arXiv subjects

M. Verbitsky

Publications and source records attributed to M. Verbitsky.

4 recordsLinked to original sources

arXiv2003

Immersion theorem for Vaisman manifolds

A locally conformally Kaehler (LCK) manifold is a complex manifold admitting a Kaehler covering M, with monodromy acting on M by Kaehler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on M. We prove a non-Kaehler analogue of Kodaira embedding theorem: any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold. As an application, we obtain that any Sasakian manifold has a contact immersion to an odd-dimensional sphere.

math.AG↗

arXiv2000

Period map for non-compact holomorphically symplectic manifolds

We study the deformations of a holomorphic symplectic manifold $M$, not necessarily compact, over a formal ring. We show (under some additional, but mild, assumptions on $M$) that the coarse deformation space exists and is smooth, finite-dimensional and naturally embedded into $H^2(M)$. For a holomorphic symplectic manifold $M$ which satisfies $H^1(M,{\cal O}_M) = H^2(M,{\cal O}_M)=0$, the coarse moduli of formal deformations is isomorphic to $\Spec\C[[t_1, ..., t_n]]$, where $t_1$, ... $t_n$ are coordinates in $H^2(M)$. This revised version contains one minor improvement: exposition in Subsection 5.1 has been made more detailed and rigourous.

math.AG↗

arXiv1998

Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties

We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space $V$ by an action of a finite group $G$ of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if the action of $G$ is generated by complex reflections. This is used to study the subvarieties of a Hilbert scheme of a complex torus. We show that any subvariety of a generic deformation of a Hilbert scheme of a torus is birational to a quotient of another torus by an action of a Weyl group of some semisimple Lie algebra. In Appendix, we produce counterexamples to a false theorem stated in our preprint math.AG/9801038.

math.AG↗

arXiv1998

Trianalytic subvarieties of generalized Kummer varieties

Let $X$ be a hyperkaehler manifold. Trianalytic subvarieties of $X$ are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus $T$, the Hilbert scheme $T^{[n]}$ classifying zero-dimensional subschemes of $T$ admits a hyperkaehler structure. A finite cover of $T^{[n]}$ is a product of $T$ and a simply connected hyperkaehler manifold $K^{[n-1]}$, called generalized Kummer variety. We show that for $T$ generic, the corresponding generalized Kummer variety has no trianalytic subvarieties. This implies that a generic deformation of the generalized Kummer variety has no proper complex subvarieties.

math.AG↗