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Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-Kähler varieties

Zhichao Tang, Haitao Zou

Abstract

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.

Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-Kähler varieties

Abstract

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 -adic Galois representations attached to hyper-Kähler varieties with second Betti number . 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 -adic algebraic monodromy groups in higher degrees to those in degree via the theory of Frobenius tori and the Looijenga-Lunts-Verbitsky Lie algebra.
Paper Structure (33 sections, 55 theorems, 148 equations)

This paper contains 33 sections, 55 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Abelian varieties and their relatives
  3. Hyper-Kähler varieties and the semisimple Mumford--Tate conjecture
  4. Mumford--Tate conjecture in a family
  5. Strategy of proof
  6. Rank comparison and $p$-adic methods
  7. Big monodromy and specialization
  8. Semisimplicity at Galois generic fibers
  9. Relations with previous works and applications
  10. Motivic Galois group and compatible system
  11. André motives and realizations
  12. Motivic Galois groups and Mumford--Tate conjecture
  13. Compatibility of systems of Galois representations
  14. LLV representations of hyper-Kähler varieties
  15. Looijenga--Lunts--Verbitsky Lie algebra
  16. ...and 18 more sections

Key Result

Theorem 1.2.3

Let $X$ be a hyper-Kähler variety over a finitely generated field $K$ with $b_2(X) \geq 4$. Then $(\mathop{\mathrm{MTC}}\nolimits_2)$ holds for $X$.

Theorems & Definitions (131)

  • Conjecture : $\mathop{\mathrm{MTC}}\nolimits_i$
  • Definition 1.2.1
  • Theorem 1.2.3: Tankeev, André
  • Theorem A: \ref{['thm:rank of ell algebraic monodromy groups']}
  • Theorem B: \ref{['thm:semisimple Mumford--Tate conjecture even part']}
  • Corollary 1.2.6
  • Theorem C
  • Corollary 1.3.2
  • Conjecture : Ab
  • Theorem 1.5.2
  • ...and 121 more