Variation of Hodge structures associated to a hermitian symmetric domain

Question

Let $D$ be an irreducible hermitian symmetric domain. Then there exists a variation of Hodge structures $(h_s)_{s\in D}$ on a vector space $V$ satisfying specific conditions which depend on $D$ such that the space $S=\{h_s\}$ is isomorphic to $D$. Let $X$ be a smooth projective variety and let $\pi\colon X\to D$ be a morphism such that $H^n(X_s,\mathbb{Q})$ gives back the Hodge structure associated to $D$. (The number $n$ is clearly the weight of $(h_s)$ and $X_s$ is the fiber over $s\in D$.) I think such a morphism $\pi$ and such a variety $X$ always exist although I do not really know why. I also think that there actually existst many such $X$ and $\pi$.

We know that the hermitian symmetric domains are classified by the special nodes in the Dynkin diagrams. So we can write down a complete list of their isomorphism types. The question is now the following: Can someone please write down an example of $X$ for each of the isomorphism types of $D$, such as the Grassmannian, isotropic Grassmannian, etc.?

If I consider the set $\mathcal{X}$ of all morphism $X\to D$ which give rise to the variation of Hodge structures associated to $D$. Can someone give an interesting invariant $j$ such that $j(X_s)$ is independent of $X\in\mathcal{X}$. If someone can do, what does the function $s\mapsto j(X_s)$ on $D$ look like?

Maybe there exists already something similar in the literature which you can refer to. This of course also counts as an answer.

(Please do not downvote my question and try to answer instead. Thank you.)

Answer 1

You might want to look at the book "Mumford-Tate Groups and Domains: Their Geometry and Arithmetic", by Mark Green, Phillip Griffiths, and Matt Kerr. You should also look at the recent work of Colleen Robles. These might have answers to some of your questions.

Answer 2

I think you meant to say that $\pi:X\to D$ is a family of smooth projective varieties ($X$ won't itself be a projective variety, because $D$ is noncompact). Also, let me relax the condition to allow $h_s$ to be a direct summand of $H^n(X_s)$. Then as I understand (and I don't promise that I do) from Milne's Introduction to Shimura varieties, the existence of such a family is known in cases A,B,C and unkown in some others such as $E_6, E_7$, although it is expected. The basic idea is that if $D=G/K$ and $G$ embeds into a symplectic group, then $D$ embeds into a Siegel upper half plane, and you can restrict the universal family of abelian varieties.