### A Theorem of Versality for Unfoldings of Complex Analytic Foliation Singularities.

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A normal form for small CR-deformations of the standard CR-structure on the (2n+1)-sphere is presented. The space of normal forms is parameterized by a single function on the sphere. For n>1, the normal form is used to obtain explicit embeddings into ${\u2102}^{n+1}$. For n=1, the cohomological obstruction to embeddability is identified.

We show the uniqueness of local and global decompositions of abstract CR-manifolds into direct products of irreducible factors, and a splitting property for their CR-diffeomorphisms into direct products with respect to these decompositions. The assumptions on the manifolds are finite non-degeneracy and finite-type on a dense subset. In the real-analytic case, these are the standard assumptions that appear in many other questions. In the smooth case, the assumptions cannot be weakened by replacing...

We study compact Kähler manifolds $X$ admitting nonvanishing holomorphic vector fields, extending the classical birational classification of projective varieties with tangent vector fields to a classification modulo deformation in the Kähler case, and biholomorphic in the projective case. We introduce and analyze a new class of $\mathrm{\mathit{t}\mathit{a}\mathit{n}\mathit{g}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{a}\mathit{l}\mathit{d}\mathit{e}\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{s}}$, and show that they form a smooth subspace in the Kuranishi space of deformations of the complex structure of $X$. We extend Calabi’s theorem on the structure of compact Kähler...

Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in ${\u2102}^{p}$, for some $p\>0$) or differentiable (parametrized by an open neighborhood of $0$ in ${\mathbb{R}}^{p}$, for some $p\>0$) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point $t$ of the parameter space, the fiber over $t$ of the first family is biholomorphic to the fiber over $t$ of the second family. Then, under which conditions are the...