论文标题
Kummer类型及其固定基因座的符合性品种的一组符合性。
Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci
论文作者
论文摘要
我们描述了在中间$ \ ell $ -ADIC共同体上的galois动作，该共同体是光滑，投影四倍的$ k_a（v）$，这些$ k_a（v）$是在Abelian Surface $ a $ a $ a abel v $ v $ abelian Surface $ a $ a $ a $ a $ a sheaves sheaves上的阿尔巴内斯形态的纤维中发生的。我们显示此动作取决于$ h^2_ {ét}（a _ {\ bar {k}}}，\ Mathbb {q} _ \ ell（1））$和子组$ g_a（v）\ leqslant（a \ leqslant（a \ times \ times \ hat {a p {a a a}）$ v $，这概括了Hassett和Tschinkel [HT13]在$ \ Mathbb {C} $上进行的分析。结果，在数字字段上，我们提供了一个条件，在该条件下，$ k_2（a）$和$ k_2（\ hat {a}）$不是同等的。 $ g_a（v）$的点对应于$ k_a（v）$。超过$ \ mathbb {c} $，已知它们是符合性的，并包含在地图$ \ mathrm {aut}（aut}（k_a（v））\ to \ mathrm {o}（h^2（h^2（k_a（k_a（k_a（v），\ mathbb {z}）））））中）$。我们描述了所有品种的内核$ k_a（v）$ dimension至少$ 4 $。 当$ k_a（v）$在特征0的字段上是一个四倍时，互动的定点基因座包含K3表面，其周期类别占中间同胞的很大一部分。我们检查了固定的基因座$ k_a（0，l，s）$以上的$ \ mathbb {c} $，其中$ l $是$（1,3）$ - 极化，发现在Lagrangian的Lagrangian振动下，将K3表面纤维化为$ k_a（0，l，s）$。
We describe the Galois action on the middle $\ell$-adic cohomology of smooth, projective fourfolds $K_A(v)$ that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surface $A$ with Mukai vector $v$. We show this action is determined by the action on $H^2_{ét}(A_{\bar{k}},\mathbb{Q}_\ell(1))$ and on a subgroup $G_A(v) \leqslant (A\times \hat{A})[3]$, which depends on $v$. This generalizes the analysis carried out by Hassett and Tschinkel [HT13] over $\mathbb{C}$. As a consequence, over number fields, we give a condition under which $K_2(A)$ and $K_2(\hat{A})$ are not derived equivalent. The points of $G_A(v)$ correspond to involutions of $K_A(v)$. Over $\mathbb{C}$, they are known to be symplectic and contained in the kernel of the map $\mathrm{Aut}(K_A(v))\to \mathrm{O}(H^2(K_A(v),\mathbb{Z}))$. We describe this kernel for all varieties $K_A(v)$ of dimension at least $4$. When $K_A(v)$ is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed loci in fourfolds $K_A(0,l,s)$ over $\mathbb{C}$ where $l$ is a $(1,3)$-polarization, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of $K_A(0,l,s)$.
