Deformations of Banach Coverings of Complex Manifolds

Abstract

Introduction. In this paper we investigate branched coverings of compact complex manifolds and, in particular, we answer certain questions connected with the deformation theory of branched coverings. Let M and W be compact complex manifolds, 311 a branched k-sheeted covering of W. The Main Theorem of ? 4 gives sufficient conditions on M and W (expressed as cohomological conditions on W) for every small deformation of M to be a I-sheeted covering of W. Our methods apply to branched coverings of the following type: M is a compact submanifold of a projective line bundle, F#, over the compact manifold W; the map F# -W makes M a branched covering of W. We show in ? 1 that cyclic coverings of W (and hence two-sheeted coverings) are of this type. Theorem 4. 1 gives a sufficient condition (involving cohomology groups on M and FP) that every small deformation of M is a branched covering of W. In ? 2 the spectral sequence for a fibre space is applied to express cohomology groups on F# in terms of cohomology groups on W. These results are used in ? 4 to reduce the condition of Theorem 4. 1 to a condition involving cohomology groups on W. If W= Pn, we obtain very explicit results (? 5 and ? 6) since strong results are available on cohomology groups on pn. In ? 7 we compute the number of moduli for cyclic coverings of P*, (n ?>2). Finally, in ? 6, we give some examples of branched coverings of P2 having small deformations which are not branched coverings of p2. Essential use is made of Theorem 1. 1 in which we show that a branched covering of a projective algebraic manifold is itself projective algebraic. We refer to the papers of Kodaira and Spencer (in particular [14] ) for those matters concerning deformatiois of complex structures. lirzebruch's book [10] will serve as a general introduction to bundles, Chern classes, duality, etc. By vector bundle we will always mean a fibre whose fibre is Cq and whose structure group is GL (q, C). Line bundle will mean a with q = 1. If B is a on a manifold, (B)