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Invariants in symplectic geometry via holomorphic curves. (Invariants en géométrie symplectique via les courbes holomorphes.) (English) Zbl 1007.53066

Dumas, François (ed.) et al., Nouveaux invariants en géométrie et en topologie. Paris: Société Mathématique de France, Panor. Synth. 11, 1-59 (2001).
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.”
In 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.
The 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.”
The third section discusses the space of modules of stable applications of 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.
In the fourth section the author studies some properties which Gromov-Witten invariants should have [cf. M. Kontsevich and 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.
In the short last section the author gives a quick review of some other invariants.
For the entire collection see [Zbl 0981.53001].

MSC:

53D35 Global theory of symplectic and contact manifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H15 Families, moduli of curves (analytic)
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