Title: Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics
Abstract: Using results from the theory of harmonic maps,
Kotschick proved that a closed hyperbolic four-manifold cannot admit a
complex structure. We give a new proof which instead relies on
properties of Einstein metrics in dimension four. The benefit of this
new approach is that it generalizes to prove that another class of
aspherical four-manifolds (graph manifolds with positive Euler
characteristic) also fail to admit complex structures. This is joint
work with Luca Di Cerbo.