JongHae Keum, Fake quadric surfaces

A smooth projective complex surface S is called a Q-homology quadric if it has the same Betti numbers as the smooth quadric surface. Let S be a Q-homology quadric. Then its cohomology lattice is of rank 2, (even or odd) unimodular. By the classification of surfaces, S is either rational or of general type. In the latter case, S is called a fake Q-homology quadric (fake quadric, in short). There is an unsolved question raised by Hirzebruch: does there exist a surface of general type which is homeomorphic to the smooth quadric surface? I will report recent progress on these surfaces.