Berman, Robert J. Kähler-Einstein metrics and Archimedean zeta functions. (English) Zbl 1528.32037 Hujdurović, Ademir (ed.) et al., European congress of mathematics. Proceedings of the 8th congress, 8ECM, Portorož, Slovenia, June 20–26, 2021. Berlin: European Mathematical Society (EMS). 199-251 (2023). Summary: 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 a 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. A weaker zero-free property is also shown to be relevant for the Calabi-Yau equation. 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.For the entire collection see [Zbl 1519.00033]. MSC: 32Q20 Kähler-Einstein manifolds 14J45 Fano varieties 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 11S40 Zeta functions and \(L\)-functions 14G40 Arithmetic varieties and schemes; Arakelov theory; heights Keywords:Kähler-Einstein metric; Fano variety; random point process; Langlands \(L\)-functions; Arakelov geometry × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] L. Alvarez-Gaumé, J.-B. Bost, G. Moore, P. Nelson, and C. Vafa, Bosonization on higher genus Riemann surfaces. Comm. Math. Phys. 112 (1987), no. 3, 503-552 · Zbl 0647.14019 [2] K. Aomoto, On the complex Selberg integral. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 152, 385-399 Zbl 0639.33002 MR 916224 · Zbl 0639.33002 [3] T. Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. (2) 102 (1978), no. 1, 63-95 Zbl 0374.53022 MR 494932 · Zbl 0374.53022 [4] R. J. 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