Skip to main content

Real uniruled manifolds

Objective

The following list of questions describe the four main directions which I want to develop.
1) Topology of real uniruled manifolds.
May the connected sum of two closed hyperbolic manifolds of dimension at least three be Lagrangian embedded in a uniruled symplectic manifold? Being able to answer to this question through the negative using the symplectic field theory introduced by Eliashberg-Givental and Hofer requires to understand pseudo-holomorphic curves in the cotangent bundle of such a connected sum. For this purpose, one needs some understanding of closed geodesics on such manifolds. Conversely, what are the simplest real three-dimensional projective manifolds which have hyperbolic or SOL manifolds in their real loci?
2) Enumerative problems in real uniruled manifolds.
Is it possible to extract integer valued invariants from the count of real rational curves of given degree in the projective three-space (for instance) which interpolate an adequate number of real lines? Same question in dimensions greater than three for curves passing through points.
3) Lagrangian strings in symplectic manifolds.
I would like to investigate the interactions between closed Lagrangian strings and open Lagrangian strings in symplectic manifolds. These strings -which I recently introduced- interact through holomorphic disks both punctured on their boundaries and interiors. What can be the analogous TQFT associated to coherent sheaves on complex projective manifolds? How are these strings related to Gromov-Witten invariants?
4) Volume of linear systems of real divisors.
The theory of closed positive currents provides probabilistic informations on the topology of real hypersurfaces in Kähler manifolds. I want to push a work in progress as far as possible in this subject.

Call for proposal

ERC-2010-StG_20091028
See other projects for this call

Host institution

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Address
Rue Michel Ange 3
75794 Paris
France
Activity type
Research Organisations
EU contribution
€ 932 626
Principal investigator
Jean-Yves Welschinger (Mr.)
Administrative Contact
Sandrine Magnetto (Dr.)

Beneficiaries (1)

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France
EU contribution
€ 932 626
Address
Rue Michel Ange 3
75794 Paris
Activity type
Research Organisations
Principal investigator
Jean-Yves Welschinger (Mr.)
Administrative Contact
Sandrine Magnetto (Dr.)