Complex surfaces and indefinite metrics

In a joint work with J. Davidov, O. Mushkarov and M. Yotov we considered some geometric structures on compact complex surfaces that are related to split quaternions and lead to existence of indefinite anti-selfdual metrics. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler (or hypersymplectic) , are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. Every compact complex surface admitting a para-hyperhermitian structure has vanishing first Chern class and, unlike the definite case, many of these surfaces carry infinite dimensional families of such structures. In the talk will be explained the relation among the different structures together with several examples and some open problems.