On Hermitian Geometry of Complex Surfaces

Locally conformally Kähler metrics on certain non-Kählerian surfaces

“…Our Theorem 1 has some contact with the papers [8] and [5], where it is also shown that some Enoki surfaces admit lcK metrics. However, in those papers the authors construct lcK metrics with further symmetries: they are anti-self-dual and they are invariant by a circle action.…”

“…However, in those papers the authors construct lcK metrics with further symmetries: they are anti-self-dual and they are invariant by a circle action. Metrics like that cannot exist on general Kato or Enoki surfaces, and so our results cannot be recovered by the anti-self-dual approach of [8] and [5]. Our special construction in Sect.…”

“…Our special construction in Sect. 2 overlaps with [8] and [5], at least for certain values of the parameter t, but we think useful to have a direct construction of lcK metrics without passing through anti-self-dual ones.…”

“…In [5] there is also a construction of lcK (and anti-self-dual) metrics on Inoue-Hirzebruch surfaces, which constitute the "most degenerate" subfamily of Kato surfaces. Theorem 2 gives lcK metrics on small deformations of InoueHirzebruch surfaces.…”

Compatible complex structures on twistor space

“…When the first Betti number is odd and the Euler characteristic is non zero, the only other possible case is that of surfaces of class V II with 0 < χ = b 2 [7], for which there is (yet) no classification. (For some existence results see [18].) Let J be a compatible almost complex structure on (Z,g).…”