A Remark about Donaldson’s Construction of Symplectic Submanifolds

On the Construction of Contact Submanifolds with Prescribed Topology

We prove the existence of contact submanifolds realizing the Poincaré dual of the top Chern class of a complex vector bundle over a closed contact manifold. This result is analogue in the contact category to Donaldson’s construction of symplectic submanifolds. The main tool in the construction is to show the existence of sequences of sections which are asymptotically holomorphic in an appropiat...

Lagrangian submanifolds and Lefschetz pencils SG/0407126 16

Given a Lagrangian submanifold in a symplectic manifold and a Morse function on the submanifold, we show that there is an isotopic Morse function and a symplectic Lefschetz pencil on the manifold extending the Morse function to the whole manifold. From this construction we define a sequence of symplectic invariants classifying the isotopy classes of Lagrangian spheres in a symplectic 4-manifold.

Donaldson’s Results on Symplectic Hypersurfaces and Lefschetz Pencils

On Contact and Symplectic Lie Algeroids

In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by m...

Generalized Geometry and Noncommutative Algebra Abstracts of Talks

A generalized complex structure may be understood as a "weakly holomorphic" Poisson structure: given an integrability condition, a generalized complex structure "integrates" to a symplectic groupoid equipped with a weak holomorphic structure, i.e., a compatible holomorphic structure on the associated stack. Similarly, generalized complex branes may be represented as weakly holomorphic coisotrop...