Arakelov inequalities and characterization of totally geodesic ball quotients in đť’śg
Abstract
We establish an Arakelov inequality for variations of Hodge structures underlying families of principally polarized complex Abelian varieties. In the equality case, this leads to a numerical characterization of certain totally geodesic ball quotients inside the moduli space of Abelian varieties. Our result extends work of Möller-Viehweg-Zuo by removing the strong positivity conditions imposed in their statement. For families over compact base spaces, our proof involves showing that the period map associated with a family of Abelian varieties factors through certain MMP operations and then generalizing the results of Möller, Viehweg, and Zuo to an appropriate singular setting. In the quasiprojective surface case, we implement a new approach that does not pass through Miyaoka-Yau-type uniformisation theorems but uses an argument going back to an idea of Mok, symmetric space theory and semistability considerations instead.
Related
- Quantitative finiteness for maximal complex totally geodesic submanifolds of non-arithmetic ball quotients
- Arakelov inequalities for fibered surfaces in positive characteristic
- Totally Geodesic Submanifolds of TeichmĂĽller Space in Complex Dimension at Least Three
- Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set
- Arithmeticity, superrigidity, and totally geodesic submanifolds of infinite-volume hyperbolic manifolds
BibTeX
@misc{costantini2026arakelov,
title = {{Arakelov inequalities and characterization of totally geodesic ball quotients in $\mathcalA_g$}},
author = {Matteo Costantini and Daniel Greb and Carolina Tamborini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37023},
eprint = {2609.37023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37023},
}
RIS
TY - UNPB
TI - Arakelov inequalities and characterization of totally geodesic ball quotients in đť’ś_g
AU - Matteo Costantini
AU - Daniel Greb
AU - Carolina Tamborini
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37023
UR - https://arxiv.org/abs/2609.37023
AB - We establish an Arakelov inequality for variations of Hodge structures underlying families of principally polarized complex Abelian varieties. In the equality case, this leads to a numerical characterization of certain totally geodesic ball quotients inside the moduli space of Abelian varieties. Our result extends work of Möller-Viehweg-Zuo by removing the strong positivity conditions imposed in their statement. For families over compact base spaces, our proof involves showing that the period map associated with a family of Abelian varieties factors through certain MMP operations and then generalizing the results of Möller, Viehweg, and Zuo to an appropriate singular setting. In the quasiprojective surface case, we implement a new approach that does not pass through Miyaoka-Yau-type uniformisation theorems but uses an argument going back to an idea of Mok, symmetric space theory and semistability considerations instead.
ER -