourhood U C C2\ {0} of the sphere 53 = dB2 onto an open set V such that I = (f(S3) does not disconnect S. Are there other surfaces ? In first section we investigate the general situation: A theorem of Donaldson [13] gives a Z-base (£,) of //2(S,Z), such that £/£, =-<%. These cohomology classes can be represented by line bundles L, such that K$Li = L2 =-1. Indeed, these line bundles generalize exceptional curves of the first kind, and since 5 is minimal, they have no sections. Over the versal deformation S-> B of S these line bundles form families £/. We propose the following conjecture which can be easily checked for surfaces with GSS:

Domaines: Variables complexes [math.CV]

Georges Dloussky. ON SURFACES OF CLASS VII WITH NUMERICALLY ANTICANONICAL DIVISOR. American Journal of Mathematics, 2006. ⟨hal-01255762⟩