New explicit formulas for Faltings’ delta-invariant


In this paper we give new explicit formulas for Faltings’ \(\delta \)-invariant in terms of integrals of theta functions, and we deduce an explicit lower bound for \(\delta \) only in terms of the genus and an explicit upper bound for the Arakelov–Green function in terms of \(\delta \). Furthermore, we give a canonical extension of \(\delta \) and the Zhang–Kawazumi invariant \(\varphi \) to the moduli space of indecomposable principally polarised complex abelian varieties.

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The contents of this paper largely coincide with my Ph.D. thesis. I would like to thank my advisor Gerd Faltings for introducing me into Arakelov theory and for his suggestion to study the \(\delta \)-invariant. I also would like to thank Rafael von Känel and Robin de Jong for useful discussions and Michael Rapoport for a remark on an early version of this paper.

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Correspondence to Robert Wilms.

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Wilms, R. New explicit formulas for Faltings’ delta-invariant. Invent. math. 209, 481–539 (2017).

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Mathematics Subject Classification

  • 14G40
  • 14H42
  • 14H55
  • 14K25
  • 11G30