Formality of Donaldson submanifolds

We introduce the concept of s-formal minimal model as an extension of formality. We prove that any orientable compact manifold M, of dimension 2n or (2n-1), is formal if and only if M is (n-1)-formal. The formality and the hard Lefschetz property are studied for the symplectic manifolds constructed by Donaldson with asymptotically holomorphic techniques. This study permits us to show an example of a Donaldson symplectic manifold of dimension eight which is formal simply connected and does not satisfy the hard Lefschetz theorem.Comment: 24 pages, no figures, Latex2e; v3. statement of Lemma 2.7 correcte

An 8-dimensional non-formal simply connected symplectic manifold

A non-formal simply connected compact symplectic manifold of dimension 8 is constructed.Comment: 8 pages, 1 figure; v2. exposition greatly improved; v3. final version. To appear in Annals of Mathematic

Nilmanifolds with a calibrated G_2-structure

We introduce obstructions to the existence of a calibrated G_2-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g, then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G_2-structure.Comment: 21 pages; v2: added some introductory details on G_2 structures in Section 2, exposition improved. To appear in Differential Geometry and its Application