Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics

Abstract

We show that extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure. This result stems from a new proof of the fact that a closed real-hyperbolic 4-manifold cannot support a complex structure. Finally, we construct infinitely many extended graph 4-manifolds with positive Euler characteristic which support almost complex structures.

A typo in the proof of Theorem 1 is corrected. 10 pages, no figures. To appear in J. Geometric Analysis