On a Compact Complex Manifold In $\mathcal C$ without Holomorphic 2-Forms

Abstract

We shall obtain the following partial result (Proposition 2): For X as above let /: X*-+Y be (a holomorphic model of) an algebraic reduction of X. Then fl(/)=£(/)=0 where a(f) (resp. &(/)) is the algebraic (resp. Kummer) dimension of / (cf. [6]). In particular the irregularity q(X%) of any smooth fiber X% vanishes. Moreover X* is not bimeromorphic to a K3 surface. The result is used in [6]. The arrangement of this note is as follows. We gather some preliminary material in Section 1. Also the relation of our problem with some fundameutal problems on the theory of compact Kahler manifolds and manifolds in C will be explained. In Section 2 we prove that a smooth fiber space of complex tori always admits a Kahler polarization, provided that it can be compactified to a morphism of compact complex manifolds in C. Finally in Section 3 we shall show Proposition 2 mentioned above using the results obtained in Section 2 and in [6] [71. In this note complex manifolds are assumed to be paracompact and connected. For a surjective morphism h: X-+Y of complex manifolds we shall write dim h=dim X— dim Y. A fiber space is a proper surjective morphism with connected fibers.