This is the special issue dedicated to Professor Yum-Tong Siu on the occasion of his 60th birthday.

The conformal space \frac M was introduced by Dirac in 1936. It is an algebraic manifold with a spin structure and possesses naturally an invariant Lorentz metric. By carefully studying the birational transformations of \frac M, we obtain explicitly the transition functions of the spin bundle over \frac M. Since the transition functions are closely related to the propagation in physics, we get a kind of solutions of the Dirac equation by integrals constructed from the propagation. Moreover, we prove that the invariant Lorentz metric together with one of such solutions satisfies the Einstein-Dirac combine equation

Let X be a complex projective n-dimensional manifold of general type whose canonical system is composite with a pencil. If the Albanese map is generically finite, but not surjective, or if the irregularity is strictly larger than n and the image of X in Alb(X) is of Kodaira dimension one, then the geometric genus p_g(F) of a general fibre F of the canonical map is one and the latter factors through the Albanese map. The last part of this result holds true for any threefold with q(X) ≥ 5.

In this paper we describe the Einstein-Kähler metric for the Cartan-Hartogs domains which are the special case of the Hua domains. First of all, we reduce the Monge-Ampére equation for the metric to an ordinary differential equation in the auxiliary function X = X(z,w). This differential equation can be solved to give an implicit function in X. Secondly, for some cases, we obtained explicit forms of the complete Einstein-Kähler metrics on Cartan-Hartogs domains which are the non-homogeneous domains.

For a uniruled projective manifold, we prove that a general rational curve of minimal degree through a general point is uniquely determined by its tangent vector. As applications, among other things we give a new proof, using no Lie theory, of our earlier result that a holomorphic map from a rational homogeneous space of Picard number 1 onto a projective manifold different from the projective space must be a biholomorphic map.

For a large class of Banach spaces X we prove the following. If Ω ⊂ X is open and pseudoconvex, and E → Ω is a locally trivial holomorphic Banach bundle, then the sheaf cohomology groups H^q(Ω, E) vanish for q ≥ 1. We also give an application concerning neighborhoods of complex submanifolds.

Let D be a bounded domain in Cⁿ, and let u ∈ C²(D) be a real valued-function.

The main purpose of this paper is to provide some criteria for the integrality and arithmeticity of a cocompact lattice corresponding to a complex ball of dimension two.

To determine cohomology groups of holomorphic vector bundles or more general coherent analytic sheaves on complex manifolds is very important in several complex variables and complex geometry. For example, Cartan-Serre's theorem B and Kodaira's vanishing theorem are fundamental respectively in the studies of two classes of complex manifolds: Stein manifolds and projective algebraic manifolds. Deformation and moduli space of complex structures are also closely related to the determination of the cohomology groups of the bundles according to Kodaira-Spencer's theory

We prove geodesic completeness and global conectivity for a step 2k + 2 subRiemannian manifold. Using complex Hamiltonian mechanics we also calculate some subRiemannian distances.

We prove a conjecture from [BK2] that the multi-dimensional vector addition formula for Baker-Akhiezer functions obtained there characterizes Jacobians among principally polarized abelian varieties. We also show that this addition formula is equivalent to Gunning's multisecant formula for the Kummer variety obtained in [Gu2].

We then use this addition formula to obtain cubic relations among theta functions that characterize the locus of hyperelliptic Jacobians among irreducible abelian varieties. In genus 3 our equations are equivalent to the vanishing of one theta-null, and thus are classical (see [M], [P]), but already for genus 4 they appear to be new.

The main purpose of this article is to increase the efficiency of the tools introduced in [B.Mg. 98] and [B.Mg. 99], namely integration of meromorphic cohomology classes, and to generalize the results of [B.Mg. 99]. They describe how positivity conditions on the normal bundle of a compact complex submanifold Y of codimension n + 1 in a complex manifold Z can be transformed into positivity conditions for a Cartier divisor in a space parametrizing n-cycles in Z .

As an application of our results we prove that the following problem has a positive answer in many cases :

Let Z be a compact connected complex manifold of dimension n+p. Let Y ⊂ Z a submanifold of Z of dimension p-1 whose normal bundle N _Y|Z is (Griffiths) positive. We assume that there exists a covering analytic family (X _s ) _s∈S of compact n-cycles in Z parametrized by a compact normal complex space S.

Is the algebraic dimension of Z ≥ p ?