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.

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.

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$--forms. We show that the spectral function of $\Box^{(q)}_b$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_b$ in $]0,\infty[$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szeg\H{o} kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szeg\H{o} kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded into $\mathbb C^N$, for some $N\in\mathbb N$.

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

It is proved that if the twistor space of a self-dual four-manifold of positive scalar curvature contains a real effective divisor of degree two, then the four-manifold is diffeomorphic to the connected sum n\mathbb {C}{P^ 2} of n complex projective planes for some n. It follows that if the four-manifold is known to be homeomorphic to n\mathbb {C}{P^ 2} , then it is also diffeomorphic to n\mathbb {C}{P^ 2} .