Misha Verbitsky

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A locally conformally Kähler (l.c.K.) manifold is a complex manifold admitting a Kähler covering M̃ , with each deck transformation acting by Kähler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on M̃ . We prove a structure theorem for compact Vaisman manifolds. Every compact Vaisman(More)
Let G be a finite group acting on a symplectic complex vector space V . Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by “symplectic reflections”, i.e. symplectomorphisms with fixed space of codimension 2 in V . Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic.(More)
A hypercomplex manifold is a manifold equipped with a triple of complex structures I, J,K satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics, and prove a quaternionic analogue of(More)
Let (M, I) be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (socalled Nijenhuis tensor) N : Λ0,1(M)−→ Λ(M) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold (M, I) is called nearly Kähler if it admits a Hermitian(More)
It is well known that a Sasakian manifold is equipped with a strictly pseudoconvex CR-structure. Let M be a strictly pseudoconvex CR-manifold. We show that M admits a Sasakian metric g if and only if M admits a proper, transversal CR-holomorphic action by S, or, equivalently, an S-invariant CR-holomorphic embedding to an algebraic cone. This embedding is(More)