Equicontinuous families of meromorphic mappings with values in compact complex surfaces

Abstract

We prove that a family of meromorphic mappings from a bidisc to a compact complex surface, which are equicontinuous in a neighborhood of the boundary of the bidisc, has the volumes of its graphs locally uniformly bounded.

Weak normality of families of meromorphic mappings and bubbling in higher dimensions

“…When dimension n of the source U is two and X is Kähler the volumes of the graphs of a weakly converging sequence are still bounded, see [18]. The same is true if is X an arbitrary compact complex surface (and again dim U = 2), see [32]. We shall say more about this in Section 6.…”

“…Moreover, it was proved in [32] that volumes of weakly converging sequence are bounded also in the case when X is any compact complex surface. The proof uses Kaähler case separately and then the fact that a non-Kähler surface has only finitely many rational curves.…”

“…Indeed, for Kähler X it was proved in [18] using Skoda's potentials. In [32] its was proved for non-Kähler X using that fact that such X can contain only finitely many rational curves as well as existence of certain dd c -exact (2, 2)-forms.…”