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.

Notes on the $P = W$ conjecture, I: filtrations.

Here I want to make some notes on the P = W conjecture of Mark de Cataldo, Tamas Hausel, and Luca Migliorini.

Take $C$ some fixed base curve, and let $M_{Dol}(C; n, d)$ be the moduli space of rank $n$ semistable Higgs bundles of degree $d$.  On the other hand let $M_{B}(C; n, d)$ be the moduli space of local systems on $C \setminus \bullet$ with monodromy $e^{2 \pi i d/ n}$ around $\bullet$.

Nonabelian Hodge theory identifies these spaces; when $n, d$ are coprime, both spaces are smooth and the identification is a diffeomorphism.  In particular we have:

$$H^*(M_{Dol}(C; n, d), \mathbb{Q}) = H^*(M_{B}(C; n, d), \mathbb{Q})$$

The cohomology carries various additional structures on each side, and one can ask whether these behave at all reasonably under this identification.

There is no a-priori reason to expect such a thing: the identification between the spaces is not even complex analytic.  The $P = W$ conjecture asserts an identification of two such filtrations:

$$P_k H^* (M_{Dol}(C; n, d), \mathbb{Q}) = W_{2k} H^*(M_{B}(C; n, d), \mathbb{Q})$$

Above the perverse Leray filtration on the LHS is taken with respect to the Hitchin fibration.

Let me note down some basic facts about these filtrations, and some others.  We take the usual convention for denoting filtrations: lower subscripts denote an increasing filtration, and upper subscripts a decreasing filtration.


The weight filtration. According to Deligne, the cohomology of a complex algebraic variety $X$ carries in general an increasing filtration,

 $$ 0 = W_{-1} H^j(X, \mathbb{Q}) \le W_0 H^j(X, \mathbb{Q})  \le \cdots \le W_{2j} H^j (X, \mathbb{Q}) = H^j (X, \mathbb{Q}) $$

It's normalized by the condition that when $X$ is smooth and proper, $W_k H^* (X, \mathbb{Q}) = \bigoplus H^{\le k}(X, \mathbb{Q}) $.


If $f: X \to Y$ is any map of algebraic varieties, the induced pullback preserves the weight filtration: $f^* (W_i H^* (Y, \mathbb{Q}) ) \subset
W_i H^* (X, \mathbb{Q})$.  In fact it preserves this filtration strictly, which is the even stronger property that

$$ f ^* W_i H^*(Y,\mathbb{Q})  = W_i H^* (X, \mathbb{Q} ) \cap f^* H^*(Y, \mathbb{Q})$$

Because the cup product can be interpreted as a pullback from the diagonal and weights are compatible with the Kunneth decomposition, one learns $W_i \cup W_j \subset W_{i+j}$.  (There is nothing to be learned from the strictness, since any class can be written as itself cup the unit.)

The weights on $H^i(X, \mathbb{Q})$ begin at $i$ if $X$ is smooth, and end at $i$ when $X$ is proper.

The Hodge filtration.  Brought to you again by Deligne, this is a decreasing filtration on the cohomology of a complex algebraic variety.

$$ H^i(X, \mathbb{C}) = F^0 H^i(X, \mathbb{C}) \ge F^1 H^i(X, \mathbb{C}) \ge \cdots $$

When $X$ is smooth and projective, there's the Hodge decomposition $H^n(X, \mathbb{C}) = \bigoplus H^{p,n-p}(X, \mathbb{C})$, and in this case $F^{p} H^n(X, \mathbb{C}) = \bigoplus H^{\ge p, \cdot}(X, \mathbb{C})$.

In general, the Hodge and weight filtrations interact in the following way: the Hodge filtration makes $Gr^W_k H^{*}(X, \mathbb{C})$ into a pure Hodge structure of weight $k$.  This just means that, for every $k, p$, the following is a direct sum decomposition:

$$Gr^W_k H^{*}(X, \mathbb{C}) = F^p \left(
Gr^W_k H^{*}(X, \mathbb{C}) \right) \oplus  \overline{F^{k + 1 - p}} \left(
Gr^W_k H^{*}(X, \mathbb{C}) \right) $$

This means in particular that if $p + q > n$, we have $F^p \cap \overline{F}^q \cap W_n = 0$.  This gives us a Hodge grading on a piece of the complex cohomology: if

$${}^{p,q}H^{i}(X, \mathbb{C}) := F^{p}
H^{i}(X, \mathbb{C}) \cap \overline{F}^q
H^{i}(X, \mathbb{C}) \cap W_{p+q}H^{i}(X, \mathbb{C})$$

we have

$$\bigoplus_{i,p,q} {}^{p,q} H^i(X, \mathbb{C}) < H^*(X, \mathbb{C})$$

There are now nonvanishing groups where $p + q \ne i$, and also this sum is generally much smaller than $H^*(X, \mathbb{C})$.


The perverse Leray filtration.  If $B$ is some algebraic variety, then according to Beilinson, Bernstein, and Deligne ([BBD]), there is something called the perverse $t$-structure on its derived category of constructible sheaves $Con(B)$.  In particular, for integers $k$, there are "truncation functors" ${}^p \tau_{\le k} : Con(B) \to Con(B)$ and ${}^p \tau_{\ge k}: Con(B) \to Con(B)$, which fit into the truncation exact triangle.  That is, for any $F \in Con(B)$, we have

$$ {}^p \tau_{\le k} F \to F \to {}^p \tau_{\ge k + 1} F \xrightarrow{[1]} $$

The $\tau_{\le k}$ give an increasing filtration in the derived category of $F$.  That is, we have maps

$$0 \to \cdots \to \tau_{\le k - 1} F \to \tau_{\le k} F \to \tau_{\le k+1} F \to \cdots \to F$$

This gives rise to the definition of the perverse filtration.  It's an increasing filtration given by

$$ P_k H^*(B, F) := \mathrm{Image}(H^*(B, \tau_{\le k} F) \to H^*(B, F))$$

We do not know very much about this filtration.  One thing we do know is a rather different looking characterization in terms of taking hyperplane slices on $B$.

A map of algebraic varieties $f : X \to B$ gives rise to a perverse Leray filtration on the cohomology of $X$.  We shift the grading a bit, and denote

$$P_k H^*(X, \mathbb{Q}) := P_{k + \dim B} H^*(B, f_* \mathbb{Q})$$

In the setting when $X \to B$ is proper, the decomposition theorem of [BBD] asserts that in fact there is a splitting $f_* \mathbb{Q} = \bigoplus {}^p \mathcal{H}^i(f_* \mathbb{Q})$, which gives rise to a splitting of the perverse filtration upon taking cohomology.  This splitting is not canonical, but can be made so (in various different ways!) after choosing a relatively ample line bundle.

(Conversely, the perverse filtration plays a role in Mark and Luca's proof of the decomposition theorem using ``only Hodge theory''.)