Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-Kähler varieties
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914344141324288 |
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| author | Tang, Zhichao Zou, Haitao |
| author_facet | Tang, Zhichao Zou, Haitao |
| contents | In this paper, we establish two main results concerning the Mumford-Tate conjecture for hyper-Kähler varieties. First, we prove the conjecture for the semisimplified $\ell$-adic Galois representations attached to hyper-Kähler varieties with second Betti number $b_2 \geq 4$. As a direct consequence, we deduce that the Hodge conjecture implies the Tate conjecture for powers of hyper-Kähler varieties. Second, we show that the Mumford-Tate conjecture for hyper-Kähler varieties is invariant under deformation. The proofs rely on comparing the ranks of $\ell$-adic algebraic monodromy groups in higher degrees to those in degree $2$ via the theory of Frobenius tori and the Looijenga-Lunts-Verbitsky Lie algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_19835 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-Kähler varieties Tang, Zhichao Zou, Haitao Algebraic Geometry Number Theory 14J20, 14J42, 14F20 In this paper, we establish two main results concerning the Mumford-Tate conjecture for hyper-Kähler varieties. First, we prove the conjecture for the semisimplified $\ell$-adic Galois representations attached to hyper-Kähler varieties with second Betti number $b_2 \geq 4$. As a direct consequence, we deduce that the Hodge conjecture implies the Tate conjecture for powers of hyper-Kähler varieties. Second, we show that the Mumford-Tate conjecture for hyper-Kähler varieties is invariant under deformation. The proofs rely on comparing the ranks of $\ell$-adic algebraic monodromy groups in higher degrees to those in degree $2$ via the theory of Frobenius tori and the Looijenga-Lunts-Verbitsky Lie algebra. |
| title | Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-Kähler varieties |
| topic | Algebraic Geometry Number Theory 14J20, 14J42, 14F20 |
| url | https://arxiv.org/abs/2602.19835 |