Extension of Positive Line Bundles and Meromorphic Maps.
This note is an attempt to describe a part of the historical development of the research on separately holomorphic functions.
We first exhibit counterexamples to some open questions related to a theorem of Sakai. Then we establish an extension theorem of Sakai type for separately holomorphic/meromorphic functions.
New examples of extremal Kähler metrics are given on blow-ups of parabolic ruled surfaces. The method used is based on the gluing construction of Arezzo, Pacard and Singer [5]. This enables to endow ruled surfaces of the form with special parabolic structures such that the associated iterated blow-up admits an extremal metric of non-constant scalar curvature.
We provide a new proof of a result of X.X. Chen and G.Tian [5]: for a polarized extremal Kähler manifold, the minimum of the modified K-energy is attained at an extremal metric. The proof uses an idea of C. Li [16] adapted to the extremal metrics using some weighted balanced metrics.
We study different notions of extremal plurisubharmonic functions.