Four-fold Massey products in Galois cohomology

In this paper, we develop a new necessary and sufficient condition for the vanishing of 4-Massey products of elements in the mod-2 Galois cohomology of a field. This new description allows us to define a splitting variety for 4-Massey products, which is shown in the Appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such 4-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.Comment: Final version: several corrections made throughout the paper; some sections reorganized; will appear in Compositio Mathematic

Leibniz algebroids, twistings and exceptional generalized geometry

We investigate a class of Leibniz algebroids which are invariant under diffeomorphisms and symmetries involving collections of closed forms. Under appropriate assumptions we arrive at a classification which in particular gives a construction starting from graded Lie algebras. In this case the Leibniz bracket is a derived bracket and there are higher derived brackets resulting in an $L_\infty$-structure. The algebroids can be twisted by a non-abelian cohomology class and we prove that the twisting class is described by a Maurer-Cartan equation. For compact manifolds we construct a Kuranishi moduli space of this equation which is shown to be affine algebraic. We explain how these results are related to exceptional generalized geometry.Comment: 58 page

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.Comment: 21 pages, no figure

Triple Massey products over global fields

Let $K$ be a global field which contains a primitive $p$-th root of unity, where $p$ is a prime number. M. J. Hopkins and K. G. Wickelgren showed that for $p=2$, any triple Massey product over $K$ with respect to $\mathbb{F}_p$, contains 0 whenever it is defined. We show that this is true for all primes $p$.Comment: The final version of this paper appeared in Documenta Mathematica, Vol. 20 (2015) 1467-148

On formality of Sasakian manifolds

We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this we produce a method of constructing simply connected K-contact non-Sasakian manifolds. On the other hand, for every $n \geq 3$, we exhibit the first examples of simply connected compact Sasakian manifolds of dimension $2n + 1$ which are non-formal. They are non-formal because they have a non-zero triple Massey product. We also prove that arithmetic lattices in some simple Lie groups cannot be the fundamental group of a compact Sasakian manifold.Comment: 22 pages, no figures; v2. some corrections; v3. Accepted in J. Topolog