Kyoto Journal of Mathematics

Canonical Kähler metrics and arithmetics: Generalizing Faltings heights

Yuji Odaka

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Abstract

We extend the Faltings modular heights of Abelian varieties to general arithmetic varieties, show direct relations with the Kähler–Einstein geometry, the minimal model program, and Bost–Zhang’s heights and give some applications. Along the way, we propose the “arithmetic Yau–Tian–Donaldson conjecture” (the equivalence of a purely arithmetic property of a variety and its metrical property) and partially confirm it.

Article information

Source
Kyoto J. Math., Volume 58, Number 2 (2018), 243-288.

Dates
Received: 4 October 2015
Revised: 15 September 2016
Accepted: 20 September 2016
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1520046287

Digital Object Identifier
doi:10.1215/21562261-2017-0023

Mathematical Reviews number (MathSciNet)
MR3799703

Zentralblatt MATH identifier
06896955

Subjects
Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]

Keywords
Faltings heights Kähler–Einstein metrics Arakelov geometry

Citation

Odaka, Yuji. Canonical Kähler metrics and arithmetics: Generalizing Faltings heights. Kyoto J. Math. 58 (2018), no. 2, 243--288. doi:10.1215/21562261-2017-0023. https://projecteuclid.org/euclid.kjm/1520046287


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