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- Misha Verbitsky
- 2001

Let M be a hypercomplex Hermitian manifold, (M, I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form Ω on (M, I). It is well known that M is hyperkähler if and only if the form Ω is closed. The M is called… (More)

- Liviu Ornea, Misha Verbitsky
- 2003

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)

- Misha Verbitsky
- 1999

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)

- Misha Verbitsky
- 2005

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)

- Misha Verbitsky
- 2002

Let M be a K3 surface or an even-dimensional compact torus. We show that the category of coherent sheaves on M is independent from the choice of the complex structure, if this complex structure is generic.

- Misha Verbitsky
- 2004

Let (M, I) be a compact Kähler manifold admitting a hypercomplex structure (M, I, J,K). We show that (M, I, J,K) admits a natural HKT-metric. This is used to construct a holomorphic symplectic form on (M, I).

- Liviu Ornea, Misha Verbitsky
- 2006

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)

- Misha Verbitsky
- 1995

We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such man-ifolda are also known as complex manifolds of hyperkk ahler type. We obtain that a complex manifold of hyperkk ahler type is Mirror dual to itself. The Mirror Conjecture is stated (following Kontsevich, ICM talk) as the equivalence of certain… (More)

- D . Kaledin, Misha Verbitsky
- 2008

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(M). For a holomorphic symplectic manifold M which satisfies H1(OM ) = H 2(OM ) =… (More)