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

Abstract

We study the Mumford-Tate conjecture for hyper-Kähler 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.

39 pages; Proof of Theorem A has been shorten with new ideas; Remove the restriction on second Betti number; Section 4 and 5 are simplified accordingly;