Invariants in symplectic geometry via holomorphic curves (Q2780635)

This is a survey article dedicated to the problem of finding invariants for symplectic manifolds. It is a paper rich in ideas and extremely instructive. The author introduces important notions and ideas, illustrates them with well chosen examples, formulates fundamental theorems, gives only sketches of proofs, if any, pin-pointing main ideas, but avoiding technicalities. Sometimes she proves weaker but instructive versions of these theorems or for particular important classes of examples. Quoting the author: ``I have tried to give some ideas, particularly those concerning the difficulties of the subject and the possible strategies to overcome them.'' The shortest review could be: ``The author has well succeeded in her aims.'' NEWLINENEWLINENEWLINEIn the first section the concepts of soft (mous) and coarse (grossier) invariants for symplectic manifolds are introduced. For symplectic manifolds, due to the Darboux theorem, there are no local invariants. The last part of this section is dedicated to the introduction of the concept of pseudo-holomorphic curves in symplectic manifolds. NEWLINENEWLINENEWLINEThe second section of the survey is concerned with pseudo-holomorphic curves and an introduction to Gromov-Witten invariants. The aim of this section is best explained by the author herself: ``The aim is to convince the reader that pseudo-holomorphic curves may be useful in symplectic geometry by showing the way to use them to forbid a big ball to embed itself into a fine cylinder.'' NEWLINENEWLINENEWLINEThe third section discusses the space of modules of stable applications of \textit{M. Kontsevich} [cf. Prog. Math. 129, 335-368 (1995; Zbl 0885.14028)]. The section contains the definition, some simple examples, the definition of the topology of this space, the discussion of its compactness as well as of the smoothness. Virtual fundamental classes are also studied. NEWLINENEWLINENEWLINEIn the fourth section the author studies some properties which Gromov-Witten invariants should have [cf. \textit{M. Kontsevich} and \textit{Yu. Manin}, Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)]. Finally, some of these invariants are calculated for simple but important symplectic manifolds. NEWLINENEWLINENEWLINEIn the short last section the author gives a quick review of some other invariants.NEWLINENEWLINEFor the entire collection see [Zbl 0981.53001].