TY - JOUR

T1 - On the Mumford–Tate conjecture for hyperkähler varieties

AU - Floccari, Salvatore

N1 - Funding Information: I am most grateful to Ben Moonen and Arne Smeets for their careful reading and the many comments, which substantially improved this text. I am also thankful to the anonymous referee for his/her, comments.

PY - 2022/7

Y1 - 2022/7

N2 - We study the Mumford–Tate conjecture for hyperkähler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O’Grady’s ten dimensional example, and all of their self-products. For an arbitrary hyperkähler variety whose second Betti number is not 3, we prove the Mumford–Tate conjecture in every codimension under the assumption that the Künneth components in even degree of its André motive are abelian. Our results extend a theorem of André.

AB - We study the Mumford–Tate conjecture for hyperkähler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O’Grady’s ten dimensional example, and all of their self-products. For an arbitrary hyperkähler variety whose second Betti number is not 3, we prove the Mumford–Tate conjecture in every codimension under the assumption that the Künneth components in even degree of its André motive are abelian. Our results extend a theorem of André.

KW - Hodge theory

KW - Hyperkähler varieties

KW - Motives

KW - Mumford–Tate conjecture

UR - http://www.scopus.com/inward/record.url?scp=85106459041&partnerID=8YFLogxK

U2 - 10.1007/s00229-021-01316-4

DO - 10.1007/s00229-021-01316-4

M3 - Article

AN - SCOPUS:85106459041

VL - 168

SP - 309

EP - 324

JO - Manuscripta mathematica

JF - Manuscripta mathematica

SN - 0025-2611

IS - 3-4

ER -