Blowing up and desingularizing constant scalar curvature K\"{a}hler manifolds

This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already carry constant scalar curvature Kaehler metrics.Comment: Greatly revised version. 43 page

Infinite geodesic rays in the space of Kahler potentials

In this paper we explore the connection between special degenerations of algebraic manifolds and geodesics in the space of Kahler metrics. We provide a new and general geometric construction of nontrivial solutions for the geodesic equation. We show how to associate to any special nontrivial degeneration a geodesic of inifite length.Comment: 17 page

On cscK resolutions of conically singular cscK varieties

In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups

On the resolution of extremal and constant scalar curvature Kaehler orbifolds

In this paper we give sufficient conditions on a compact orbifold with an extremal Kaehler metric to admit a resolution with an extremal Kaehler metric. We also complete the Kaehler constant scalar curvature case.Comment: This paper, together with the next one by the same authors, contains more general results than arxiv.org/abs/1402.5919 (by the same authors), which will then be withdraw