International audienceWe propose new tools based on basic lattice theory to calculate the integral cohomology of the quotient of a manifold by an automorphism group of prime order. As examples of applications, we provide the Beauville--Bogomolov forms of some irreducible symplectic orbifolds; we also show a new expression for a basis of the integral cohomology of a Hilbert scheme of two points on a surface

The Beauville-Bogomolov lattice is computed for a simplest singular symplectic manifold of dimension 4, obtained as a partial desingularization of the quotient S [2] /ι, where S [2] is the Hilbert square of a K3 surface S and ι is a symplectic involution on it. This result applies, in particular, to the singular symplectic manifolds of dimension 4, constructed by Markushevich-Tikhomirov as compactifications of families of Prym varieties of a linear system of curves on a K3 surface with an anti-symplectic involution.

ABSTRACT. Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact 3−folds, called building blocks, satisfying a stability condition 'at infinity'. Such bundles are known to parametrise solutions of the Yang-Mills equation over the G2−manifolds obtained from asymptotically cylindrical Calabi-Yau 3−folds studied by Kovalev, Haskins et al. and Corti et al..The most important tool is a generalisation of Hoppe's stability criterion to holomorphic bundles over smooth projective varieties X with Pic X ≃ Z l , a result which may be of independent interest.Finally, we apply monads to produce a prototypical model of the curvature blow-up phenomenon along a sequence of asymptotically stable bundles degenerating into a torsion-free sheaf.

We propose a generalization of Verbitsky's global Torelli theorem in the framework of compact Kähler irreducible holomorphically symplectic orbifolds by adapting Huybrechts' proof [12]. As intermediate step needed, we also provide a generalization of the twistor space and the projectivity criterion based on works of Campana [6] and Huybrechts [17] respectively. Reminders and complements2.1 Definition of orbifolds Definition 2.1. A n-dimentional orbifold is a connected paracompact complex space X such that for every point x ∈ X, there exists an open neighborhood U and a triple (V, G, π) such that V is

We describe the integral cohomology of the generalized Kummer fourfold giving an explicit basis, using Hilbert scheme cohomology and tools developed by Hassett and Tschinkel. Then we apply our results to a IHS variety with singularities, obtained by a partial resolution of the generalized Kummer fourfold quotiented by a symplectic involution. We calculate the Beauville-Bogomolov form of this new variety, presenting the first example of such a form that is odd.