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E-polynomial of SL(2,C)-character varieties of complex curves of genus 3

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g=3 into $ SL(2,C), and also of the moduli space of twisted representations. The case of genus g=1,2 has already been done in [http://arxiv.org/abs/1106.6011]. We follow the geometric technique introduced in [http://arxiv.org/abs/1106.6011], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.


Introduction
Let X be a smooth complex projective curve of genus g ≥ 1, and let G be a complex reductive group. The G-character variety of X is defined as the moduli space of semisimple representations of π 1 (X) into G, that is, For complex linear groups G = GL(n, C), SL(n, C), the representations of π 1 (X) into G can be understood as G-local systems E → X, hence defining a flat bundle E which has deg E = 0. A natural generalization consists of allowing bundles E of non-zero degree d. The G-local systems on X correspond to representations ρ : π 1 (X − {p 0 }) → G, where p 0 ∈ X is a fixed point, and ρ(γ) = d n Id, γ a loop around p 0 , giving rise to the moduli space of twisted representations is the moduli space parameterizing flat bundles, i.e., (E, ∇), where E → X is an algebraic bundle of degree d and rank n (and fixed determinant in the case G = SL(n, C)), and ∇ is an algebraic connection on X − {p 0 } with a logarithmic pole at p 0 with residue − d n Id. The Riemann-Hilbert correspondence [3,18] gives an analytic isomorphism (but not an algebraic isomorphism) D d (G) ∼ = M d (G). The Dolbeault moduli space H d (G) is the moduli space of G-Higgs bundles (E, Φ), consisting of an algebraic bundle E → X of degree d and rank n (also with fixed determinant in the case G = SL(n, C)), and a homomorphism Φ : E → E ⊗ K X , known as the Higgs field (in the case G = SL(n, C), the Higgs field has trace 0). In this situation the theory of harmonic bundles [2,17] gives a homeomorphism M d (G) ∼ = H d (G).
When gcd(n, d) = 1 these moduli spaces are smooth and the underlying differentiable manifold is hyperkähler, but the complex structures do not correspond under the previous isomorphisms. The cohomology of these moduli spaces has been computed in several particular cases, mostly from the point of view of the Dolbeault moduli space H d (G). Poincaré polynomials for G = SL(2, C) were computed by Hitchin in his seminal paper on Higgs bundles [11] and for G = SL(3, C) by Gothen in [8]. More recently the case of G = GL(4, C) has been solved in [6], and a recursive formula for the motive of the moduli space of Higgs bundles of arbitrary rank and degree coprime to the rank has been given in [7]. In particular, this gives the Betti numbers of the character variety for arbitrary coprime rank and degree.
Hausel and Rodriguez-Villegas started the computation of the E-polynomials of G-character varieties focusing on G = GL(n, C), SL(n, C) and PGL(n, C) using arithmetic methods inspired by the Weil conjectures. In [9] they obtained the E-polynomials of M d (G) for G = GL(n, C) in terms of a simple generating function. Following these methods, Mereb [16] studied the case of SL(n, C) giving an explicit formula for the E-polynomial in the case G = SL(2, C), while for SL(n, C) these polynomials are given in terms of a generating function. He proved also that the E-polynomials for SL(n, C) are palindromic and monic.
Logares, Newstead and the second author [12] introduced a geometric technique to compute the Epolynomial of character varieties by using stratifications and also handling fibrations which are locally trivial in the analytic topology but not in the Zariski topology. The main results of [12] are explicit formulae for the E-polynomials for character varieties for G = SL(2, C) and g = 1, 2. Actually, the geometric technique allows for dealing with "twisted" character varieties in which the holonomy around the puncture is not diagonizable (in this case there is no correspondence with a De Rham or Dolbeault moduli space as mentioned above).
The new input consists of using the map M(SL(2, C)) → C 3 given by (A 1 , B 1 , A 2 , B 2 , A 3 , B 3 ) → (t 1 , t 2 , t 3 ), t i = tr([A i , B i ]), to stratify conveniently the space of representations in SL(2, C) 6 . This leads us to analyse a fibration over a two-dimensional basis (the quadric C in Figure 1), for which we have to compute the E-polynomial. Therefore the arguments of [12] about Hodge monodromy fibrations over one-dimensional bases have to be extended to general bases (see Theorem 2). In the case of M(SL(2, C)) we also have to deal with reducible representations. The arguments used in [12,Section 8.2.2] to compute the E-polynomial of the irreducible locus give an increasingly large number of strata as the genus increases. So here we use an alternative route (suggested by the ideas in [14]), consisting of computing the E-polynomial of the irreducible locus by subtracting the E-polynomial of the subset of reducible representations.
which means that the E-polynomial of the invariant part of the cohomology is the polynomial accompanying T , and the E-polynomial of the non-invariant part is the polynomial accompanying N . This monodromy is of interest from the point of view of Mirror Symmetry [10] and answers a question raised to the authors by Tamás Hausel.
After finishing and submitting a first version of this paper, the authors have managed to complete the computation of E-polynomials of character varieties for surfaces of general genus g ≥ 3 in [15]. The case provided here cannot however be subsumed in [15], since it serves as the starting point of induction on the genus carried there. Moreover, this specific case of g = 3 has special features and has to be treated by an ad-hoc method (it is the only case that needs the analysis of Hodge monodromy representations over a 2-dimensional basis).
2.1. E-polynomials. We start by reviewing the definition of E-polynomials. A pure Hodge structure of weight k consists of a finite dimensional complex vector space H with a real structure, and a decomposition H = k=p+q H p,q such that H q,p = H p,q , the bar meaning complex conjugation on H. A Hodge structure of weight k gives rise to the so-called Hodge filtration, which is a descending filtration F p = s≥p H s,k−s . We define Gr p F (H) := F p /F p+1 = H p,k−p . A mixed Hodge structure consists of a finite dimensional complex vector space H with a real structure, an ascending (weight) filtration . . . ⊂ W k−1 ⊂ W k ⊂ . . . ⊂ H (defined over R) and a descending (Hodge) filtration F such that F induces a pure Hodge structure of weight k on each Gr W k (H) = W k /W k−1 . We define H p,q := Gr p F Gr W p+q (H) and write h p,q for the Hodge number h p,q := dim H p,q .
Let Z be any quasi-projective algebraic variety (maybe non-smooth or non-compact). The cohomology groups H k (Z) and the cohomology groups with compact support H k c (Z) are endowed with mixed Hodge structures [4,5]. We define the Hodge numbers of Z by . The Hodge-Deligne polynomial, or E-polynomial is defined as When h k,p,q c = 0 for p = q, the polynomial e(Z) depends only on the product uv. This will happen in all the cases that we shall investigate here. In this situation, it is conventional to use the variable q = uv. If this happens, we say that the variety is of balanced type. For instance, e(C n ) = q n .
The key property of Hodge-Deligne polynomials that permits their calculation is that they are additive for stratifications of Z. If Z is a complex algebraic variety and Z = n i=1 Z i , where all Z i are locally closed in Z, then e(Z) = n i=1 e(Z i ). There is another useful property that we shall use often: if there is an action of Z 2 on X, then we have polynomials e(X) + , e(X) − , which are the E-polynomials of the invariant and anti-invariant parts of the cohomology of X, respectively. More concretely, e(X) + = e(X/Z 2 ) and e(X) − = e(X) − e(X) + . Then if Z 2 acts on X and on Y , we have the equality (see [12,Proposition 2.6

Fibrations.
We have to deal with fibrations that are locally trivial in the analytic topology. We want to compute the E-polynomial of the total space in terms of the polynomials of the base and the fiber. The fibration defines a local system H k c , whose fibers are the cohomology groups H k The fibers possess mixed Hodge structures, and the subspaces W t (H k c (F b )) are preserved by the holonomy. We will assume from now on that F is of balanced type, so . Associated to the fibration, there is a monodromy representation: Suppose that the monodromy group Γ = im (ρ) is an abelian and finite group. Then H k,p,p c (F ) are modules over the representation ring R(Γ). So there is a well defined element, the Hodge monodromy representation, As the monodromy representation (4) has finite image, there is a finite covering B ρ → B such that the pull-back fibration has trivial monodromy.
Let S 1 , . . . , S N be the irreducible representations of Γ (there are N = #Γ of them, and all of them are one-dimensional). These are generators of R(Γ) as a free abelian group. We write the Hodge monodromy representation of (5) as Theorem 2. Suppose that B ρ is of balanced type. Then Z is of balanced type. Moreover, there are polynomials s 1 (q), . . . , s N (q) ∈ Z[q] (only dependent on B, B ρ and Γ, but not on the fibration or the fiber) such that e(Z) = a 1 (q)s 1 (q) + . . . + a N (q)s N (q), for R(Z) = a 1 (q)S 1 + . . . a N (q)S N .
Proof. The Leray spectral sequence of the fibration (3) has E 2 -term and abuts to H k c (Z). By [1], E l,m j (Z) has a mixed Hodge structure for j ≥ 2, and the differentials d j are compatible with the mixed Hodge structure. Therefore e(Z) = e(H * c (Z)) = e(E * , * 2 (Z)). The mixed Hodge structure associated to π in (6) is the product mixed Hodge structure E l,m 2 (Z ) = H l c (B ρ ) ⊗ H m c (F ). By our assumption, B ρ and F are of balanced type, so Z is of balanced type. There is a map E l,m 2 (Z ) → E l,m 2 (Z), which preserves the mixed Hodge structures. This map is surjective, so Z is of balanced type.
By definition, The local systems S i → B are 1-dimensional and have a mixed Hodge structure. When we pull-back S i → B to B ρ , we get trivial local systems. Hence H l c (B ρ ) H l c (B, S i ) and it has the induced mixed Hodge structure. So e(H * c (B, H * c (F ))) = e(H * c (B, R(Z))) = a i (q)e(H * c (B, S i )). We define s i (q) = e(H * c (B, S i )) to have the statement.
satisfying the property that e(R(Z)) = e(Z).
A trivial application of Theorem 2 happens when π : Z → B is a fibre bundle with fibre F such that the action of π 1 (B) on H * c (F ) is trivial. Then R(Z) = e(F )T , where T is the trivial local system and e(Z) = e(F )e(B) (this result appears in [12,Proposition 2.4]).
The hypothesis that the action of π 1 (B) on H * c (F ) is trivial holds in particular in the following cases: • B is irreducible and π is locally trivial in the Zariski topology.
• π is a principal G-bundle with G a connected algebraic group.
• Z is a G-space with isotropy H < G such that G/H → Z → B is a fiber bundle, and G is a connected algebraic group.
The formula in Theorem 2 says that e(B) = (q − ) + e p=2 s ip . Therefore Another case that we are going to use later is the following.
Corollary 4. Let (3) be a fibration over B = C * × C * with finite monodromy, such that F is of balanced type. Then Z is also of balanced type and its E-polynomial is given by Proof. First note that π 1 (B) = Z × Z, so the monodromy Γ, being a quotient Z × Z Γ, must be abelian. Moreover, there is some n > 0 such that Z n × Z n Γ, and the covering associated to Z n × Z n is of balanced type (is C * × C * again). Hence B ρ is of balanced type, since there are coverings C * × C * → B ρ → C * × C * . Now let S be an irreducible representation of R(Γ). If S = T , the trivial representation, then e(T ) = e(B) = (q − 1) 2 .
If S is a non-trivial representation, then take the coveringB → B associated to S. Again there are are surjections, but the composition is an isomorphism (multiplication by n). Therefore e(R(B)) = e(B) = (q − 1) 2 , and e( e p=2 S ip ) = 0. This happens for all choices of S, so it must be e(S) = 0, for any irreducible non-trivial S.

2.3.
Building blocks. We need to recall some E-polynomials from [12] which serve as building blocks for the E-polynomials of character varieties for any genus g ≥ 2. They are basically associated to the case of g = 1. First, we have that e(SL(2, C)) = q 3 − q and e(PGL(2, C)) = q 3 − q. Consider the following subsets of SL(2, C): • W 0 := conjugacy class of 1 0 0 1 . It has e(W 0 ) = 1.
Now consider the map Note that f is equivariant under the action of SL(2, C) by conjugation on both spaces. We stratify X = SL(2, C) 2 as follows: We also introduce the varieties f −1 (C) for fixed C ∈ SL(2, C) and define accordingly There is a fibration PGL(2, C)/D → X 4,λ → X 4,λ , and e(X 4,λ ) = (q 2 + q)e(X 4,λ ).

E-polynomial of the twisted character variety
We start by computing the E-polynomial of the twisted character variety M 1 = M 1 (SL(2, C)) for a curve X of genus 3. This space can be described as the quotient Notice that we only need to consider a geometric quotient, since all representations are irreducible: if there was a common eigenvector v for ( To compute the E-polynomial of the moduli space, we will stratify the space in locally closed subvarieties and compute the E-polynomial of each stratum, since E-polynomials are additive with respect to stratifications. We stratify the space W according to the possible values of the traces of the commutators. To simplify the notation, we write [A 1 , We are interested in the following condition: if ξ 1 , ξ 2 , ξ 3 share an eigenvector v with eigenvalue λ i in each case, then in a suitable basis , and therefore, as ξ 1 ξ 2 ξ 3 = − Id, we obtain that λ 1 λ 2 λ 3 = −1. Other possibility is that v has eigenvalue λ 1 for ξ 1 , λ 2 for ξ 2 and λ −1 3 for ξ 3 , yielding λ 1 λ 2 λ −1 3 = −1, etc. Working out all possibilities, we have that ξ 1 , ξ 2 , ξ 3 can share an eigenvector when (14) λ Equivalently, in terms of the traces, when This defines a (smooth) quadric C ⊂ C 3 , depicted in Figure 1.
Proof. We have already seen the if part. For the only if part, note that if (t 1 , t 2 , t 3 ) satisfies (15) then the eigenvalues λ 1 , λ 2 , λ 3 satisfy one of the four relations in (14). Changing some λ i by λ −1 i if necessary, we can suppose that λ 3 = −λ 1 λ 2 .
Suppose that some ξ i is diagonalizable. Without loss of generality, we suppose it is ξ 1 , and choose a basis with Then we have the equation We get the equations t 2 = a + d and t 3 = tr ξ 3 = tr ξ −1 , it must be a = λ 2 , d = λ −1 2 ; thus ad = 1 and hence bc = 0, which implies that the matrices share an eigenvector. If λ 1 = ±1, then ξ 1 = ± Id, and ξ 2 = ±ξ 3 , so the matrices share their eigenvectors. Now suppose that all ξ i are not diagonalizable, so they are of Jordan type. Let v i be the only eigenvector (up to scalar multiples) of ξ i . If ξ i do not share an eigenvector, then v 1 , v 2 is a basis, on We stratify the space W according to the traces of ξ 1 , ξ 2 , ξ 3 . Consider the planes t i = ±2 and the quadric C above. Then consider as well the intersections of these seven subvarieties. This gives the required stratification. We shall compute the E-polynomials of the chunk of W lying above each of these strata, starting by the lower-dimensional ones (points) and going up in dimension.
We shall use some of the polynomials computed in Sections 8-12 in [12] which correspond to some of the strata of the spaces of representations for the case of a curve of genus g = 2. We shall point out which stratum we use each time.
3.4. The quadric C. One eigenvector. We now deal with the part of W lying over the quadric C, with t 1 , t 2 , t 3 = ±2. As we saw in Lemma 5, this corresponds to the case that ξ 1 , ξ 2 , ξ 3 share (at least) one eigenvector.
We deal now with the case where ξ 1 , ξ 2 and ξ 3 share just one eigenvector (up to scalar multiples), which we denote by v. Using it as the first vector of a basis that diagonalizes ξ 1 , we can arrange that where λ = λ 1 , µ = λ 2 are the eigenvalues of ξ 1 , ξ 2 associated to the eigenvector v. Our choice of basis for the above expressions of ξ 1 , ξ 2 , ξ 3 gives us a slice for the PGL(2, C)-action, since their stabilizer is trivial. So we shall only need to multiply by e(PGL(2, C)) after computing the E-polynomial of (16). This set can be regarded as a fibration To compute its E-polynomial, we would like to extend the fibration to the six curves λ, µ, λµ = ±1 and apply Corollary 4. We cannot extend the fibration; however we can extend the local system definining the Hodge monodromy fibration. By (8), the Hodge monodromy fibration R(X 4 ) is trivial over λ = ±1, and it is of order 2 over λ = 0. Consider the projections: . The Hodge monodromy fibration R(Z) can be extended (locally trivially) over the lines λ = ±1, µ = ±1 and λµ = ±1, to a Hodge monodromy fibrationR(Z) overB = C * × C * . Moreover, the monodromy around λ = 0 and µ = 0 is of order two. The corresponding group is Γ = Z 2 × Z 2 and the representation ring is generated by representations T, N 1 , N 2 , N 12 = N 1 ⊗ N 2 , where T is the trivial representation, N 1 is the representation with non-trivial monodromy around the origin of the first copy of C * , and N 2 is the representation with non-trivial monodromy around the origin of the second copy of C * .

we can obtain three pullback bundles
As a consequence, if we write R(X 4 /Z 2 ) = aT + bS 2 + cS −2 + dS 0 , as in (10), the Hodge monodromy representation is . To obtain the E-polynomial of the total space, we need to substitute each representation by its associated E-polynomial, by Theorem 2.
3.6. Generic case. Let us deal finally with the case (t 1 , t 2 , t 3 ) ∈ C, t i = ±2, which corresponds to the open subset U in (C * ) 3 defined by those representations whose ξ 1 , ξ 2 , ξ 3 are diagonalizable and do not share an eigenvector. Choosing a basis that diagonalizes ξ 1 , note that, if we write ξ 2 = a b c d , then 1 c −λ 1 a and bc = 0 (see Lemma 5). Conjugating by a diagonal matrix, we can assume that b = 1. As t 2 = a + d and t 3 = −λ 1 a − λ −1 1 d, we have that a, d are determined by the values of (t 2 , t 3 ); and c is determined by the equation det ξ 2 = ad − bc = 1. We see that for fixed (λ 1 , t 2 , t 3 ), a, b, c, d are fully determined, and so are ξ 2 , ξ 3 .
Consider the Z 2 × Z 2 × Z 2 -cover given by (λ 1 , λ 2 , λ 3 ) → (t 1 , t 2 , t 3 ) over (C − {0, ±1}) 3 . LetĒ be the pull-back fibration. The fiber over (λ 1 , λ 2 , λ 3 ) is isomorphic to The second and third copies of Z 2 act as the standard Z 2 -action on the second and third copies of X 4 , respectively. The action of the first copy of Z 2 is more delicate: it acts as conjugation Then under the isomorphism Θ, it acts by conjugation by Q −1 P 0 Q (resp. S −1 P 0 S) on the second (resp. third) factor of X 4 . This matrix is easily computed to be diagonal, so the action is (homologically) trivial.
This agrees with the result in [16], obtained by arithmetic methods.

Hodge monodromy representation for the genus 2 character variety
We introduce the following sets associated to the representations of a genus 2 complex curve, and give the E-polynomials computed in [12]: [12,Section 9]. [12,Section 11].
If we take the quotient by the Z 2 -action there is another fibration We are interested in the Hodge monodromy representations R(Y 4 ) and R(Y 4 /Z 2 ).
Proof. We follow the stratification Y 4,λ0 = 7 i=1 Z i given in [12,Section 10], and study the behaviour of each stratum when λ varies in C − {0, ±1} to obtain the Hodge monodromy representation of Y 4 .
We look at the strata: • Z 1 , corresponding to t 1 = ±2, t 2 = ±2. In this case, both ν, δ are of Jordan type. If we take the basis given by {u 1 , u 2 }, where u 1 is an eigenvector for ν and u 2 an eigenvector for δ, then for certain x, y ∈ C * . Now, since δν −1 = λ 0 0 λ −1 , we obtain that λ + λ −1 = 2 − xy. We can fix x = 1 by rescaling the basis, so y is fixed and there is no monodromy around the origin. Therefore where T is the trivial representation.
In this situation, bc = 0, so there are three possibilities: either b = c = 0 (in which case ν = Id) or b = 0, c = 0 or b = 0, c = 0 (in either case there is a parameter in C * and ν is of Jordan type).
The factor (q − 1) corresponds to the fact that now bc = 0, so there is the extra freedom given by C * .
We want to compute the Hodge monodromy representation of Y 4 /Z 2 . We have the following.
Proof. The Hodge monodromy representation R(Y 4 /Z 2 ) lies in the representation ring of the fundamental group of C − {±2}. Under the double cover C − {0, ±1} → C − {±2}, it reduces to R(Y 4 ). By Proposition 7, R(Y 4 ) is of order 2. Hence R(Y 4 /Z 2 ) has only monodromy of order 2 over the loops γ ±2 around the points ±2. This is the statement of the lemma.