Bi-Hermitian metrics on Kato surfaces
  We investigate bi-Hermitian metrics on compact complex surfaces with odd first Betti number producing new examples with connected anti-canonical divisor using the general construction of \cite{abd15}. The result is a complete classification for all \it unbranched \rm Kato surfaces and a classification up to logarithmic deformation for \it intermediate \rm surfaces. 
A. Fujiki, M. Pontecorvo
math.DG