Let $(X, \omega)$ be a weakly pseudoconvex Kähler manifold, $Y \subset X$ a closed submanifold defined by some holomorphic section of a vector bundle over $X$, and $L$ a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer $k \geq 0$, any section of the jet sheaf $L \otimes {\cal O}_{X}/{\cal I}_{Y}^{k+1}$, which satisfies a certain $L^{2}$ condition, can be extended into a global holomorphic section of $L$ over $X$ whose $L^{2}$ growth on an arbitrary compact subset of $X$ is under control. In particular, if $Y$ is merely a point, this gives the existence of a global holomorphic function with an $L^{2}$ norm under control and with prescribed values for all its derivatives up to order $k$ at that point. This result generalizes the $L^{2}$ extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of sections of a line bundle. A technical difficulty is to achieve uniformity in the constant appearing in the final estimate. To this end, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds.

The main result is a general vanishing theorem for the Dolbeault cohomology of an ample vector bundle obtained as a tensor product of exterior powers of some vector bundles. It is also shown that the conditions for the vanishing given by this theorem are optimal for some parameter values.

We study geometry of manifolds endowed with a Grassmann structure which depends on symmetries of their curvature. Due to this reason a complete decomposition of the space of curvature tensors over tensor product vector spaces into simple modules under the action of the group $G = GL(p, \R) \otimes GL(q, \R)$ is given. The dimensions of the simple submodules, the highest weights and some projections are determined. New torsion-free connections on Grassmann manifolds apart from previously known examples are given. We use algebraic results to reveal obstructions to the existence of corresponding connections compatible with some type of normalizations and to enlighten previously known results from another point of view.

Let $H({\bf D}_{n})$ be the space of holomorphic functions on the unit polydisk ${\bf D}_{n}$, and let ${\cal L}^{p, q}_{\alpha}({\bf D}_{n})$, where $p, q > 0$, $\alpha = (\alpha_{1}, \dots, \alpha_{n})$ with $\alpha_{j} > -1$, $j = 1, \dots, n$, be the class of all measurable functions $f$ defined on ${\bf D}_{n}$ such that

$\int_{[0, 1)^{n}} M^{q}_{p}(f, r) \prod_{j=1}^{n} (1-r_{j})^{\alpha_{j}} dr_{j} < \infty$,

where $M_{p}(f, r)$ denote the $p$-integral means of the function $f$. Denote the weighted Bergman space on ${\bf D}_{n}$ by ${\cal A}^{p, q}_{\alpha}({\bf D}_{n}) = {\cal L}^{p, q}_{\alpha}({\bf D}_{n}) \cap H({\bf D}_{n})$. We provide a characterization for a function $f$ being in ${\cal A}^{p, q}_{\alpha}({\bf D}_{n})$. Using the characterization we prove the following result: Let $p > 1$, then the Cesàro operator is bounded on the space ${\cal A}^{p, p}_{\alpha}({\bf D}_{n})$.

The first aim in this article is to give some sufficient conditions for a family of meromorphic mappings of a domain $D$ in $\mathbf{C}^{n}$ into $\mathbf{P}^{N}(\mathbf{C})$ omitting hypersurfaces to be meromorphically normal. Our result is a generalization of the results of Fujimoto and Tu. The second aim is to investigate extending holomorphic mappings into the compact complex space from the viewpoint of the theory of meromorphically normal families of meromorphic mappings.

In this paper, we give explicit combinatorial descriptions for toric extremal contractions under the relative setting, where varieties are not complete. It is well-known that the complete case is settled by using Reid's wall theory which can not be applied to the non-complete case. Therefore, we can achieve them by using the notion of extremal primitive relations. As applications, we can generalize some of Mustaţă's results related to Fujita's conjecture on toric varieties for the relative case.

Some results for the Bergman functions in unbounded domains are shown. In particular, a class of unbounded Hartogs domains, which are Bergman complete and Bergman exhaustive, is given.

We study the problem whether a given $2$-dimensional symmetric stable distribution with index $\alpha$ $(0 < \alpha \leq 1)$ is determined by its $1$-dimensional projections in some specified directions. We give some conditions for the affirmative answer and for the negative answer.