  Let M be a compact oriented even-dimensional manifold. This note constructs a
compact symplectic manifold S of the same dimension and a map f from S to M of
strictly positive degree. The construction relies on two deep results: the
first is a theorem of Ontaneda that gives a Riemannian manifold N of tightly
pinched negative curvature which admits a map to M of degree equal to one; the
second is a result of Donaldson on the existence of symplectic divisors. Given
Ontaneda's negatively curved manifold N, the twistor space Z is symplectic. The
manifold S is then a suitable multisection of the twistor space, found via
Donaldson's theorem.
