Kähler-Einstein metrics, Archimedean Zeta functions and phase transitions

Abstract: While the existence of a unique Kähler-Einstein metrics on a canonically polarized manifold $X$ was established already in the seventies there are very few explicit formulas available (even in the case of complex curves!). In this talk I will give a non-technical introduction to a probabilistic approach to Kähler-Einstein metrics, which, in particular, yields canonical approximations of the Kähler-Einstein metric on $X$. The approximating metrics in question are expressed as explicit period integrals and the conjectural extension to the case of a Fano variety leads to some intriguing connections with Zeta functions and the theory of phase transitions in statistical mechanics.

algebraic geometrydifferential geometrysymplectic geometry

Audience: researchers in the topic

( video )

Geometria em Lisboa (IST)

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