## The theta invariants and the volume function on arithmetic varieties

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Mounir Hajli
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## Abstract:

We introduce a new arithmetic invariant for hermitian line bundles on arithmetic varieties. We use this invariant to measure the variation of the volume function with respect to the metric. We apply the theory developed here to the study of the arithmetic geometry of toric varieties. As an application, we obtain a generalized Hodge index theorem for hermitian line bundles which are not necessarily toric. When the metrics are toric, we recover some results due to Burgos, Phillippon, Sombra and Moriwaki.## References

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## Additional Information

**Mounir Hajli**- Affiliation: School of Mathematical Sciences, Shanghai Jiao Tong University, People’s Republic of China
- MR Author ID: 1049494
- ORCID: 0000-0002-7971-5337
- Email: hajli@sjtu.edu.cn
- Received by editor(s): March 16, 2022
- Received by editor(s) in revised form: September 9, 2022, and October 10, 2022
- Published electronically: January 4, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 2237-2256 - MSC (2020): Primary 14G40
- DOI: https://doi.org/10.1090/tran/8849
- MathSciNet review: 4549705