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, 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, 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.

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, 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, 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.

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 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, 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, 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.

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, 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, 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.