嗅得手植棠梨初发轻黄蕊
读了读 Topos of Music, 感觉我又想学数学了..
这篇文章是老李推的 Automorphic functions on moduli spaces of bundles on curves of bundles on curves over local field: A survey. 这是 Alexander Braverman 和 David Kazhdan 的文章, 具体内容是 Kazhdan 在 ICM 2022 上的 plenary talk.
我们现在要解决的问题就是: 如何在 Stack 上定义 function.
对于 algebraic variety over a local field , 是一个拓扑空间 endowed with a natural topology.
Defn 2.1: A function is smooth if
(a) non-archimedian and is locally constant.
(b) archimedian and (locally) there exists a closed embedding where smooth var. over and a -function s.t. .
于是, 我们首先得到空间 space of smooth function, 以及 subspace of the compact-supports.
对于 line bundle over , 刨掉 zero section 得到 -torsor.
现在, 我们 set
这里的 表示的是 $F^\ast $ 作为群在 上的某个作用. 我们得到的 就是一个 locally trivial -bundle. 同样地, 我们可以定义其 smooth section by , 以及 subspace of the compact-supports.
最后, 如果不想 take trivial bundle, 我们可以很自然地想到 line bundle of top differential form. 取 , 则是我们所熟悉的, 有一个自然 Hermitian product 结构的 subspace. 自然地, 能够考虑 为其 Hilbert space completion.
现在, 我们希望 extend 如上 defn 到 a class of algebraic stacks 上.
Defn 2.3. An algebraic stack is admissible, if there exists a presentation of as , where smooth variety, affine alg. gp. Denote projection.
将 表现为 quotient 被称作一个 admissible presentation.
Remark 2.4. 所有 smooth admissible stack of finite type 总可以表示为 , with smooth. 所有 admissible stack 都是自动 locally of finite type的.
Defn 2.5. (1) For non-archimedian, is an admissible stack of finite type over . Choose an admissible presentation for some variety and set
where the latter stands for -coinvariants.
(2) For non-archimedian, locally of finite type. Then we can write as a direct limit of open substack of finite type. .
(3) For archimedian, we make an analogous definition but take coinv. in the cat. of topological spaces where endowed with Frechet topology.
Prop. 2.6. is an admissible stack of finite type, then does not depend on a choice of an admissible presentation .
这是原文中提到的一个特例: stack over 的情况.
对于 non-archimedian. 我们来 construct 一些 elements in .
考虑 , defined over , regular scheme over . 此外, assume defined over , 于是我们可以 define obvious map by
现在, 对于 , 我们有 complex line bundle 拥有 canonical trivialization on . 于是, 考虑此前定义的 (我认为这里应该是记错了, 这里的 为什么不是 ?) 我们得到
复合两个映射, 我们得到
未完待续