论文标题
关于超卡勒品种的共同体的加洛伊斯表示
Galois representations on the cohomology of hyper-Kähler varieties
论文作者
论文摘要
我们表明，在字段$ k \ subset \ subset \ mathbb {c} $上，$ b_2（x）> 6 $在$ k \ subset \ mathbb {c} $上的安德烈动机$ x $ 6 $由其组件以$ 2 $为准。更准确地说，我们证明，如果$ x_1 $和$ x_2 $是变形等效的超级kähler品种，则具有$ b_2（x_i）> 6 $，并且是否存在hodge等轴测$ f \ colon h^2（x_1，x_1，x_1，x_1，\ nathb {q} {q}） $ x_1 $和$ x_2 $的有限扩展为$ k $，在存在非平凡的奇数共同体学的情况下，是同构的。结果，$ x_1 $和$ x_2 $的埃塔尔共同体的Galois表示形式也是同构的。我们证明了有限场上品种的类似结果，该品种可以将其提升为Mumford-tate猜想是正确的Hyper-Kähler品种。
We show that the André motive of a hyper-Kähler 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-Kähler 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 André 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 étale 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-Kähler varieties for which the Mumford--Tate conjecture is true.
