A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. In the published version of "Mapping class group and a global Torelli theorem for hyperkähler manifolds" I made an error based on a wrong quotation of Dennis Sullivan's famous paper "Infinitesimal computations in topology". I claimed that the natural homomorphism from the mapping class group to the group of automorphims of cohomology of a simply connected Kähler manifold has finite kernel. In a recent preprint [KS], Matthias Kreck and Yang Su produced counterexamples to this statement. Here I correct this error and other related errors, observing that the results of "Mapping class group and a global Torelli theorem" remain true after an appropriate change of terminology.

A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold M with the deck transformation group acting conformally on M. If M admits a holomorphic flow, acting on M conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifold M with potential, dim M ≥ 3, can be embedded into a Hopf manifold, thus improving similar results for Vaisman manifolds Ornea and Verbitsky

A nilmanifold is a quotient of a nilpotent group G by a cocompact 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 satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkähler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any G-invariant HKT-metric on a nilmanifold is balanced with respect to all associated complex structures.

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. We prove a quaternionic analogue of A.D. Aleksandrov and Chern-Levine-Nirenberg theorems.

A quaternionic version of the Calabi problem on the Monge-Ampère equation is introduced, namely a quaternionic Monge-Ampère equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n, H), uniqueness (up to a constant) of a solution is proven, as well as the zero order a priori estimate. The existence of a solution is conjectured, similar to the Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of the complex Hessian equation, making sense on any complex manifold.

Let M be an irreducible holomorphically symplectic manifold. We show that all faces of the Kähler cone of M are hyperplanes Hi orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kähler cone is a connected component of a complement of the positive cone to the union of all Hi. We provide several characterizations of the MBM-classes. We show the invariance of MBM property by deformations, as long as the class in question stays of type (1, 1). For hyperkähler manifolds with Picard group generated by a negative class z, we prove that ±z is Q-effective if and only if it is an MBM class. We also prove some results towards the Morrison-Kawamata cone conjecture for hyperkähler manifolds.

Abstract. Let M be a hypercomplex Hermitian manifold, (M, /) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linearalgebraic construction produces a canonical nowhere degenerate (2 7 0 . Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics. We exploit a remarkable analogy between the de Rham DG-algebra of a Kahler manifold and the Dolbeault DG-algebra of an HKT-manifold. The supersymmetry of a Kahler manifold X is given by an action of an 8-dimensional Lie superalgebra g on A*(X), containing the Lefschetz £L(2)-triple, the Laplacian and the de Rham differential. We establish the action of # on the Dolbeault DG-algebra A*' 0 (M, /) of an HKT-manifold. This is used to construct a canonical Lefschetz-type 5L(2)-action on the space of harmonic spinors of M.

Abstract. A locally conformally Kähler (l.c.K.) manifold is a complex manifold admitting a Kähler coveringM , 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 onM . We prove a structure theorem for compact Vaisman manifolds. Every compact Vaisman manifold M is fibered over a circle, the fibers are Sasakian, the fibration is locally trivial, and M is reconstructed from the Sasakian structure on the fibers and the monodromy automorphism induced by this fibration. This construction is canonical and functorial in both directions.