Constant scalar curvature Kähler metrics on fibred complex surfaces. (English) Zbl 1085.53064

Fine, Joel

J. Differ. Geom. 68, No. 3, 397-432 (2004).

The main purpose of the paper is to prove the existence of constant scalar curvature Kähler metrics on certain complex surfaces. More precisely, the author proves the following theorem: “If \(X\) is a compact connected complex surfaces admitting a holomorphic submersion onto a high genus complex curve with fibres of genus at least two, then, for all large \(r\), the Kähler class \(k_r\) contains a constant scalar curvature metric.”
The main technique used in the proof is an adiabatic limit. First, by fitting together the constant scalar curvature Kähler metrics on the fibres of \(X\) and a large multiple of a metric on the base, a family of approximate solutions is constructed. Then the analysis is developed necessary to show the existence of a genuine solution in the same cohomology class. Finally, the above theorem is extended to certain higher dimensional fibrations.

Reviewer: Constantin Călin (Iaşi)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C56	Other complex differential geometry

Keywords:

holomorphic submersion; adiabatic limit