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

Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics