PreprintCC BY 4.0arXiv
The Mumford--Tate group of a very general Lagrangian fiber
Abstract
For a primitive symplectic variety X admitting a holomorphic Lagrangian fibration f: X → B, we classify the possible Mumford--Tate groups of the very general fiber. We extract as a consequence that the discriminant of f has codimension one in B. In the special case when X is a hyper-Kähler manifold which is general among those admitting a Lagrangian fibration, and b2(X) ≥ 7, we further show that the very general fiber of f is Hodge-generic. Finally, as an application of our classification, we confirm an expectation of Li--Tosatti that when f is not isotrivial, the special Kähler metric on B has positive holomorphic sectional curvature on the regular values of f.
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BibTeX
@misc{varvak2026mumfordtate,
title = {{The Mumford--Tate group of a very general Lagrangian fiber}},
author = {Edward Varvak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02481},
eprint = {2610.02481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02481},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Mumford--Tate group of a very general Lagrangian fiber
AU - Edward Varvak
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02481
UR - https://arxiv.org/abs/2610.02481
AB - For a primitive symplectic variety X admitting a holomorphic Lagrangian fibration f: X → B, we classify the possible Mumford--Tate groups of the very general fiber. We extract as a consequence that the discriminant of f has codimension one in B. In the special case when X is a hyper-Kähler manifold which is general among those admitting a Lagrangian fibration, and b_2(X) ≥ 7, we further show that the very general fiber of f is Hodge-generic. Finally, as an application of our classification, we confirm an expectation of Li--Tosatti that when f is not isotrivial, the special Kähler metric on B has positive holomorphic sectional curvature on the regular values of f.
ER -