Ivancevic_Applied-Diff-Geom

632 Applied Differential Geometry: A Modern IntroductionThese generalizations of SH 1 (M, ω) are closely connected to Reznikov’sFutaki type characters [Reznikov (1997)]. It is not yet clear what is themost natural disconnected extension of Ham(M, ω).4.12.1.4 StabilityAnother important property of Hamiltonian bundles is stability [Lalondeand McDuff (2002)]. A symplectic (resp. Hamiltonian) bundle π : P → Bwith fiber (M, ω) is said to be stable if π may be given a symplectic (resp.Hamiltonian) structure with respect to any symplectic form ω ′ on M thatis sufficiently close to (but not necessarily cohomologous to) ω, in such away that the structure depends continuously on ω ′ .Using Moser’s homotopy argument, it is easy to prove that any symplecticbundle is stable. The following characterization of Hamiltonian stabilityis an almost immediate consequence of above theorem:A Hamiltonian bundle π : P → B is stable iff the restriction mapH 2 (P, R) → H 2 (M, R) is surjective.The following result is less immediate: Every Hamiltonian bundle isstable.The proof uses the (difficult) stability property for Hamiltonian bundlesover S 2 that was established in [Lalonde et al. (1999); McDuff (2000)] aswell as the (easy) fact that the image of the evaluation map π 2 (Ham(M))−→ π 2 (M) lies in the kernel of [ω].4.12.1.5 Cohomological SplittingWe next extend the splitting results of [Lalonde et al. (1999); McDuff(2000)], which prove that the rational cohomology of every Hamiltonianbundle π : P → S 2 splits additively, i.e., there is an additive isomorphismH ∗ (P ) ∼ = H ∗ (S 2 ) ⊗ H ∗ (M).For short we will say in this situation that π is c−split. This is a deepresult, that requires the use of Gromov–Witten invariants for its proof. 16The results of the present subsection provide some answers to the following16 Recall that Gromov–Witten invariants are rational numbers that count pseudo–holomorphic curves meeting prescribed conditions in a given symplectic manifold. Theseinvariants may be packaged as a homology or cohomology class in an appropriate space,or as the deformed cup product of quantum cohomology. They have been used to distinguishsymplectic manifolds that were previously indistinguishable. They also play acrucial role in closed type IIA string theory.