Notes on the P = W conjecture, II: the cohomology of character varieties

So what do we actually know about the cohomology of the (twisted) character variety $M = M_B(C, n, d)$?

Some information is collected in this article of Tamás Hausel and Fernando Rodriguez-Villegas; I review some of it here.

It will be convenient to fix a basis for $H^*(C, \mathbb{Q})$.  We write:

• $1 \in H^0(C, \mathbb{Q})$ for the identity, 
• $e_\eta \in H^1(C, \mathbb{Q})$ for a basis,
• $[C] \in H^2(C, \mathbb{Q})$ for the fundamental class.

Tautological classes. As with many moduli spaces, cohomology classes can be constructed from tautological objects.

As a warm up, consider $n = 1$.  Note that a locally constant GL(1) bundle on a curve with one puncture necessarily extends across the puncture, since the monodromy around that puncture is a commutator, hence 1.  The moduli space is $M_B(C, 1, d) = (\mathbb{C}^*)^{2g}$, and it comes with a map $M \times C \to BGL(1)$.  This map is algebraic in the first factor and locally constant in the second; in any case we write $c_1 \in H^*(M \times C)$ for the pullback of the tautological generator of $H^*(BGL(1), \mathbb{Q})$.  It is possible to see that this has a Kunneth decomposition of the form

$$c_1 = \sum e_\eta \otimes \epsilon_{1,\eta} \in H^1(C, \mathbb{Q}) \otimes H^1((\mathbb{C}^*)^{2g}, \mathbb{Q})$$

(One way to see this is by passing across the (abelian) nonabelian isomorphism, which turns
$ (\mathbb{C}^*)^{2g})$ into the cotangent bundle of the Jacobian, and identifies the tautological bundle with the Poincare bundle.)

Since $H^1((\mathbb{C}^*)^{2g}, \mathbb{C}) = {}^{1,1} H^1((\mathbb{C}^*)^{2g}, \mathbb{C}) $, we know that the classes $\epsilon$ have homogenous Hodge type.  It's easy to see however that $c_1$ cannot -- it's a self-conjugate element of  ${}^{1,2}H^2 \oplus {}^{2,1} H^2$ -- which is no contradiction, since the map $C \times (\mathbb{C}^*)^{2g} \to BGL(1)$ is not algebraic.

Now for general $n$, taking determinants give maps $M_B(C, n, d) \to M_B(C, 1, d)$; pulling back the $\epsilon$ classes, we get classes which we give the same name on $M_B(C, n, d)$.

On the other hand, a GL(n) bundle with scalar monodromy around the puncture can be extended to a PGL(n) bundle over the whole curve, we get a tautological principal PGL(n) bundle over $M \times C$.  It is coherent in the $M$ factor and constructible in the $C$ factor, but in any case has an underlying topological bundle $\mathbb{U} \to M \times C$.  This gives a map $M \times C \to \mathrm{\bullet}/PGL(n)$, which is algebraic in the first factor, and locally constant in the second.   Pulling back the tautological classes in $H^*(BPGL(n), \mathbb{Q}) = \mathbb{Q}[\overline{c}_2, \overline{c}_3, \ldots, \overline{c}_n]$ give classes (which we give the same names) in $H^*(M \times C, \mathbb{Q})$.

We take their Kunneth components:

$$\overline{c}_k = (1 \otimes \beta_k) \oplus \left(\sum e_\eta \otimes \phi_{k, \eta} \right) \oplus ([C] \otimes \alpha_k)$$

To determine the weights of these classes, take a triangulation and write $\Delta_C$ as the simplicial scheme with the topology of $C$; it's a finite simplicial scheme.  Now the map $M_B(C; n, d) \times \Delta_C \to PBGL(n)$ is algebraic, and hence the pullback preserves Hodge structures.  On the other hand, the Hodge structure on $H^*(\Delta_C, \mathbb{Q})$ is the trivial Hodge structure, essentially because $\Delta_C$ is finite.  It follows immediately that $\beta_k, \phi_{k,\eta}, \alpha_k$ have the same homogenous Hodge type $(k,k)$ as $\overline{c}_k$.

(This argument for the weights is written up in this note.  The original argument of Tamás and Fernando for the weights of the $\phi_k$ are more complicated, though perhaps not essentially different; they moreover do not quite manage to determine the weights of the $\alpha_k$.  Those arguments however do not explicitly require the use of simplicial schemes.)

In sum we have classes:

• $\epsilon_{1,\eta} \in {}^{1,1} H^1(M_B(C, n, d), \mathbb{C})$
• $\alpha_{k} \in {}^{k,k} H^{2k-2} (M_B(C, n, d), \mathbb{C})$
• $\phi_{k,\eta} \in {}^{k,k} H^{2k-1} (M_B(C, n, d), \mathbb{C})$
• $\beta_{k} \in {}^{k,k} H^{2k} (M_B(C, n, d), \mathbb{C})$

Generation.  It is a theorem of Markman that the above classes generate the cohomology; he however proves this across the nonabelian Hodge theorem on the Hitchin moduli space using an identification of it with a moduli space of sheaves, which we will discuss next time.