A new object is introduced - the "Fischer bundle". It is, formally speaking, an Hermitean bundle of infinite rank over a bounded symmetric domain whose fibers are Hilbert spaces whose elements can be realized as entire analytic functions square integrable with respect to a Gaussian measure ("Fischer spaces"). The definition was inspired by our previous work on the "Fock bundle". An even more general framework is indicated, which allows one to look upon the two concepts from a unified point of view....

We investigate the Bergman kernel function for the intersection of two complex ellipsoids $\{(z,w_1,w_2)\in {ℂ}^{n+2}:|z_1{|}^2+\cdots +|z_n{|}^2+|w_1{|}^q<1,|z_1{|}^2+\cdots +|z_n{|}^2+|w_2{|}^r<1\}.$ We also compute the kernel function for $\{(z_1,w_1,w_2)\in {ℂ}^3:|z_1{|}^{2/n}+|w_1{|}^q<1,|z_1{|}^{2/n}+|w_2{|}^r<1\}$ and show deflation type identity between these two domains. Moreover in the case that $q=r=2$ we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.

We compute the Bergman kernel functions of the unbounded domains ${\Omega }_p=(z^{\text{'}},z)\in ℂ²:z>p\left(z^{\text{'}}\right)$, where $p\left(z^{\text{'}}\right)={\left|z^{\text{'}}\right|}^{\alpha }/\alpha$. It is also shown that these kernel functions have no zeros in ${\Omega }_p$. We use a method from harmonic analysis to reduce the computation of the 2-dimensional case to the problem of finding the kernel function of a weighted space of entire functions in one complex variable.

In this note we compute the Bergman kernel of the unit ball with respect to the smallest norm in ${ℂ}^n$ that extends the euclidean norm in ${ℝ}^n$ and give some applications.

We establish $L^p$-estimates for the weighted Bergman projection on a nonsingular cone. We apply these results to the weighted Fock space with respect to the minimal norm in ℂⁿ.

We introduce the Bloch space for the minimal ball and we prove that this space can be identified with the dual of a certain analytic space which is strongly related to the Bergman theory on the minimal ball.

It is proved that if $M$ is a weakly 1-complete Kähler manifold with only one end, then $H_c^1(M,𝒪)=0$ or there exists a proper holomorphic mapping of $M$ onto a Riemann surface.

The Briançon-Skoda number of a ring $R$ is defined as the smallest integer k, such that for any ideal $I\subset R$ and $l\ge 1$, the integral closure of $I^{k+l-1}$ is contained in $I^l$. We compute the Briançon-Skoda number of the local ring of any analytic irreducible planar curve in terms of its Puiseux characteristics. It turns out that this number is closely related to the Milnor number.