论文标题
绝对Hodge和$ \ Ell $ - ADIC单片
Absolute Hodge and $\ell$-adic Monodromy
论文作者
论文摘要
令$ \ mathbb {v} $为$ k $ -variety $ s $上的hodge结构的动机变化，让$ \ mathcal {h} $是关联的$ k $ -k $ -Algebraic hodge bundle，lit $ f textrm {aut}（aut}（\ natuy}（\ mathbb {C} $ a自动）。绝对的hodge cumenture预测，给定的hodge vector $ v \ in \ mathcal {h} _ {\ mathbb {c}，s} $上方$ s \ in s（\ mathbb {c}）$上的$ s \ in s（\ mathbb {c}）$ in INS $ \ \ \ \ thebb {v} _ {v} _ { \ Mathcal {h} _ {\ Mathbb {C}，s_σ} $是hodge，位于$ \ mathbb {v} _ {s_σ} $中。在我们有代数子变量$ z \ subset s _ {\ mathbb {c}} $中，其中包含$ s $的情况下，我们研究了这个问题。使用$ \ Mathbf {h} _z $和$ \ Mathbf {h} _ {Z_σ} $来自复杂的理论和$ \ ell $ -ADIC本地系统，我们建立了一个标准，我们建立了一个标准，暗示对$ \ Mathbf}的$ v $的绝对hodge undgeure to $ v $对$ v $的$ v $构成。然后，我们使用我们的标准来建立绝对Hodge猜想的新案例。
Let $\mathbb{V}$ be a motivic variation of Hodge structure on a $K$-variety $S$, let $\mathcal{H}$ be the associated $K$-algebraic Hodge bundle, and let $σ\in \textrm{Aut}(\mathbb{C}/K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector $v \in \mathcal{H}_{\mathbb{C}, s}$ above $s \in S(\mathbb{C})$ which lies inside $\mathbb{V}_{s}$, the conjugate vector $v_σ \in \mathcal{H}_{\mathbb{C}, s_σ}$ is Hodge and lies inside $\mathbb{V}_{s_σ}$. We study this problem in the situation where we have an algebraic subvariety $Z \subset S_{\mathbb{C}}$ containing $s$ whose algebraic monodromy group $\mathbf{H}_Z$ fixes $v$. Using relationships between $\mathbf{H}_Z$ and $\mathbf{H}_{Z_σ}$ coming from the theories of complex and $\ell$-adic local systems, we establish a criterion that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic condition on $\mathbf{H}_{Z}$. We then use our criterion to establish new cases of the absolute Hodge conjecture.
