There are many deep connections between symplectic and complex geometry, ranging from the phenomena of mirror symmetry to consequences of the fact that a symplectic manifold is a flabby kind of Kähler manifold in which the complex structure has lost its integrability. This talk will explore the latter relationship, explaining, for example, how the infinite number of complex structures on the product of projective lines (the famous family of Hirzebruch surfaces) is reflected in the symplectic world. Kähler manifolds have many special topological properties (such as restrictions on their fundamental groups) that often are not shared by symplectic manifolds. On the other hand there are very interesting situations in which there is some symplectic rigidity. We will discuss some recent work in this area.

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