1,039 research outputs found
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
Exact -structures on unimodular Lie algebras
We consider seven-dimensional unimodular Lie algebras
admitting exact -structures, focusing our attention on those with
vanishing third Betti number . We discuss some examples,
both in the case when , and in the case when the Lie
algebra is (2,3)-trivial, i.e., when both
and vanish. These examples are solvable, as
, but they are not strongly unimodular, a necessary
condition for the existence of lattices on the simply connected Lie group
corresponding to . More generally, we prove that any
seven-dimensional (2,3)-trivial strongly unimodular Lie algebra does not admit
any exact -structure. From this, it follows that there are no compact
examples of the form , where is a
seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra,
is a co-compact discrete subgroup, and is an exact
-structure on induced by a left-invariant one on .Comment: Final version; to appear in Monatshefte f\"ur Mathemati
- …