  In this paper, we study the topological properties on complex polynomial
Hamiltonian differential systems of degree $n+1$ having an isochronous center
of Morse type. Firstly, we prove that if on the level curve containing an
isochronous center there is only one singular point, then the vanishing cycle
associated to this center is a zero homology cycle on the Riemann surface of a
generic level curve. This result provides a positive answer to an open question
proposed by L. Gavrilov for a very large class of Hamiltonian systems.
Secondly, we obtain a necessary condition for isochronicity that the
$(n+1)$-degree part of the Hamiltonian function must have a factor with
multipicity no less than $(n+1)/2$. Thirdly, we show that if the vanishing
cycle associated to an isochronous Morse center is a zero homology cycle on the
Riemann surface of a generic level curve, then the real Hamiltonian systems do
not admit any isochronous center at all when the degree of the Hamiltonian
functions is odd. Therefore we consequently prove that a positive answer to L.
Gavrilov's question implies the conjecture proposed by X. Jarque and J.
Villadelprat is true.
