Einstein Metrics, Harmonic Forms, and Symplectic Manifolds.

Given a smooth compact 4-manifold M  which admits Einstein metrics, is its moduli space of Einstein metrics connected?  While the corresponding question in higher dimensions typically has a negative answer, there are interesting 4-manifolds M for which the answer is known to be affirmative. One important class of 4-manifolds for which we do not know the answer, however,  consists of the underlying 4-manifolds of the del Pezzo surfaces. In this lecture, I will explain a  result which provides interesting partial information concerning this case.