Scalar curvature and projective embeddings. II.

*(English)*Zbl 1159.32012From the introduction: This is a sequel to part I of this paper [the author, J. Differ. Geom. 59, No. 3, 479–522 (2001; Zbl 1052.32017)], which studied connections between the differential geometry of complex projective varieties and certain specific ’balanced’ embeddings in projective space. The original plan was that this sequel would be a lengthy paper, discussing various extensions and ramifications of the ideas studied in part I. However, this plan has been modified in the light of subsequent developments. On the one hand, Mabuchi has extended the results of part I to the case where the varieties have infinitesimal automorphisms. On the other hand, Phong and Sturm have sharpened some of the arguments in part I. They also explain the relation of the ideas to the Deligne pairing and the Chow norm, and to earlier work of Zhang, which the author was unfortunately not aware of when writing part I. These developments mean that some of the results planned for the sequel are now redundant while, on the other hand, the exposition of all the different points of view has grown into a daunting task. Thus, instead, this sequel is a short paper devoted to the proof of one result which is quite an easy consequences of the main theorem in the previous paper.

The main result of the paper is that, under certain hypotheses, a Kähler metric of constant scalar curvature minimises the Mabuchi functional. The method uses finite-dimensional approximations involving projective embeddings.

The main result of the paper is that, under certain hypotheses, a Kähler metric of constant scalar curvature minimises the Mabuchi functional. The method uses finite-dimensional approximations involving projective embeddings.