Proof of the Arnold conjecture for surfaces and generalizations to certain Kähler manifolds (Q1085499)

Given a compact symplectic manifold (P,\(\omega)\) and a smooth function H: S'\(\times P\to {\mathbb{R}}: (t,p)\mapsto H_ t(p)\), \(S'={\mathbb{R}}/{\mathbb{Z}}\), a periodic family of (''exact Hamiltonian'') vector fields \(X_ t\) is given by \(\omega (\cdot,X_ t)=dH_ t(\cdot)\). The author presents a proof for the Arnold conjecture in the case of surfaces which is stated in the following way: ''On a compact surface \((F_ g,\omega)\) of genus \(g\geq 1\) with volume form \(\omega\) every exact Hamiltonian vector field of period 1 possesses at least 3 solutions of period 1. If all the 1-periodic solutions are nondegenerate, then there exist at least \(2(g+1)\) of them.'' Some generalizations to certain Kähler manifolds are also presented.