We consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P. We assume that the general fiber is isomorphic to a product of two elliptic curves. We prove that such a hyper-Kähler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface. 1. Preliminaries First we define our main ob...

We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of harmonic one-forms. For example, this is the case for complete Kähler manifolds for which the symplectic form has an appropriate decay at infinity. This extends ...

We show that surface bundles over surfaces with base and fiber of genus at least 2 have non-vanishing simplicial volume. The simplicial volume ||M ||, introduced by Gromov [3], is a homotopy invariant which measures the complexity of the fundamental class of an oriented manifold M . It is determined by the classifying map of the universal covering, and tends to be non-zero for large manifolds o...

Let π : X → B be a holomorphic submersion between compact Kähler manifolds of any dimensions, whose fibres and base have no non-zero holomorphic vector fields and whose fibres admit constant scalar curvature Kähler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature Kähler metric on X. The condition involves the CM-line bundle—a certai...

We address the problem of computing the fundamental group of a symplectic S-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian S-action. Several examples are presented to illustrate our main results.

We address the problem of computing the fundamental group of a symplectic S1-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian S1-action. Several examples are presented to illustrate our main results.

Forward These notes are based on five 1.5 hour lectures on torus actions on contact mani-folds delivered at the summer school on Symplectic Geometry of Integrable Hamil-tonian Systems at Centre de Recerca Matemàtica in Barcelona in July 2001. Naturally the notes contain more material that could have been delivered in 7.5 hours. I am grateful to Carlos Curràs-Bosch and Eva Miranda, the organizer...

We give a brief survey of the main results of [C04] and [C11], devoted to the bimeromorphic structure of compact Kähler manifolds X. Such manifolds are decomposed by means of iterated fibrations into elementary components, which are orbifold pairs with a canonical bundle either positive, negative, or torsion. Towers of ‘torsion and negative’ components build however the new (unconditional) clas...

Extending the work of [7] on groups definable in compact complex manifolds and of [1] on strongly minimal groups definable in nonstandard compact complex manifolds, we classify all groups definable in nonstandard compact complex manifolds showing that if G is such a group then there are a linear algebraic group L, a definably compact group T , and definable exact sequence 1→ L → G → T → 1.

We extend the deenition of analytic and Reidemeister torsion from closed compact Riemannian manifolds to compact Riemannian manifolds with bound