arXiv subjects
Publications and source records attributed to M. Pontecorvo.
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.
We show that in a certain subfamily of the Kuranishi family of any half Inoue surface the algebraic dimensions of the fibers jump downwards at special points of the parameter space showing that the upper semi-continuity of algebraic dimensions in any sense does not hold in general for families of compact non-Kaehler manifolds. In the Kaehler case, the upper semi-continuity always holds true in a certain weak sense.
We show that any hyperbolic Inoue surface (or Inoue-Hirzebruch surface of even type) admits anti-self-dual bihermitian structures. The same result also holds for any of its small deformations as far as its anti-canonical system is non-empty. Similar results are obtained for parabolic Inoue surfaces. Our method also yields anti-self-dual hermitian metrics on any half Inoue surface. We use the twistor method of Donaldson-Friedman for the proof.