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On the height of the universal abelian variety
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In this paper we extend the arithmetic intersection theory of adelic divisors on quasiprojective varieties developed by X. Yuan and S. W. Zhang to cover certain adelic arithmetic divisors that are not nef nor integrable. The key concept used in this extension is the relative finite energy introduced by T. Darvas, E. Di Nezza, and C. H. Lu. As an application, we prove that the line bundle of Siegel--Jacobi forms on the universal abelian variety endowed with its invariant hermitian metric is not integrable but we compute its arithmetic self-intersection number using the new extension. The techniques developed in this paper can be applied in many other situations like mixed Shimura varieties or the moduli space of stable marked curves.
Forward citations
Cited by 2 Pith papers
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Pure extension of the theta divisor over the moduli space of abelian varieties
The pure weight-2 extension of the universal theta divisor is the Zariski closure twisted by div θinv, where θinv is the difference of the tropical and smooth tropical Riemann theta functions; Moret-Bailly's key formu...
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Concave transforms of compactified S-metrized divisors
For big compactified S-metrized divisors on quasi-projective varieties over adelic curves, concave transforms are constructed and shown to satisfy arithmetic volume and Hilbert-Samuel formulas.
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