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arXiv · 2602.19835

Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-K\"ahler varieties

Zhichao Tang·Haitao Zou
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Abstract

We study the Mumford-Tate conjecture for hyper-K\"ahler varieties. We identify the Mumford-Tate group with a Levi factor of the connected total $\ell$-adic monodromy group. It follows that the Mumford-Tate conjecture holds after semisimplification in every cohomological degree. We call this the semisimple Mumford-Tate conjecture. As applications, we derive a Hodge-to-Tate implication for powers, prove deformation invariance of the Mumford-Tate conjecture, establish the $\ell$-adic Nagai conjecture for Type I reduction, and extend Hui-Larsen's hyperspecial maximality theorem from degree two to total cohomology. The proof combines Pink's generation theorem for weak Hodge cocharacters with a multiplicity-weighted direct-sum construction and a rigidity argument for the graded cohomology algebra.

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Zhichao Tang, Haitao Zou. 2026-02-23. Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-K\"ahler varieties. https://arxiv.org/abs/2602.19835

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