# What are the advantages to viewing Hermitian symmetric domains as a moduli for Hodge structures?

I am reading Milne's An Introduction to Shimura Varieties. At the moment, I am unable to see the forest from the trees with Deligne's approach.

Here's the basic set-up. Let $$X$$ be a Hermitian symmetric domain. Milne claims these are all simply connected (I know of no proof), he considers variation of Hodge structures in which the local systems $$H^n(V_s,\mathbb{Q})$$ are constant (recall each of these carry a Hodge decomposition / structure and these are required to ary continuations with $$s\in X$$).

Going backwards, Milne shows how to give the set of Hodge structures on a vector space $$V$$ with some conditions the structure of a Hermitian symmetric domain.

All questions below are more or less related / different aspects of a bigger question: Why variation of Hodge structures for Shimura variety?

Question 1: What are the advantages to this point of view? What are the things a student should be looking out for?

Question 2: Going backwards, let me try an example and hopefully someone has insight. Let $$X$$ be the Hermitian symmetric domain $$\mathcal{H}_1$$, the upper half plane. What are the associated variation of Hodge structures one studies here?

Question 3: Shimura varieties play a role in the Langlands program. How does / Does the variation of Hodge structures enter play in this story as well?

Edit: I am convinced, due to the lack of good references for Shimura varieties, that if I find the correct answer to this, it is worth providing a detailed write-up later. If I have time, I will make this questions more precise.

• This is not my area of expertise, but you might get some insight by reading some of Phillip Griffiths’s seminal work on VHS. Jul 24, 2023 at 19:02
• @TedShifrin Thanks for the suggestion. I'm not sure which paper of Griffith's that would be? I looked it up and I only see Loring Tu's exposition on the theory: publications.ias.edu/sites/default/files/variationsofhodge.pdf Jul 24, 2023 at 19:31
• Volume 3 of Selected Works by Phillip A. Griffiths has all the relevant papers. The motivation for VHS comes from his various papers on "Periods of Integrals on Algebraic Manifolds." An excellent summarizing article appeared in the Bull. Amer. Math. Soc. 76 (1970), 228-296. More modern exposition on VHS appears in "Infinitesimal Variations of Hodge Structure" (I,II,III), the first two with coauthors Carlson, Green, and Harris, and Harris respectively, all appearing in 1983. Jul 24, 2023 at 21:35