Extremal Kahler metrics on projective bundles over a curve

          @misc{ scivideos_PIRSA:09050031,
            doi = {10.48660/09050031},
            url = {https://pirsa.org/09050031},
            author = {Apostolov, Vestislav},
            keywords = {},
            language = {en},
            title = {Extremal Kahler metrics on projective bundles over a curve},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2009},
            month = {may},
            note = {PIRSA:09050031 see, \url{https://scivideos.org/index.php/pirsa/09050031}}
          }
          

Abstract

I will discuss the existence problem of extremal Kahler metrics (in the sense of Calabi) on the total space of a holomorphic projective bundle P(E) over a compact complex curve. The problem is not solved in full generality even in the case of a projective plane bundle over CP^1. However, I will show that sufficiently ``small'' Kahler classes admit extremal Kahler metrics if and only if the underlying vector bundle E can be decomposed as a sum of stable factors. This result can be viewed as a ``Hitchin-Kobayashi correspondence'' for projective bundles over a curve, but in the context of the search for extremal Kahler metrics. The talk will be based on a recent work with D. Calderbak, P. Gauduchon and C. Tonnesen-Friedman.