Abstract:

This is a joint work with Joshua Jordan. In 1998 Beauville gave a classification of complex surfaces wit split tangent bundle.  We study the pluri-closed metrics on these surfaces, which include some classical non-Kahler examples such as Hopf surfaces. We determine a proper cohomology group $H$ in which both the pluri-closed metric and its Bismut Ricci form live following discussions in Streets-Tian and others. We are able to characterize $H$ and solve the prescribing Bismut Ricci form problem completely. The setting can be compared to the usual Kahler setting, with some twists. We raise some further questions regarding the metric collapsing, which can be compared to corresponding problems in Riemann and Kahler geometry.