TY - JOUR

T1 - Galois representations on the cohomology of hyper-Kähler varieties

AU - Floccari, Salvatore

N1 - Funding Information: I am most grateful to Ben Moonen and Arne Smeets for their help and encouragement. I wish to thank Lie Fu for many useful discussions around the topics of this paper. I also thank the referee for his/her careful review. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

PY - 2022/5

Y1 - 2022/5

N2 - We show that the Andr\'{e} motive of a hyper-K\"{a}hler variety \(X\) over a field \(K \subset \mathbb{C}\) with \(b_2(X)>6\) is governed by its component in degree \(2\). More precisely, we prove that if \(X_1\) and \(X_2\) are deformation equivalent hyper-K\"{a}hler varieties with \(b_2(X_i)>6\) and if there exists a Hodge isometry \(f\colon H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})\), then the Andr\'e motives of \(X_1\) and \(X_2\) are isomorphic after a finite extension of \(K\), up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the \'{e}tale cohomology of \(X_1\) and \(X_2\) are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-K\"{a}hler varieties for which the Mumford--Tate conjecture is true.

AB - We show that the Andr\'{e} motive of a hyper-K\"{a}hler variety \(X\) over a field \(K \subset \mathbb{C}\) with \(b_2(X)>6\) is governed by its component in degree \(2\). More precisely, we prove that if \(X_1\) and \(X_2\) are deformation equivalent hyper-K\"{a}hler varieties with \(b_2(X_i)>6\) and if there exists a Hodge isometry \(f\colon H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})\), then the Andr\'e motives of \(X_1\) and \(X_2\) are isomorphic after a finite extension of \(K\), up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the \'{e}tale cohomology of \(X_1\) and \(X_2\) are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-K\"{a}hler varieties for which the Mumford--Tate conjecture is true.

KW - math.AG

KW - 14C30, 14F20, 14J20, 14J32

KW - Galois representations

KW - Motives

KW - Hodge theory

KW - Hyper-Kähler varieties

UR - http://www.scopus.com/inward/record.url?scp=85122292164&partnerID=8YFLogxK

U2 - 10.1007/s00209-021-02923-3

DO - 10.1007/s00209-021-02923-3

M3 - Article

VL - 301

SP - 893

EP - 916

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1

ER -