Stability, energy functionals, and Kähler-Einstein metrics

Details Stability, energy functionals, and Kähler-Einstein metrics Stability, energy functionals, and Kähler-Einstein metrics

2002-03-24

arxiv.org/abs/math/0203254v1

Mathematics

Differential Geometry

PlainTEX file, 28 pages

Scientific paper

An explicit seminorm $||f||_{#}$ on the vector space of Chow vectors of projective varieties is introduced, and shown to be a generalized Mabuchi energy functional for Chow varieties. The singularities of the Chow varieties give rise to currents supported on their singular loci, while the regular parts are shown to reproduce the Mabuchi energy functional of the corresponding projective variety. Thus the boundedness from below of the Mabuchi functional, and hence the existence of K\"ahler-Einstein metrics, is related to the behavior of the current $[Y_s]$ and the seminorm $||f||_{#}$ along the orbits of $SL(N+1,{\bf C})$.

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