THEOREM 1. Let M be a compact two dimensional complex manifold with zero Euler number. Suppose there is a basis {a,, aZ, a3, a3 of the first real cohomology group H(M, R) such that the cup product a, U u2 U (Ye U LY~ is not zero. Then either M is biholomorphic to the complex torus or M is covered by the euclidean space.

In this paper we prove new embedding results for compactly supported deformations of CR submanifolds of Cn+d: We show that if M is a 2-pseudoconcave CR submanifold of type (n,d) in Cn+d, then any compactly supported CR deformation stays in the space of globally CR embeddable in Cn+d manifolds. This improves an earlier result, where M was assumed to be a quadratic 2-pseudoconcave CR submanifold of Cn+d. We also give examples of weakly 2-pseudoconcave CR manifolds admitting compactly supported CR deformations that are not even locally CR embeddable.

We show that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.

Let X1 and X2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann (CR) manifolds of dimensions 2m − 1 and 2n − 1 in Cm+1 and Cn+1, respectively. We introduce the Thom- Sebastiani sum X = X1⊕X2 which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in Cm+n+2. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1 for all n > 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X1 ⊕X2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly, we show that if X = X1 ⊕ X2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X1 and X2 provided that X2 admits a transversal holomorphic S1-action.

In this paper, we develop an approach to the study of compact K~ hler manifolds which admit mappings of everywhere maximal rank into quotients of polydiscs, eg into Riemann surfaces or products of them. One main tool will be a detailed study of the harmonic maps in the corresponding homotopy classes (for definition and general properties of harmonic maps between Riemannian manifolds see [3]). Starting with a result of Siu, we prove in Sect. 2 that the local level sets of the components of these mappings are analytic subvarieties of the domain. This, together with a generalization of the similarity principle of Bers and Vekua which is proved in the appendix and a residue argument, enables us to give conditions involving the Chern and K~ ihler classes of the considered manifolds, under which this harmonic map is of maximal rank everywhere and, in case domain and image have the same dimension, in