Rotation numbers of Hamiltonian isotopies in complex projective spaces
Résumé
In this paper, we prove the existence of symplectic invariantsmwe call them rotation numbersmfor the time-1 map of a Hamiltonian isotopy in EP"-1. These invariants bear interesting properties related to the geometry of the fixed-point set of the symplectic map. They are obtained by lifting the problem to the linear space IE", and then using invariant 9eneratin9 functions to define them as solutions of a certain finite-dimensional min-max method. 1. Statement of the result. The space E" (n > 2) is endowed with its standard Euclidean structure, denoted by (.,). We take its standard symplectic form to be f(z, z') (iz, z') for all z, z' in Let S 2n-1 be the unit sphere in En; the group S acts on S2n-, with quotient space S 2n-/S EPn-. We denote by n S 2n-IEP-1 the projection and by i" S 2n-IE the inclusion. There is a unique symplectic structure co on tEP "-1 that satisfies the relation i*f n'co (see [12]). An easy computation shows that, for the case n 2, the symplectic area of IEP equals n. Let h (ht)t[o,] be a time-dependent Hamiltonian on IEPn-l, and (t)t[0,1] be its associated Hamiltonian isotopy. Recall that (0t)t[0 is obtained by integrating the vector field (Xt)t[0,1] defined by dht ix, co for ] in [0, 1]. A number, the action of x, is associated in a standard way to any fixed point x of 0 as follows. Consider the contractible loop y (t t(x),t [0, 1]) in pn-1, and choose any 2-disc D such that OD y. Then the symplectic area of D is a real number defined modulo n, and is denoted by a(x). The action c(x) of x is 1 a(x) + ht(tPt(x)) dt e ]R/Z. c(x) .= Our result is now the following theorem. THEOREM 1.1. There exist n rotation numbers 0 < tl < < tn < 1 with the following properties. (a) The image in S-IR/Z of the sequence (q,..., tn) depends only on the iso-topy (t)te[o,1], up to a cyclic shift of the indices and to a global rotation of S1. It also depends continuously (for the C 1-topology) on the isotopy.