# Which mapping class group representations come from algebraic geometry?

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?

• If infinite dimensional representations are ok you can take the ring of functions $\mathcal O(Rep(\pi_1(\Gamma_g),G))$ of the representation variety into some algebraic group $G$. Or you could take the homology of the representation variety. Nov 4, 2014 at 14:26
• This question already has nontrivial answers, but trivial answers are useful, too. (1) The tensor product of the symplectic representation with itself is a new PVHS. (2) Every finite index subgroup gives a permutation representation, which is a (boring weight 0) PVHS. Even if they factor through $Sp_{2g}(\mathbb Z)$, they don't come from the defining representation. And the group is residually finite, so many don't. Dec 17, 2014 at 16:09

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 $M_g$.

• Re: abelian not being essential - true, but far from easy, see (on arxiv) the recent paper of Larsen, Lubotzky, Malestein, Grunewald. Nov 5, 2014 at 5:07

Kontsevich constructed a family of varieties over moduli space with interesting cohomology in the middle dimension. I don't think anyone proved that it is not the symplectic representation, but Kontsevich conjectured that is faithful.

• Is it clear they're actually representations of the mapping class group? I think I might have asked Kontsevich back in 2006 and I recall he shrugged. Nov 5, 2014 at 2:37
• Yes, it works. There are two difficulties. The first is that there are lots of choices; you just have to make all of them. That is: take the universal curve over the moduli stack of double covers of a genus $g$ curve ramified in $n=4g-4$ points. This maps to $M_g$, so its cohomology is a local system there. The second problem is the stackiness: when you take cohomology, it's like taking $\mathbb Z/2$ invariants, but the interesting cohomology all has action by $-1$. But you can just take the tensor square, or some other ad hoc option. Nov 5, 2014 at 2:57
• It would be interesting if you could flesh out the details of this approach. Nov 5, 2014 at 4:43

See:

On a function on the mapping class group of a surface of genus 2 Ryoji Kasagawa (Topology and its Applications, 2000), and the closely related beautiful paper of Looijenga. (Prym Representations of Mapping Class Groups, Geom. Ded. 1997).