Suppose I have a finite set of points in $\mathbb{P}^1$ (over the complex numbers), and suppose that at each point, I am given a [**Edit:** quasi-unipotent] conjugacy class in $Sp(2g,\mathbb{Z})$ for $g$ a fixed positive integer. Then near each point, I have an analytic neighborhood where I can construct a family of complex tori whose $H^1$ varies according to the monodromy. Furthermore, if there are representatives of the conjugacy classes whose product is $1 \in Sp(2g, \mathbb{Z})$, then I can at least glue these local families into a $C^\infty$ family of $2g$-dimensional tori over the punctured $\mathbb{P}^1$.


**First question:** If we have representatives whose product is identity, does there exist a family of abelian varieties over the punctured line whose $H^1$ has the specified monodromy at the points? I think we can do this by choosing a variation of Hodge structure on the corresponding local system of rank $2g$ groups, and taking a quotient, but it's all a bit cloudy to me.


**Second question:** Given a particular puncture and its assigned conjugacy class in $Sp(2g,\mathbb{Z})$, does there exist a family of genus $g$ curves over a small neighborhood of the puncture whose $H^1$ has monodromy in the specified conjugacy class? (I suppose I should just ask this about the punctured affine line.) In the $C^\infty$ world, this can be done with a [mapping torus](http://en.wikipedia.org/wiki/Mapping_torus), but I don't know how it works holomorphically.


**Third question:** If the answers to the first two questions are "yes", can we make our family of abelian varieties out of Jacobians of genus $g$ curves? Up to some finite cover problem, I think this is basically asking if there exists a family of genus $g$ curves whose $H^1$ has the specified monodromy.


From [David Brown's answer here](https://mathoverflow.net/questions/835/algorithms-for-semistable-reduction-of-families-of-curves/904#904), it seems too much to ask for an explicit minimal model curve over the punctures when $g > 2$, but I'm just hoping for existence away from those points.



Does there exist a family of curves (or abelian varieties) on the punctured line with specified monodromy on H^1?
There are several restrictions. First, the existence of potential semistable reduction (Grothendieck-Mumford) implies that your representative $\sigma$ at every puncture must be quasi-unipotent of level 2, i.e, there exists a positive integer $N$ such that $(\sigma^N-1)^2=0$. Second, the Zariski closure $G\subset Sp\_{2g,Q}$ of the global monodromy group $\Gamma \subset Sp(2g,Z)$ must satisfy the following properties (Deligne).


1. Its identity component $G^0$ is semisimple.
2. All absolutely simple quotients of $G^0$ (over the field $C$ of complex numbers) are classical algebraic groups (A\_r,B\_r,C\_r,D\_r) and ``their" natural nontrivial irreducible subrepresentations in $C^{2g}$ are (fundamental) minuscule, i.e., the corresponding set of weights is an orbit of the Weyl group.


