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Kähler-Einstein metrics, canonical random point processes and birational geometry

Paper i proceeding
Författare Robert Berman
Publicerad i ALGEBRAIC GEOMETRY: SALT LAKE CITY 2015, PT 1
ISBN 978-1-4704-3577-6
Publiceringsår 2018
Publicerad vid Institutionen för matematiska vetenskaper
Språk en
Länkar dx.doi.org/10.1090/pspum/097.1/0166...
Ämnesord complex differential geometry, normal crossing divisors, monge-ampere, equations, q-fano varieties, statistical-mechanics, ricci curvature, line bundles, k-stability, manifolds, singularities
Ämneskategorier Matematik

Sammanfattning

In the present paper and the companion paper (Berman, 2017) a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and shown to converge in probability towards a canonical measure, coinciding with the canonical measure of Song-Tian and Tsuji. In the case of a variety X of general type we obtain as a corollary that the (possibly singular) Kahler-Einstein metric on X with negative Ricci curvature is the limit of a canonical sequence of quasi-explicit Bergman type metrics. In the opposite setting of a Fano variety X we relate the canonical point processes to a new notion of stability, that we call Gibbs stability, which admits a natural algebro-geometric formulation and which we conjecture is equivalent to the existence of a Kahler-Einstein metric on X and hence to K-stability as in the Yau-Tian-Donaldson conjecture.

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