06/03/2012, 10:30 — 11:30 — Room P4.35, Mathematics Building
Daniele Sepe, CAMGSD/IST
Lecture II - Topological and symplectic classification
A theorem due to Liouville, Mineur and Arnol'd states that if a
Lagrangian fibration admits a compact and connected fibre , then
the fibre is diffeomorphic to a torus, nearby fibres are also tori
and there exists a symplectomorphism between a neighbourhood of
and the zero section of the cotangent bundle to the torus which
preserves the fibrations. In this lecture, a generalisation of this
theorem for complete Lagrangian fibrations is proved, using the
natural fibrewise action of the cotangent bundle to the base on the
total space of the fibration. This construction allows to develop a
topological (in fact, smooth) and symplectic classification theory
for such fibrations, which yields two topological invariants, the
period net and Chern class, and one symplectic characteristic
class, the Lagrangian Chern class.
See also
https://www.math.tecnico.ulisboa.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf
