# Shimura varieties and Maximal conditions

Working with Shimura varieties, I have been convinced to call them (or the families giving rise to them especially in $A_{g}$) somehow the "maximal" families. The motivation of this, has been for example the following results:

Consider a subvariety $S$ of $A_{g}$ over which there exists a variation of Hodge structures $\mathcal{V}$ (One can equip $A_{g}$ with a level structure $m$ for $m\geq 3$)

1) The (generic) monodromy group $Mon^{0}$ is a normal subgroup of $MT^{der}$, the generic Mumford-Tate group of $\mathcal{V}$. If $S$ is a Shimura subvariety, by a result of Andre, the monodromy group attains it's maximal bound, i.e. $Mon^{0}=MT^{der}$.

2) Viehweg-Zuo results on Arakelov equalities for Shimura curves: If $S$ is a Shimura curve in $A_{g}$, then $S$ attains the maximal bound in Arakelov inequality.

My question is: Are there other results that support this interpretation of Shimura families as the maximal ones (in some particular sense of $"maximality"$ as above)?

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Just a remark : your property (1) i.e $Mon^0 = MT^{der}$ does not characterise Shimura subvarieties at all - indeed it holds for any subvariety of $A_{g,m}$ provided it contains a CM point. (I am assuming, by $Mon^0$ you mean the neutral component of the Zariski closure of the monodromy representation attached to the smooth locus of your subvariety) – user42721 Dec 10 '13 at 13:33