Let $\Gamma\_g$ be the mapping class group of a closed oriented surface $\Sigma$ of genus $g$. There is a natural surjection $t \colon \Gamma\_g \to \mathrm{Sp}(2g,\mathbf Z)$ which sends a mapping class to the induced action on $H^1(\Sigma,\mathbf Z)$. Composing $t$ with any representation of the symplectic group produces a large number of linear representations of $\Gamma\_g$. 


These are only a small fraction of all representations of the mapping class groups. Others can for instance be obtained from 3D TQFTs or by from different constructions involving lower central series. My question is however whether the symplectic representations are the only ones that can be defined "algebro-geometrically". 


Let me ask a more concrete question. A representation of $\Gamma\_g$ is the same as a local system on the moduli space of curves of genus $g$, $M\_g$. For a representation which factors through $\mathrm{Sp}(2g,\mathbf Z)$ this local system underlies a polarized variation of Hodge structure, since it is pulled back from a PVHS on the Shimura variety parametrizing principally polarized abelian varieties of genus $g$. Is the converse true - if a local system (say with $\mathbf Q$ coefficients) on $M\_g$ underlies a PVHS, is it isomorphic to one of the symplectic local systems?



Which mapping class group representations come from algebraic geometry?
Dan,


Although I'm no longer very active on MO, I thought I'd make a few comments, since your question is an interesting one (and you're not anonymous).


The paper of Looijenga referenced in Igor's answer would show that there are "algebro-geometric" representations of $\Gamma\_g$ which **don't** factor through $Sp(2g,\mathbb{Z})$. In summary, he takes a finite abelian [but this shouldn't be essential] Galois topological covering $\tilde \Sigma\to \Sigma$ and looks at the finite index subgroup $\tilde \Gamma\_g\subset \Gamma\_g$ of elements which lift to $\tilde \Sigma$ and act trivially on the Galois group. The point is that $\tilde \Gamma\_g$ will act on $H^1(\tilde \Sigma)$, and in this way he gets new representations (of the subgroup, but you can always induce up to $\Gamma\_g$). To see that this comes from a PVHS over the stack $M\_g$,
 consider the moduli stack $\tilde M$ parameterizing maps $f:\tilde C\to C$ of curves which topologically the same as $\tilde \Sigma\to \Sigma$. We have a map $\pi:\tilde M\to M\_g$ sending $\tilde C\to C$ to $C$. Let $V$ be the VHS on $\tilde M$ with $H^1(\tilde C)$ as it's fibre. Then $\pi\_\*V$ is a VHS on $M\_g$ which gives rise to Looijenga's.


If you relax "comes from algebraic geometry" to allow monodromies of motivic variations of mixed Hodge structures, and I don't see why you wouldn't, then there are even more interesting possibilities gotten by looking at the (generally) singular spaces of semistable vector bundles over the universal curve of $M\_g$.


