### ${\u2102}^{*}$-actions on ${\u2102}^{3}$ are linearizable.

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In this paper we consider the map L defined on the Bergman space [...] of the right half plane [...] .

For each 2-dimensional complex torus $T$, we construct a compact complex manifold $X\left(T\right)$ with a ${\u2102}^{2}$-action, which compactifies ${\left({\u2102}^{*}\right)}^{4}$ such that the quotient of ${\left({\u2102}^{*}\right)}^{4}$ by the ${\u2102}^{2}$-action is biholomorphic to $T$. For a general $T$, we show that $X\left(T\right)$ has no non-constant meromorphic functions.

We describe the integral cohomology rings of the flag manifolds of types Bₙ, Dₙ, G₂ and F₄ in terms of their Schubert classes. The main tool is the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex algebraic groups, recovering thereby the results of R. Marlin.

In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...

Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection...

The group SU(1,d) acts naturally on the Hilbert space $L\xb2\left(Bd{\mu}_{\alpha}\right)(\alpha >-1)$, where B is the unit ball of ${\u2102}^{d}$ and $d{\mu}_{\alpha}$ the weighted measure ${(1-|z\left|\xb2\right)}^{\alpha}dm\left(z\right)$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...