An invitation to Kähler–Einstein metrics and random point processes

Berman, Robert J.

doi:10.4310/sdg.2018.v23.n1.a2

Cited by 8 publications

(29 citation statements)

References 67 publications

(163 reference statements)

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“…There are works e.g. [3][4][5][6][7] by Berman,[8][9][10][11][12][13][14][15] by Ferrari, Flurin, Klevtsov, Song, Zelditch. On the other hand, probabilistic aspects of the centre of mass μX (g) does not seem to have been actively investigated in the aforementioned works, which is the focus of the present paper.…”

Section: Introduction and The Statement Of The Main Resultsmentioning

confidence: 99%

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Expected Centre of Mass of the Random Kodaira Embedding

Hashimoto

2022

J Geom Anal

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Let $$X \subset \mathbb {P}^{N-1}$$ X ⊂ P N - 1 be a smooth projective variety. To each $$g \in SL (N , \mathbb {C})$$ g ∈ S L ( N , C ) which induces the embedding $$g \cdot X \subset \mathbb {P}^{N-1}$$ g · X ⊂ P N - 1 given by the ambient linear action we can associate a matrix $$\bar{\mu }_X (g)$$ μ ¯ X ( g ) called the centre of mass, which depends nonlinearly on g. With respect to the probability measure on $$SL (N , \mathbb {C})$$ S L ( N , C ) induced by the Haar measure and the Gaussian unitary ensemble, we prove that the expectation of the centre of mass is a constant multiple of the identity matrix for any smooth projective variety.

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Section: Introduction and The Statement Of The Main Resultsmentioning

confidence: 99%

“…in terms of the notation (7). We then define an N × N hermitian matrix μ p (g) ∈ √ −1u(N ) whose (i, j)th entry is given by…”

Section: Definitionmentioning

confidence: 99%

Expected Centre of Mass of the Random Kodaira Embedding

Hashimoto

2022

J Geom Anal

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“…51. Fekete points on complex manifolds have also generated some interest in complex geometry [46][47][48][49] .…”

Section: B Long Range Case S < Dmentioning

confidence: 99%

Coulomb and Riesz gases: The known and the unknown

Lewin

2022

Preprint

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We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in R d interacting with the Riesz potential ±|x| −s (resp. − log |x| for s = 0). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g. in random matrix theory). The main question addressed in the article is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case s < d. For the convenience of the reader we give the detail of what is known in the short range case s > d. In the last part we discuss phase transitions and mention what is expected on physical grounds.

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“…We start by providing some background on Kähler-Einstein metrics and recapitulating the probabilistic approach to Kähler-Einstein metrics; the reader is referred to the survey [14] for more background and [15] for relations to the Yau-Tian-Donaldson conjecture.…”

Section: Introductionmentioning

confidence: 99%

“…Thus one virtue of the probabilistic approach is that it yields canonical approximations of the Kähler-Einstein metric on X, expressed as essentially explicit period type integrals formulas (see formula 1.4 below). These are reminiscent of the aforementioned few explicit formulas for Kähler-Einstein metrics, involving hypergeometric integrals (see [14,Section 2.1]).…”

Section: Introductionmentioning

confidence: 99%

Kähler-Einstein metrics and Archimedean zeta functions

Berman¹

2021

Preprint

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While the existence of a unique Kähler-Einstein metric on a canonically polarized manifold X was established by Aubin and Yau already in the 70s there are only a few explicit formulas available. In previous work a probabilistic construction of the Kähler-Einstein metric was introduced -involving canonical random point processes on X -which yields canonical approximations of the Kähler-Einstein metric, expressed as explicit period integrals over a large number of products of X. Here it is shown that the conjectural extension to the case when X is a Fano variety suggests a zero-free property of the Archimedean zeta functions defined by the partition functions of the probabilistic model. The convergence in the case of log Fano curves is settled, exploiting relations to the complex Selberg integral in the orbifold case. Some intriguing relations to the zero-free property of the local automorphic L-functions appearing in the Langlands program and arithmetic geometry are also pointed out. These relations also suggest a natural p-adic extension of the probabilistic approach.

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An invitation to Kähler–Einstein metrics and random point processes

Cited by 8 publications

References 67 publications

Expected Centre of Mass of the Random Kodaira Embedding

Expected Centre of Mass of the Random Kodaira Embedding

Coulomb and Riesz gases: The known and the unknown

Kähler-Einstein metrics and Archimedean zeta functions

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