LECTURE 5
Multi-Valued Perturbations
In this lecture we return to the proof of the Arnold conjecture. We shall assume
throughout that (M,w) satisfies (4) with a negative factor, i.e.
c1(v) = Tw(v)
for all smooth maps v : 82 ___. M and some constant T < 0, where c 1 ( v) = J v c 1
and w(v) = J vw. This negative monotonicity condition implies that every J-
holomorphic sphere in M has negative Chern number. The first section explains
the difficulties which arise in the compactness theorem from the presence of multiply
covered J-holomorphic spheres with negative Chern number. In the spring of 1996
several groups of researches found methods to overcome these difficulties. The
first paper which appeared was by Fukaya-Ono [14]. Other approaches are due to
Liu-Tian [28], Ruan [42], Siebert [51], and Hofer-Salamon [20, 21, 22, 23]. The
approach discussed here was developed in [22, 23].
Section 5.2 describes an axiomatic setup for multi-valued perturbations of the
8-equation, which destroy multiply covered J-holomorphic spheres. The existence
of such pertubations is proved in Section 5.3. Section 5.4 deals with the resulting
moduli spaces. They are no longer smooth manifolds, but instead are branched
manifolds with rational weights. Section 5.5 deals with the compactness problem
for these moduli spaces. Section 5.6 discusses the definition of the Gromov-Witten
invariants, and Section 5. 7 describes how the constructions of this lecture lead to
Floer homology over the rationals. The analytical details are given in [22, 23].
5.1. J-holomorphic spheres with negative Chern number
Fix an almost complex structure J E .J(M,w) and denote by M(k; J) the modulispace of J-holomorphic spheres v: 82 ___. M with Chern number c 1 (v) = k. Under
our assumptions this moduli space is only nonempty when k < 0. For a generic
almost complex structure J the subsetM^8 (k ; J) c M(k; J)
of simple spheres is a smooth manifold of dimension
dimM^5 (k; J) = 2n + 2k.
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