is biholomorphic equivalent to the Satake compactification of H2/Γ′ for a certain subgroup Γ′ ⊂ Sp(2, Z). This variety has 96 singularities which correspond to certain zero-dimensional cusps and these singularities are ordinary double points (nodes). In the paper [CM] it has been pointed out that the results of [GN] imply that a (projective) small resolution of this variety is a rigid Calabi-Ya...

We show that a twistor space of a self-dual metric on 4CP with U(1)-isometry is not Moishezon iff there is a C∗-orbit biholomorphic to a smooth elliptic curve, where the C∗-action is the complexification of the U(1)action on the twistor space. It follows that the U(1)-isometry has a two-sphere whose isotropy group is Z2. We also prove the existence of such twistor spaces in a strong form to sho...

We investigate the vertical foliation of the standard complex contact structure on Γ \ Sl(2,C), where Γ is a discrete subgroup. We find that, if Γ is nonelementary, the vertical leaves on Γ \Sl(2,C) are holomorphic but not regular. However, if Γ is Kleinian, then Γ \ Sl(2,C) contains an open, dense set on which the vertical leaves are regular, complete and biholomorphic to C∗. If Γ is a uniform...

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in C are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between pseudoconvex domains. The proofs are ...

The Chern-Moser normal form and its analog on finite type hypersurfaces in general do not respect symmetries. Extending the work of N. K. Stanton, we consider the local equivalence problem for symmetric Levi degenerate hypersurfaces of finite type in C. The results give for all such hypersurfaces a complete normalization which respects the symmetries. In particular, they apply to tubes and rigi...

In order to get estimates on the solutions of the equation ∂̄u = ω on Stein manifold, we introduce a new method the ”raising steps method”, to get global results from local ones. In particular it allows us to transfer results form open sets in C to open sets in a Stein manifold. Using it we get L − L results for solutions of equation ∂̄u = ω with a gain, s > r, in strictly pseudo convex domains i...

Let X $X$ be a compact complex space in Fujiki's Class C $\mathcal {C}$ . We show that the group Aut ( ) $\operatorname{Aut}(X)$ of all biholomorphic automorphisms has Jordan property: there is (Jordan) constant J = $J J(X)$ such any finite subgroup G ⩽ $G\leqslant \operatorname{Aut}(X)$ an abelian H $H\leqslant G$ with index [ : ] $[G:H]\leqslant J$ This extends, quite different method, result...