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The purpose of this paper is to study holomorphic approximation and approximation of $$\overline{\partial }$$ -closed forms in complex manifolds of complex dimension $$n\ge 1$$ . We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the $${{\mathcal {C}}}^\infty $$ -smooth and the $$L^2$$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.

Let V be a compact and irreducible complex space of complex dimension v whose regular part is endowed with a complete Hermitian metric h. Let $$\pi :M\rightarrow V$$ be a resolution of V. Under suitable assumptions on h we prove that $$\begin{aligned} H^{v,q}_{2,\overline{\partial }}( \mathrm {reg} (V),h)\cong H^{v,q}_{\overline{\partial }}(M),\quad q=0,\ldots ,v. \end{aligned}$$ Then we show that the previous isomorphism applies to the case of Saper-type Kahler metrics, as introduced by Grant Melles and Milman, and to the case of complete Kahler metrics with finite volume and pinched negative sectional curvatures.

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2n-1$ in $\mathbb{C}^{N}$. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $n\ge 3$ and $N=n+1$, Yau found a necessary and sufficient condition for the interior regularity of the Harvey-Lawson solution to the complex Plateau problem by means of Kohn--Rossi cohomology groups on $X$ in 1981. For $n=2$ and $N\ge n+1$, the first and third authors introduced a new CR invariant $g^{(1,1)}(X)$ of $X$. The vanishing of this invariant will give the interior regularity of the Harvey-Lawson solution up to normalization. For $n\ge 3$ and $N>n+1$, the problem still remains open. In this paper, we generalize the invariant $g^{(1,1)}(X)$ to higher dimension as $g^{(\Lambda^n 1)}(X)$ and show that if $g^{(\Lambda^n 1)}(X)=0$, then the interior has at most finite number of rational singularities. In particular, if $X$ is Calabi--Yau of real dimension $5$, then the vanishing of this invariant is equivalent to give the interior regularity up to normalization., Comment: arXiv admin note: substantial text overlap with arXiv:1203.1380

We study the almost complex manifold and its symmetry algebra, the set of germs of infinitesimal automorphisms. The main result is that there is a certain condition to a torsion bundle under which the dimension of the symmetry algebra can not exceed 4 in the case of complex dimension 2. We will give an example which attains the maximal dimension 4, and another example without non-degeneracy whose symmetry algebra is of infinite dimension.

Let D be a J-pseudoconvex region in a smooth almost complex manifold (M, J) of real dimension four. We construct a local peak J-plurisubharmonic function at every point p ∈ bD of finite D’Angelo type. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashi hyperbolicity of D at p. In case the point p is of D’Angelo type less than or equal to four, or the approach is nontangential, we provide sharp estimates of the Kobayashi pseudometric.

Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as Kahler and hyperbolic geometries are concerned. In the second part of the paper, we give algebraic and topological obstructions to the existence of a geometrically 2-formal Kahler metric, at the level of the second cohomology group. A strong interaction with almost Kahler geometry is to be noted. In complex dimension 3, we list all the possible values of the second Betti number of a geometrically 2-formal Kahler metric.

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds (see Theorem 1.1). The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinitesimal pseudometric and on the scaling method in almost complex manifolds.

We show that the closed stability index of an excellent henselian local ring of real dimension d≥2 with real closed residue field is Open image in new window When d=2 it is shown that the value of Open image in new window can be either 2 or 3 and give characterizations of each of these values in terms of the relation of A with its normalization and in terms of the real spectrum of A.

Here we consider complex manifolds M which are holomorphically separable, i.e., the global holomorphic functions (9 (M) separate the points of M. We investigate T n (n-torus) actions on M, where T"={(ei~ ei~ 0eN"}, and n is the complex dimension of M. We assume that the action is effective, i.e., if 0ET ~ is nonzero, then there exists z e M with O.z+z. Our final assumption on the action is one of smoothness; the map T n x M ~ M given by (0, z ) , O . z is C 1 in both variables and holomorphic in z. The most obvious examples of such actions are given by Reinhardt domains in ~", with the usual action O.z=(ei~ . . . . , e i ~ The standard Reinhardt action may also be changed as follows: if A is an algebraic automorphism of T n, then O.az=(AO).z. An obvious question that arises is whether all T"-actions can arise in this way. Our main results are contained in the following four theorems.