论文标题
动机镜子对称性和$χ$独立于Higgs捆绑在任意特征中
Motivic mirror symmetry and $χ$-independence for Higgs bundles in arbitrary characteristic
论文作者
论文摘要
我们证明，$ \ mathrm {sl} _n $和$ \ m m i \ m mathrm {pgl} _n $ -higgs捆绑等级的（扭曲的）动机（扭曲的Orbifold）动机（SL} _n $和$ \ Mathrm {pgl} _n $ -higgs捆绑在一个平稳封闭的范围内，该曲线是在一个封闭式的平稳曲线上，因为该曲线是无处不在的，即无处不在的态度。 The equality of twisted orbifold Hodge numbers of these moduli spaces was conjectured by Hausel and Thaddeus and recently proven by Groechenig, Ziegler and Wyss via $p$-adic integration and then by Maulik and Shen using the decomposition theorem, an analysis of the supports of $D$-twisted Hitchin fibrations and vanishing cycles.我们以零特征的证明将毛利克和沉的几何思想与贝蒂实现的亚伯动机的保守性结合在一起。要应用后者，我们证明相关动机是阿贝利安。特别是，我们证明了$ \ mathrm {sl} _n $ -higgs moduli Space的动机是Abelian，这是我们以前在$ \ mathrm {gl} _n $ -case中的工作。然后，我们使用动机附近的周期来从特征零中推断出积极特征的结果。使用相同的想法，我们证明了$ \ mathrm {gl} _n $ -higgs捆绑包的动机$χ$独立。
We prove that the (twisted orbifold) motives of the moduli spaces of $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles of coprime rank and degree on a smooth projective curve over an algebraically closed field in which the rank is invertible are isomorphic in Voevodsky's triangulated category of motives. The equality of twisted orbifold Hodge numbers of these moduli spaces was conjectured by Hausel and Thaddeus and recently proven by Groechenig, Ziegler and Wyss via $p$-adic integration and then by Maulik and Shen using the decomposition theorem, an analysis of the supports of $D$-twisted Hitchin fibrations and vanishing cycles. Our proof in characteristic zero combines the geometric ideas of Maulik and Shen with the conservativity of the Betti realisation on abelian motives; to apply the latter, we prove that the relevant motives are abelian. In particular, we prove that the motive of the $\mathrm{SL}_n$-Higgs moduli space is abelian, building on our previous work in the $\mathrm{GL}_n$-case. We then use motivic nearby cycles to deduce the result in positive characteristic from that in characteristic zero. Using the same ideas, we prove motivic $χ$-independence for $\mathrm{GL}_n$-Higgs bundles.
