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- 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. symplectomor-phisms with fixed space of codimension 2 in V. Symplectic resolutions are always semismall. A crepant resolution of V /G is always… (More)

- 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 canoni-cal 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˜M , 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 oñ M. We prove a structure theorem for compact Vaisman manifolds. Every compact Vaisman… (More)

A nilmanifold is a quotient of a nilpotent group G by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a G-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of G-invariant complex structures which… (More)

- Misha Verbitsky
- 2003

We construct examples of compact hyperkähler manifolds with torsion (HKT manifolds) which are not homogeneous and not locally conformal hyperkähler. Consider a total space T of a tangent bundle over a hyperkähler manifold M. The manifold T is hypercomplex, but it is never hy-perkähler, unless M is flat. We show that T admits an HKT-structure. We also prove… (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)

- 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.

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 hy-percomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics, and prove a quaternionic analogue of… (More)

- Liviu Ornea, Misha Verbitsky
- 2003

A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering˜M , with monodromy acting oñ M by Kähler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties oñ M. We prove that any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold (C n… (More)

- Misha Verbitsky
- 2005

Let M be a compact Riemannian manifold equipped with a parallel differential form ω. We prove a version of Kähler identities in this setting. This is used to show that the de Rham algebra of M is weakly equivalent to its subquo-tient (H * c (M), d), called the pseudocohomology of M. When M is compact and Kähler and ω is its Kähler form, (H * c (M), d) is… (More)