Invariants entiers en g\'eom\'etrie \'enum\'erative r\'eelle

TL;DR

This paper explores integer-valued invariants in real enumerative geometry, comparing them to Gromov-Witten invariants, analyzing their sharpness and arithmetical properties, and examining conditions for pseudo-holomorphic discs with boundary on Lagrangian submanifolds.

Contribution

It introduces real enumerative invariants analogous to Gromov-Witten invariants, analyzes their properties, and provides criteria for the existence of pseudo-holomorphic discs.

Findings

Integer-valued invariants serve as real counterparts to Gromov-Witten invariants.

Sharpness of lower bounds provided by these invariants is analyzed.

Results on the existence or non-existence of pseudo-holomorphic discs with boundary on Lagrangian submanifolds.

Abstract

I first recall the various problems of real enumerative geometry out of which I could extract some integer valued invariants, providing some real counterpart to Gromov-Witten invariants. I then discuss sharpness of the lower bounds given by these invariants and some of their arithmetical properties. Finally, I present further results which guaranty the presence or absence of pseudo-holomorphic discs with boundary on a Lagrangian submanifold of a given symplectic manifold.