Intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve

Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization $\tilde{\mathcal{M}}_{Dol}^G$ of the moduli space $\mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,\mathbb{C}), SL(2,\mathbb{C})$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of $\tilde{\mathcal{M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal{M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal{M}_{Dol}^G$ and prove that the mixed Hodge structure on it is actually pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

space is singular and many fundamental theorems like Poincaré duality or Hard-Lefschetz theorem fail for ordinary cohomology groups. To overcome this fact one might opt for two solutions: to resolve singularities or to consider a different cohomological invariant, namely, intersection cohomology. Here we adopt both strategies, showing how they are strongly interrelated. We first show the following result.
Theorem 1.1. Let C be a curve of genus 2 and let M G Dol be the moduli space of G-Higgs bundles of rank 2 and degree 0 on C. Then there exists a symplectic desingularizationπ :M G Dol → M G Dol . This theorem was obtained independently by Bellamy and Schedler in [3] in the language of character varieties. In fact it is well known that, by the non abelian Hodge theorem [38], M G Dol is real analytic isomorphic to the character variety of representations of the fundamental group of C into G. The equivalence with our result is obtained by applying the isosingularity principle [38,Theorem 10.6], which states that two corresponding singular points on the moduli spaces admit isomorphicétale neighbourhoods. Having a symplectic desingularization is indeed a special feature of this moduli space: in fact, apart from rank 2 and degree 0 G-Higgs bundles on a genus 2 curve, the only other case in which a symplectic resolution exists is that of degree 0 G-Higgs bundles of arbitrary rank on an elliptic curve. For higher genera and ranks, the moduli spaces do not admit a symplectic desingularization, see [40] and [27].
Let us now give a sketch of the proof of these two results. The resolution is constructed using a strategy developed by O' Grady in [34] and [35] to desingularize moduli spaces of sheaves on K3 or abelian surfaces. With this procedure, O' Grady obtains two new examples of compact hyperkähler manifolds up to birational equivalence, which are usually denoted by OG10 and OG6. Such examples have dimension, respectively 10 and 6, as M GL Dol and M SL Dol . Indeed more is true: applying similar methods to those in [6] and [13], one can show that there exists a degeneration of hyperkähler manifolds from O' Grady moduli spaces to those of Higgs bundles. In view of this, the singularities of M G Dol have isomorphic local description to those of O' Grady examples, as we show in section 3.
It is easy to see that in this case Σ G is isomorphic to the symmetric product Sym 2 (Jac(C)× H 0 (K C )), thus it is a singular 8-fold with finite quotient singularities along Ω G . The description for G = SL(2, C) is analogous mutatis mutandis. In both cases, the resolution is obtained by a single blow up along the locus Σ G : a crucial point in the construction is the description of the singularities of O' Grady moduli spaces provided by Lehn and Sorger in [30].
To prove the main theorem 1.2, we first show that the Hodge structure on intersection cohomology groups is pure. To do that we extend the natural C * -action on M G Dol , given by rescaling the Higgs field, toM G Dol : this yields an isomorphism of mixed Hodge structures between the cohomology of M G Dol and that of a compact subvariety, namely the preimage (π • χ) −1 (0) of the nilpotent cone in the resolution. On the one hand the smoothness ofM G Dol implies that the weights are greater or equal than the cohomological degree; on the other hand, since the cohomology ofM G Dol is that of a compact variety, they cannot exceed it. As a result H j (M G Dol ) carries a pure Hodge structure of weight j. Since the Hodge structure on IH * (M G Dol ) is a sub-Hodge structure of that on H * (M G Dol ), it is pure as well. Indeed one can show in the same way that the Hodge structure on the intersection cohomology of moduli space of rank 2 Higgs bundles is pure in any genus. Ultimately this is due to the fact that the resolution is obtained by blowing up along C * -equivariant subsets, so that the above argument still works. This will be shown by the author in a forthcoming paper. It is likely that such result should hold also in higher rank, however the structure of the singular locus is more complicated and requires further attention.
Knowing the purity of the Hodge structure, we can obtain cohomology by computing other cohomological invariants, called E-polynomials (cfr. section 1), generally much easier to determine than Poincaré polynomials.
The E−polynomial of a variety X is defined as where h j,p,q c = dim Gr F p Gr W p+q H j c (X). Moreover, up to replace H j c (X) with IH j c (X), one can define an analogous invariant for intersection cohomology called intersection E−polynomial. E−polynomials satisfy the an additivity property: E(X) = E(Y ) + E(X \ Y ) for all Y ⊂ X. By stratifyingM G Dol and computing E−polynomials for all strata we end up with E(M G Dol ). Moreover we also compute the E−polynomials of the contributions supported on the singular loci. By subtracting them one get the E−polynomial for the intersection cohomology of M G Dol . We remark that, though the additivity property is false in general for intersection E-polynomial, our method applies anyway as we compute the intersection E-polynomial as a sum of actual E-polynomials, for which additivity property holds. Notice that, by the purity of the Hodge structure, intersection Betti numbers with compact support are given by ib j,c = p+q=j ih j,p,q c , where ih j,p,q c = dim Gr F p Gr W p+q IH j c (X). Theorem 1.2 now follows by Poincaré duality. In the smooth coprime case, the cohomology of these spaces have been widely studied: Poincaré polynomials for SL (2, C) were computed by Hitchin in his seminal paper on Higgs bundles [21], for SL(3, C) by Gothen in [20] and in rank 4 by Garcia-Prada, Heinloth and Schmitt [17]. Furthermore, in [22] Hausel and Rodriguez-Villegas derived a conjectural formula for the E-polynomials of twisted G−character varieties focusing on G = GL(n, C), SL(n, C), In [37] Schiffmann provided a closed formula for the Poincaré polynomial of the moduli spaces in any coprime rank and degree. Such formula was shown to imply the conjectural formula of Hausel and Rodriguez-Villegas by Mellit in [29].
In the singular case, Logares, Muñoz and Newstead [28] computed the E−polynomial of the character varieties of SL(2, C) and GL(2, C) on curves of genus g = 1, 2, while Martinez and Muñoz extended it to g ≥ 3. In [1] Baraglia and Hekmati gave a new proof of these, extending it to rank 3. Furthermore, they showed how to extend the approach of Hausel and Rodriguez-Villegas used for nonsingular twisted character varieties to the singular (untwisted) case.
To the author's knowledge, this is the first result of computation of intersection cohomology for Higgs bundles or character varieties. For vector bundles, where the moduli spaces involved are compact, intersection Betti numbers were computed by Kirwan in [25]. The main motivation for this work was provided by the celebrated P = W conjecture by De Cataldo, Hausel and Migliorini (see [5]) which asserts that the Weight filtration on the cohomology of the character variety corresponds via non abelian Hodge theorem to the Perverse filtration arising from the Hitchin fibration. Though the conjecture is formulated for smooth moduli spaces, it would be interesting to see whether an analogue of the P = W conjecture exists in the singular case of moduli of Higgs bundles of non coprime rank and degree and the corresponding character varieties. Indeed, for moduli spaces with a symplectic resolution, the conjecture has been proved by the author and Mauri in [15], relying on the results of this paper.
The article is organized as follows: in Section 2 we briefly review the theory of intersection cohomology and decomposition theorem; in Section 3 we describe the local geometry of the moduli space focusing on the singularities and their normal cones. In Section 4, we construct a semismall desingularization and apply the decomposition theorem to split the cohomology of the desingularization as a direct sum of the intersection cohomology of M G Dol plus some other summands supported on the singular locus. In Section 5, we extend the natural C * -action on M G Dol to the desingularization and state a localization lemma that yields to the triviality of the weight filtration both on the cohomology of the desingularization and on the intersection cohomology of M G Dol . In Sections 6 and 7, we compute the E−polynomial for the intersection cohomology of M G Dol and show that from it, by the triviality of the weight filtration, one can recover the intersection Betti numbers of M G Dol both in the case of G = SL(2, C) and G = GL(2, C).

Quick review of intersection cohomology and decomposition theorem
Pure Hodge theory allows to use analytic methods to study algebro-geometric and topological properties of a smooth algebraic variety and comes with the Hodge-Lefschetz package, which includes deep results such as Hard Lefschetz theorem, Poincaré duality and Deligne's theorem for families of projective manifolds. When working with singular or non compact varieties theorems in the Hodge-Lefschetz package fail. To overcome this fact, there are two somewhat complementary approaches: mixed Hodge theory and intersection cohomology.
In mixed Hodge theory, introduced by Deligne in [14] and [11], one still investigates the same topological invariant, namely, the cohomology groups, whereas the structure with which it is endowed changes. In particular the (p, q)-decomposition of the cohomology of smooth projective varieties is replaced by a more complicated structure. More precisely, the rational cohomology groups are endowed with an increasing filtration W • , such that the complexifications of the graded pieces admit a (p, q)decomposition.
If W k /W k−1 ⊗ C ∼ = p+q=k V k,p,q we say that a class in V k,p,q has weight k and type (p, q).
Similarly, one can define a mixed Hodge structure on compactly supported cohomology. This leads to the definition of E−polynomials.
Definition 2.2. Let X be an algebraic variety. The E-polynomial of X is defined as and satisfies the following properties: Remark 2.1. If X is smooth of complex dimension n, then mixed Hodge structures are compatible with Poincaré duality, i.e. a class in H i (X) of weight k and type (p, q) corresponds to a class in H 2n−i c (X) of weight 2n − k of type (n − p, n − q).
Remark 2.2 (Yoga of weights). In general finding the weight of a cohomology class is a nontrivial task. However, there are some fundamental weight restrictions: i) if X is nonsingular, but possibly non-compact, then weights are high, i.e.
ii) if X is compact but possibly singular then weights are low i.e.
In intersection cohomology theory, by contrast, it is the topological invariant which is changed, whereas the (p, q)-decomposition turns out to be the same. Intersection cohomology groups are defined as the hypercohomology of some complexes, called intersection complexes, that live in the derived category of constructible complexes. Intersection complexes are constructed from local systems defined on nonsingular locally closed subsets of an algebraic variety with a procedure called intermediate extension (see [2, 1.4.25, 2.1.9, 2.1.11], [18], [19]). For a beautiful introduction with also an historical point of view, we refer to [26].
There is a natural morphism H i (X) → IH i (X) which is an isomorphism when X has at worst finite quotient singularities. Intersection cohomology groups are finite dimensional, satisfy Mayer-Vietoris theorem and Künneth formula. Although they are not homotopy invariant, they satisfy analogues of Poincaré duality, Hard Lefschetz theorem and, if X is projective, they admit a pure Hodge structure. The definition of intersection cohomology is very flexible as it allows for twisted coefficients: given a local system L on a locally closed nonsingular subvariety Y of X we can define the cohomology groups IH(Y , L). Definition 2.3. Let X be an algebraic variety and let Y ⊂ X be a locally closed subset contained in the regular part of X. Let L be a local system on Y . The intersection complex IC Y (L) associated with L is a complex of sheaves on Y which extends the complex L[dim Y ] and is determined up to unique isomorphism in the derived category of constructible sheaves by the conditions Remark 2.3. Let X be an algebraic variety with regular locus X reg . When L = Q Xreg one just writes IC X for IC X (L) and call it intersection cohomology complex of X. If X is nonsingular, then Definition 2.4. Let X be an algebraic variety. The intersection cohomology groups of X are defined as In general, given any local system L supported on a locally closed subset Y of X, the cohomology groups of Y with coefficients in L are shifted hypercohomology groups of the intersection complex associated to L: Taking hypercohomology with compact support, one likewise defines intersection cohomology groups with compact support IH * c (X) and IH * c (Y , L). Remark 2.4. Here the shift is made so that for a nonsingular variety intersection cohomology groups coincide with ordinary cohomology groups.
Remark 2.5. Just as ordinary cohomology, intersection cohomology groups carry a mixed Hodge structure (see [36]). As a result it is possible to define an analogue of E−polynomial for intersection cohomology, called intersection E−polynomial: Along with theorems of Hodge-Lefschetz package, intersection cohomology groups satisfy an analogue of Deligne's theorem for projective manifolds, the decomposition theorem. The general statement of this theorem is complicated and will not be discussed here (see for example [7] for an extensive survey on the topic). Roughly speaking, the decomposition theorem asserts that, given a proper map of algebraic varieties f : X → Y , the derived pushforward of the intersection complex of X splits as a direct sum of the intersection complex of Y and other intersection complexes associated to local systems supported on some nonsingular locally closed subsets of Y . These subsets are called supports. In general it is complicated to determine the supports and the local systems appearing in the splitting. However the decomposition theorem takes a particularly simple form when dealing with a special kind of maps, namely semismall maps.
Definition 2.5. Let f : X → Y be a map of algebraic varieties. A stratification for f is a decomposition of Y into finitely many locally closed nonsingular subsets Y α such that f −1 (Y α ) → Y α is a topologically trivial fibration. The subsets Y α are called the strata of f . Definition 2.6. Let f : X → Y be a proper map of algebraic varieties. We say that f is semismall if there exists a stratification Y = Y α such that for all α For semismall maps, the only supports are the relevant strata and their contributions to the pushforward of IC X consist of nontrivial summands IC Y α (L α ), where the local systems L α are given by the top cohomology of the fibres and turn out to have finite monodromy. More precisely, let Y α be a relevant stratum, y ∈ Y α and let F 1 , . . . , F l be the irreducible (dim Y α )−dimensional components of the fibre f −1 (y). The monodromy of the F i 's defines a group homomorphism ρ α : π 1 (Y α ) → S l from the fundamental group of Y α to the group of permutations of the F i 's. The representation ρ α defines a local system L α on Y α . In this case the semisimplicity of the local system L α is an elementary consequence of the fact that the monodromy factors through a finite group, so by Maschke theorem it is a direct sum of irreducible representations. As a result, the local systems L α will be semisimple, i.e. they will be a direct sum of simple local systems. With this notation, the statement of the decomposition theorem for semismall maps is the following. For the proof we refer to [2], [36] and [8].
Theorem 2.1 (Decomposition theorem for semismall maps). Let f : X → Y be a semismall map of algebraic varieties and let Λ rel the set of relevant strata. For each Y α ∈ Λ rel let L α the corresponding local system with finite monodromy defined above. Then there exists a canonical isomorphism in the derived category of constructible sheaves Moreover this is an isomorphism of mixed Hodge structures.

Local structure of the moduli space
Consider a curve C of genus 2 and let G = GL(2, C) or SL(2, C). We define M G Dol to be the moduli space of G−Higgs bundles on C: for G = GL(2, C) these are just ordinary Higgs bundles of rank 2 and degree 0, while for G = SL(2, C) one also asks for the determinant to be trivial.
We shall briefly recall the construction by Simpson of these moduli spaces. • [38, Theorem 4.10] Fix x ∈ C and let T G be the frame bundle at x of the universal bundle V on Q × C restricted to x. Then G × GL(p, C) acts on Q G : indeed G acts as automorphisms of (V, Φ) while GL(p, C) acts on the α's. The action of GL(p, C) on Q G lifts to T G . Since this action is free and every point in T G is stable with respect to it, one can define  As it is well known (for example see [38, section 1]), the singularities of M G Dol correspond to strictly semistable bundles. If a Higgs bundle (V, Φ) is strictly semistable, then there exists a Φ-invariant line bundle L of degree 0.
Proof. Clearly, Higgs bundles in Σ G are semistable but not stable thus they lie in the singular locus. The result follows after noticing that nontrivial extensions as Higgs bundles do not appear in M G Dol as their G-orbit in R G Dol is not closed.
Observe that in both cases Ω G ⊂ Σ G . In the general case of G = GL(2, C), Σ G is parametrized by the symmetric product Sym 2 (Jac(C)×H 0 (K C )) where Z 2 acts as the involution that switches summands. Ω G is given by the fixed points of the involution and it is parametrized by Jac(C) × H 0 (K C ). In the trivial determinant case, when G = SL(2, C), Σ G ∼ = (P ic 0 (C) × H 0 (K C ))/Z 2 and Ω G consists again of the fixed points of the involution, which are the 16 roots of the trivial bundle.  [34], [35] (see also [3] and [27]). Thanks to this fact, one can copy O' Grady's method almost verbatim to obtain a desingularization of M G Dol . In this subsection the singularities of M G Dol and their normal cones are studied, leading to the construction of a desingularization and the proof of its semismallness.
Let G = SL(2, C) or GL(2, C) and let g be its Lie algebra. We shall describe the singularities of the moduli space of Higgs bundles M G Dol with G = GL(2, C). The trivial determinant case of G = SL(2, C) is analogous, provided that we replace End(V ) by End 0 (V ).
Let A i denote the sheaf of C ∞ i−forms on C. For a polystable Higgs pair (V, Φ) consider the complex Splitting in (p, q)-forms, the cohomology of this complex is equal to the hypercohomology of the double complex This means that the cohomology groups T i of (1) fit the long exact sequence   Observe that since there is a local isomorphism EndV ∼ = g, an element of T 1 can be thought of as a matrix in g with coefficient in In this interpretation, the bracket in theorem 3.3 is the Lie bracket of g coupled with the perfect pairing 3.1.1. Interpretation with extensions. It is also possible to describe the spaces T i and the graded commutator more explicitly: consider the Higgs bundle (V, Φ) as an extension The deformation theory of the above Higgs bundle is controlled by the hypercohomology of the complex and there is a long exact sequence . When considering bundles with trivial determinant and traceless endomorphisms, Ext groups there is a natural cup product, called the Yoneda product, and its associated Yoneda square

Thinking of elements in Ext 1
H (V, V ) locally as matrices of 1-forms in g, such a product coincides with the graded commutator in theorem 3.3. This is precisely the same situation described in [34, Section 1.3]: in fact, by means of decomposition (4), Yoneda product reads as

3.2.
Normal cones of Σ G and Ω G .

At the level of M GL
Dol the same holds up to quotient by the stabilizer C * of points in Σ GL . 2 Here the terms Ext i H (L −1 , L −1 ) are not considered because of the traceless condition.
Proof. We first compute Ext i H (L, L). One has This complete the first part of the proof.
To prove the second statement one needs to describe the zero locus of Yoneda square and the proof of [34, Proposition 1.4.1] applies mutatis mutandis. For ease of the reader we sketch it in terms of Higgs extensions. Consider the map Observe that, since Ext 2 is a general fact of deformation theory. For determining the fibre one can use Luna's slice theorem: let U be the normal slice to R GL Dol in v and W := U ∩ GLΣ: Since The description of the cone in R SL Dol is identical, provided that one replaces (M, ψ) by (L −1 , −ψ) and take traceless extensions, so we just state the result.
Dol the same holds up to quotient by the stabilizer C * of points in Σ SL .
and the bundle End(V ) is holomorphically trivial. One has that H 0 (End(V )) ∼ = g and a generic element of this space can be thought of as a matrix . We shall compute the Ext i H 's and the quadratic cone defined by the graded commutator. Notice that the second line of the long exact sequence (2) is the Serre dual of the first one. Now, T 0 is given by the elements in g that commute with the Higgs field, which is diagonal, and then Ext 0 e. the first map and the last map of the sequence are isomorphisms. To compute Ext 1 H (V, V ), consider the central part of the sequence (2), which in this case is

Consider now the composition of the Yoneda product on Ext
This defines a skew-symmetric bilinear form ω which is non-degenerate by Serre duality. Set Λ 1 := Ext 1 H (L, L) and There is a natural action of the automorphism group G of (V, Φ) on this space given by the composition with the adjoint representation on sl (2). Note that this action is meaningful also when G = GL(2, C): indeed, since the action of GL(2, C) factors trough PGL(2, C), the action on sl(2) by adjoint representation is well defined.
Then the normal cone to its orbit in R G Dol is Υ −1 (0) and there exist a G-equivariant isomorphism (2), Λ 1 ). At the level of M G Dol the same holds up to quotient by the stabilizer G of points in Ω. Proof. Again, the proof is similar to that of [34, Proposition 1.5.1]. We restate the idea in terms of Higgs extensions. As noticed in the previous paragraph, there are natural isomorphisms (2) and the Yoneda product on Ext 1 is the composition of the form ω on Λ 1 with the bracket of g. Hence, the Yoneda square Υ : Let Υ := Υ |Λ 1 ⊗CId V . Thanks to the self duality of sl (2) as an algebra and to the identifications there is a well defined map Υ : Hom(sl (2), is a general fact of deformation theory. The equality at the level of fibres is a consequence of Luna's slice theorem: let U be the normal slice to R GL Even though for simplicity we consider just the case of genus 2, all the results in this sections hold for arbitrary genus g ≥ 2, provided that we change dimensions of Ext groups accordingly to the genus.

Construction of the desingularization and proof of semismallness
We now briefly recall the construction of the desingularization following O' Grady's strategy. This heavily relies on the results of Lehn and Sorger [30]. The same description has been used also by Bellamy and Schleder in [3] to construct desingularizations of the character varieties of SL(2, C) and GL(2, C).

4.1.
Local model for the desingularization. Let (Λ, ω) be a symplectic 4 dimensional vector space and let sp(Λ) be the symplectic Lie algebra of (Λ, ω), i.e. the Lie algebra of the Lie group of automorphisms of Λ that preserve the symplectic form ω. Let be the subvariety of square zero matrices in Z. Observe that this implies that any A ∈ Z has rank ≤ 2.
By [30,Théorém 4.5], if v ∈ Ω SL , there exists an euclidean neighbourhood of v in M SL Dol biholomorphic to a neighbourhood of the origin in Z. The same argument shows that there exists a local analytic isomorphism between M GL Dol and Z × C 4 . Hence the local geometry of a desingularizatioñ M G Dol is encoded in the local geometry of a symplectic desingularization of Z.
Let Σ be the singular locus of Z and Ω be the singular locus of Σ. Observe that dim Z = 6, while dim Σ = 4 and dim Ω = 0. In fact, Let G ⊂ Gr(2, Λ) be the lagrangian grassmannian of 2-dimensional ω-isotropic subspaces of Λ. Notice that G is a smooth irreducible 3-dimensional quadric and set The restriction π G of the second projection of Z × G toZ makes it the total space of a 3-dimensional vector bundle, the cotangent bundle of G. In particular,Z is a smooth symplectic variety and the restriction of the first projection of Z × G f :Z → Z is an isomorphism on the locus of rank 2 matrices of Z, thus it is a resolution of singularities. The fibre of the desingularization over a point A ∈ Σ is a P 1 corresponding to the 2-dimensional lagrangian subspaces U contained in the 3-dimensional kernel of A, while the central fibre over 0 = Ω is the whole G. As Z has a A 1 singularity along Σ \ Ω and G has dimension 3, it follows that f :Z → Z is a symplectic resolution.
The resolution f :Z → Z defined above coincides with the blow up of Z along the singular locus Σ.
Given a point v ∈ Ω G , the local isomorphism is given by taking Λ = Λ 1 and ω to be the symplectic form in (5).

4.2.
Global description of the desingularization and proof of semismallness. Observe that since the blow up is a local construction, by blowing up M G Dol along Σ G one obtains a symplectic The fibres of the desingularization are as follows: (1) over smooth locus of stable Higgs bundles M G,s Dol ,π G is an isomorphism; where G is the lagrangian grassmannian of isotropic 2-dimensional subspaces in the symplectic vector space (Ext 1 H (L, L), ω). By [23, Proposition 1.2] any symplectic resolution of singularities is semismall. However one can also check semismallness by direct computation.
First of all notice thatπ G is a proper birational map and set We distinguish two cases, depending on the group G.
• If G = SL(2, C), M SL Dol is a quasi-projective variety of dimension 6. By definition, parametrizing the fixed points of the involution (L, φ) → (L −1 , −φ) consists just of 16 points, corresponding to the roots of the trivial bundle on C. On M SL,s Dol ,π SL is an isomorphism and every point has just one pre-image, thus M SL,s Dol = M SL Dol,0 . Thus it satisfies (6). Let now v ∈ Σ SL \ Ω SL . By proposition (4.2), the fibre is P 1 . Then Σ SL corresponds to the stratum M SL Dol,1 and satisfies (6). The fibre G over each one of the 16 points of Ω SL is 3-dimensional so, Ω SL = M SL Dol,3 and (6) is satisfied as well.
• If G = GL(2, C), M GL Dol is a quasi-projective variety of dimension 10. Note that The symmetric product Sym 2 (Jac(C) × H 0 (K C )) has dimension 8. The singular locus By the decomposition theorem one obtains the splitting Notice that, up to dimensional shifts, the stalks of the local systems L Σ G and L Ω G in a generic point of the corresponding stratum are isomorphic to the top cohomology groups of the fibres, which are, respectively, H 2 (P 1 ) and H 6 (G). Moreover, as the fibres ofπ G over Σ G and Ω G are irreducible, then the monodromy of L Σ G and L Ω G is trivial. Finally, since Ω G is non singular and Σ G has finite quotient singularities, intersection cohomology and cohomology coincide. One has , where the Tate shifts (−1) and (−3) correspond to the Hodge structures Q(−1) of H 2 (P 1 ) and Q(−3) of H 6 (G). Taking hypercohomology on both sides of (7) one has

Purity of the Hodge structure
The aim of this section is to show that, althoughM G Dol is non compact, the Hodge structure on its cohomology is pure, i.e. Lemma 5.1. Let ρ : C * × A n → A n be a linear action on an affine space such that all the weights of the action are positive. Denote by s 0 : SpecC → A n the inclusion of the origin in A n and by p : A n → SpecC the projection. Then for any C * -equivariant complex K of sheaves on A n one has Rp * K = s * 0 K and Rp ! K = s ! 0 K. As a corollary one has the following result.
Proposition 5.2. Let X be a smooth variety with an action of C * . Assume that f : X → A n is a proper map, equivariant with respect to a linear action of C * on the affine space A n , such that all the weights of this action are positive. Let X 0 := f −1 (0) be the fibre of f over 0 = SpecC ⊂ A n . Then there is an isomorphism of mixed Hodge structures Proof. Since f is C * -equivariant we can apply lemma 5.1 to K = Rf * Q and by base change theorem for proper maps we get: For smooth moduli spaces M Dol (d, n) of Higgs bundles of coprime rank and degree the variety X 0 := χ −1 (0) is the nilpotent cone of χ and the previous proposition yields the purity of the Hodge structure on cohomology groups of those moduli spaces. In fact, on the one hand, M Dol (d, n) is smooth so by the weight restrictions in remark 2.2 the weights of H k (M Dol (d, n)) are ≥ k. On the other hand, proposition 5.2 gives an isomorphism of mixed Hodge structures between the cohomology of M Dol (d, n) and that of the nilpotent cone, which is compact, thus the weights of H k (M Dol (d, n)) are ≤ k. Combining these two conditions one has that H k (M Dol (d, n)) has weight k.
Letχ :M G Dol → A be the composition of χ withπ G . Suppose that one can extend the C * action on M G Dol toM G Dol in a way such thatχ is equivariant. Sinceχ is a proper map, the varietyχ −1 (0) is compact. As a result the above weight trick applies toM G Dol as well proving the following result. Proof. Sinceπ G :M G Dol → M G Dol is an isomorphism on the smooth locus, to prove the lemma one needs to extend the C * action to the fibres ofπ G over the singular loci.
for all v ∈ Σ G (resp. Ω G ) and that Υ −1 (0) ⊂ T 1 . As a consequence, to describe the action on the fibres one first needs to describe it over T 1 . Consider the diagram An element λ ∈ C * acts on the diagram by scalar multiplication on the Higgs field in the commutator. As T 1 ∼ = Im α ⊕ cokerβ, λ ∈ C * acts on T 1 as scalar multiplication on Im α and as the identity on cokerβ. Since both Σ G and Ω G are C * -invariant, one can easily show that Υ −1 (0) is invariant under the C * action on T 1 . Moreover, such action commutes with that of the stabilizer, so one has a well defined action on the fibres ofπ G . Observe that by construction the mapχ is C * -equivariant.
Purity of the Hodge structure on cohomology groups yields, by Poincaré duality, purity of Hodge structure on cohomology groups with compact supports. The following lemma implies that intersection Betti numbers of M G Dol and Betti numbers ofM G Dol can be computed just by knowing E−polynomials. Lemma 5.6. Let X be a complex algebraic variety, possibly singular, and let IE(X) = p,q α p,q u p v q denote its intersection E−polynomial. If IH * (X) admits a pure Hodge structure then Remark 5.1. When X is nonsingular, IH * (X) = H * (X) so the formula in lemma 5.6 holds for E−polynomial and usual Betti numbers.
Proof of Lemma 5.6. If IH * (X) admits a pure Hodge structure then IH k (X) has weight k for any k = 0 . . . 2 dim X. Recall that intersection cohomology groups satisfy Poincaré-Verdier duality and that this isomorphism maps classes of type (p, q) in classes of type (dim X − p, dim X − q). This concludes the proof.

Computation of the intersection E−polynomial.
The aim of the next two sections is to compute the intersection E-polynomial IE(M G Dol ) for G = SL(2, C) and G = GL(2, C); as a consequence of lemma 5.6 they will give the intersection Betti numbers of the corresponding moduli spaces of Higgs bundles. Before proceeding with computations we shall describe the general strategy. Computing hypercohomology with compact support on both sides of (7), the splitting in the decomposition theorem becomes This equality holds also at the level of E−polynomials: ) and E top (Ω G ) = E(Ω G × H 6 (G)). By additivity of E−polynomials, E(M G Dol ) is given by . In order to obtain IE(M G Dol ), by (9) it is sufficient to subtract from E(M G Dol ) the contributions E top coming from the top cohomology of the fibres over the singular loci.   The proof of the theorem consists in computing the intersection E−polynomial IE(M SL Dol ) and applying lemma 5.6 to get intersection Betti numbers. By (9) one needs first to compute E(M SL Dol ) using the stratification in (10).  Dol is

It is well known that M SL,s
Dol contains the cotangent bundle of the locus S of stable vector bundles with trivial determinant as an open dense subset, but there are several stable Higgs bundles whose underlying vector bundle is not stable. This is due to the fact that not all vector subbundles of V are Higgs subbundles: for example one may consider the bundle V = K −1 C ⊕ K C where K C denotes the canonical bundle on X. This vector bundle is not stable because the subbundle K C has slope greater than the slope of V ; however K C is not a Higgs subbundle because, in order for it to be Φ invariant, Hom(K C , K −1 C ) ∼ = K −2 C should have global sections, which is not the case as it has negative degree.
To determine the E-polynomial of M SL,s Dol one constructs a suitable stratification, computes E-polynomials of all strata and sum them. In particular, as it is of its own interest, we also compute Betti numbers of the strata by systematic employment of the following well known result. Proposition 6.3 (Addivity property of compact support cohomology). Let Y be a quasiprojective variety. Let Z be a closed subset of Y and call U its complement. Then, given the inclusions , there is a long exact sequence in cohomology We stratify the locus of stable Higgs pairs with respect to the stability of the underlying vector bundle: • pairs (V, Φ) with V stable vector bundle; • pairs (V, Φ) with V strictly semistable vector bundle; • pairs (V, Φ) with V unstable vector bundle.
6.2. The stable case. We shall parametrize all stable Higgs bundles (V, Φ) where V is a stable vector bundle.
Proposition 6.4. Let S be the locus of stable vector bundles with trivial determinant. The locus of stable Higgs pairs (V, Φ) with V ∈ S is isomorphic to T * S and its E− polynomial is Proof. Clearly, if V is a stable vector bundle then (V, Φ) is a stable Higgs pair. As a consequence the locus of stable Higgs pairs with stable underlying vector bundle is isomorphic to T * S. Narasimhan and Ramanan [32] proved that the moduli space of semistable vector bundles with trivial determinant on a nonsingular projective curve C of genus 2 is isomorphic to P 3 . A semistable vector bundle V is non stable if and only if is of the form therefore strictly semistable vector bundles are parametrized by K := Jac(C)/Z 2 where Z 2 acts as the involution L → L −1 . The variety K is a compact Kummer surface with 16 singularities, corresponding to the fixed points of the involution, whose desingularization is a K3 surface obtained by blowing up K in the singular points. As a result, the locus S of stable bundles is the complement of K inside P 3 .
Observe that the cohomology of K is the Z 2 invariant part of the cohomology of Jac(C). The cohomology of the Jacobian is generated by H 1 (Jac(C)) and the Betti numbers are The action of Z 2 on the cohomology sends every generator γ of H 1 in −γ thus the even cohomology groups are all Z 2 -invariant, while the odd ones are never. As a result, the Betti numbers of K are Alternatively, one can notice that the cohomology of K differs from the one of its desingularization just in the H 2 part, which has in addition the contribution of the 16 exceptional divisors isomorphic to P 1 , and the Betti numbers of a K3 surface are Observe that since the Hodge structure on the cohomology of Jac(C) is pure and so is the cohomology of K. In particular H 0 (K) has weight 0, H 2 (K) has weight 2 and types 4(1, 1) + (2, 0) + (0, 2), and H 4 (K) has weight 4 of type (2,2). Consider now the inclusions S j / / P 3 J . As both P 3 and K are compact, we have the long exact sequence: which splits in the following sequences First consider (11): the map i ! = i * is a restriction to a hyperplane sections, therefore it is an isomorphism by Lefschetz hyperplane theorem and H 0 c (S) = H 1 c (S) = 0. For (12), i ! is the restriction of the fundamental class of P 1 inside P 2 which remains non-zero when intersecting it generically with K, so i ! is an injection. One has H 2 c (S) = 0 and H 3 c (S) = C 5 . A similar argument shows that, in (13), i ! is an isomorphism and that H 4 c (S) = H 5 c (S) = 0. Clearly, (14) shows that H 6 c (S) ∼ = C. By Poincaré duality the Betti numbers are As T * S is a vector bundle over S, it inherits the cohomology of its base space, so the compact support cohomology groups of T * S are 6.3. Strictly semistable case. Suppose V is a strictly semistable vector bundle. We would like to investigate when V occurs in a stable Higgs pair (V, Φ). Again, one has to distinguish different cases: 6.3.1. Type (i). We shall determine stable Higgs pairs having underlying vector bundle of V = L⊕L −1 with L ∈ Jac(C) such that L ∼ = L −1 . Vector bundles of this form are parametrized by K = Jac(C)/Z 2 . We denote by K 0 locus in K fixed by the involution and by K 0 := K − K 0 its complement. Then the locus of stable Higgs bundles with underlying vector bundle of type (i) will is a fibre bundle over K 0 .
Proposition 6.5. Let S 1 be the locus of stable Higgs bundles with underlying vector bundle of type (i). Then S 1 is a (C 2 × C * )-bundle over K 0 and its E−polynomial is Proof. Consider V = L ⊕ L −1 with L ∈ Jac(C) such that L ∼ = L −1 . We have that thus a Higgs field Φ ∈ H 0 (End 0 (V ) ⊗ K C ) will be of the form is stable if and only if both L and L −1 are not preserved by Φ, that is b, c = 0. Then one needs to understand when two different Higgs fields give rise to isomorphic Higgs bundles: since the automorphisms group of V is C * , two Higgs pairs (V, Φ 1 ) and (V, Therefore, stable Higgs pairs (V, Φ) with fixed underline vector bundle V are parametrized by This is an actual quotient as all the points are semistable with respect to the action of C * . Letting V vary, one obtains a (C 2 × C * )-bundle S 1 over K 0 . Considering S 1 as a sphere bundle over K 0 , the cohomology of the total space is computed by Gysin sequence. Consider the inclusions and the long exact sequence in cohomology with the same weights as the cohomology of K. The Gysin sequence splits in the following subsequences In (16) the map C → C 6 is the product by the Euler class of a nontrivial bundle, which is non zero, therefore H 1 (S 1 ) = 0 and H 2 (S 1 ) = C 5 . Recall that in this case both the cup product with the Euler class and the pushforward increase weights of (1,1). As a consequence of Poincaré duality the compact support cohomology groups of S 1 are H 10 c (S 1 ) = C of type (5,5), Proposition 6.6. Let V be a semistable vector bundle of type (ii). Then there is no Higgs field Φ such that the pair (V, Φ) is stable.
Proof. Consider the universal line bundle L → K 0 × C and let p : K 0 × C → K 0 be the projection onto the first factor. It is well known that non trivial extensions of L by L −1 are parametrized by P(R 1 p * L 2 ): as R 1 p * L 2 is a local system on K 0 of rank one, there exists a unique nontrivial extension up to isomorphism. Consider the universal extension bundle V: it fits in the short exact sequence (18) 0 → L → V → L −1 → 0 and parametrizes all the vector bundles V on C of type (ii). In order for (V, Φ) to be a stable Higgs bundle, the Higgs field must not preserve the subbundle L. By an abuse notation, we denote by K C the pullback of the canonical bundle on C under the projection K 0 × C → C. Tensoring the sequence (18) by K C and applying the covariant functor Hom(V, −) restricted to traceless endomorphisms gives Pushing forward to K 0 gives a long exact sequence In order to prove the proposition, we show that ρ = 0. Applying the contravariant functor p * Hom(−, LK C ) to (18) gives a long exact sequence Consider the fibre of (20) on a point L ∈ K 0 . One has As H 1 (L 2 K C ) = 0, then H 1 (V * LK C ) ∼ = H 1 (K C ) ∼ = C, thus R 1 p * Hom(V, LK C ) is a local system of rank 1 on K 0 × C. Now consider (19) in the fibre over L ∈ K 0 : ext As seen before, H 1 (V * LK C ) ∼ = H 1 (K C ) ∼ = C and H 0 (L −2 K C ) ∼ = C, so the map ext in (19) is either 0 or an isomorphism. However, as V is a nontrivial extension, such a map has to be non zero, thus it is an isomorphism. As a result ρ is zero. 6.3.3. Type (iii). We consider stable Higgs bundles with underlying vector bundle V = L ⊕ L with L ∼ = L −1 ∈ K 0 . Proposition 6.7. Let S 3 be the locus of stable Higgs bundles with underlying vector bundle V = O⊕O. Then the locus S 3 of stable Higgs pairs of type (iii) is the union of 16 copies of S 3 and its E-polynomial is Proof. Up to tensor by L ∈ K 0 one may restrict to the case L = O, so that V is just the trivial bundle (2) and the Higgs field is of the form The bundle is not stable if and only if Φ is conjugate to an upper triangular matrix of elements of H 0 (K C ). As the action of SL(2, C) on H 0 (K C ) ⊗ sl (2) is trivial on the first factor, one can consider Φ ∈ H 0 (K C )⊗ sl (2) .
One can interpret the locus of simultaneously triangulable matrices (A, B) ∈ sl(2) ⊕ sl(2) as a quartic hypersurface in C 6 with coordinates (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) with equation given by annihilation of det[A, B]. where SL(2, C) acts as the simultaneous conjugation on the matrices A and B as in (21).
Corollary 6.9. The locus of stable Higgs bundles of type (iii) is isomorphic to 16 copies of S 3 , one for each point of K 0 .
Our strategy to compute the cohomology of (C 6 − Q) is the following: 1) we decompose Q as a disjoint union of the close set Q 0 = f −1 (0) and its open complement Q−Q 0 = f −1 (C − {0}); 2) we compute the cohomology with compact support of both Q 0 and Q − Q 0 and use the additivity property to compute the cohomology with compact support of Q; 3) again, as C 6 = Q ⊔ (C 6 − Q), by the additivity property of the cohomology with compact support one can compute the cohomology of C 6 − Q.
Observe that α, β, γ are the minors of order 2 of the matrix Also, fixing (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 ) and the corresponding point (α, β, γ) ∈ C, one notice that both (x 1 , x 2 , x 3 ) and (y 1 , y 2 , y 3 ) are orthogonal to (α, γ 2 , β 2 ), i.e. they satisfy the equations 2αx 1 + γx 2 + βx 3 = 0 2αy 1 + γy 2 + βy 3 = 0 If (α, β, γ) = (0, 0, 0), say β = 0, then Substituting these values in (22) and annihilating the minors of order two gives three equations all identical to Since S 3 is nonsingular connected but not compact, . We now consider stable Higgs bundles of type (iv) and we prove the following result. Thus there exists a universal extension bundle on Let p : P 1 × C → P 1 be the projection: as in the type (ii) case one can tensor the short exact sequence above by K C , apply the covariant functor Hom(V, −) and pushforward to P 1 and end up with the long exact sequence ext Stable Higgs bundles are those with Higgs field in the complement of the kernel of the map or, equivalently, the complement of the image of p * Hom(V, K C ) in p * (End 0 (V)⊗K C ). First notice that p * Hom(V, K C ) ∼ = p * K C , which is a vector bundle of rank 2 and similarly that R 1 p * Hom(V, K C ) ∼ = R 1 p * K C . As the extension is nontrivial, the map ext is non zero and its kernel has rank 1. Starting from (27), tensoring with K C , applying the functor Hom(−, O) restricted to traceless endomorphisms and pushing forward to P 1 one obtains another long exact sequence Notice that since R 1 p * K C has rank 1 and the map ext is non zero, the last map is surjective. Hence, the cokernel of p * Hom(V, K C ) → p * K C has rank 1 and consequently p * Hom(V, K C ) has rank 3. Returning to (28), we conclude that p * End 0 (V ) ⊗ K C is a vector bundle of rank 4, thus the locus of stable pairs is fibrewise the complement of a hyperplane. Finally, automorphisms have to be taken into account: the group of automorphisms of a nontrivial extension of O by itself is the additive group (C, +) ⊂ SL(2, C), and an element t ∈ (C, +) acts on the Higgs field Φ as Lemma 6.11. S 4 is a C 2 -bundle over a C * -bundle over P 1 . All bundles are Zariski locally trivial.
Proof. Let A be the kernel of the extension map in (28), minus the zero section: thus A is a C * -bundle over P 1 . One can think of p * (End 0 (V) ⊗ K C ) − p * Hom(V, K C ) as vector bundle of rank 3 over A.
Similarly, the kernel of the extension map of (29) gives rise to a vector bundle A over A of rank 1 and the map p * Hom(V) → p * End 0 (V) ⊗ K C ) lifts to a (C, +)-equivariant map of vector bundles over A whose kernel has rank 2. The automorphism action of (C, +) on A is linear and given by a → a + tc, hence the quotient A/C is A itself. As the map above is equivariant, one has that [p * (End 0 (V) ⊗ K C ) − p * Hom(V, K C )]/C → A/C ∼ = A is a vector bundle of rank 2 over A.
Corollary 6.12. The locus of stable Higgs bundles of type (iv) is isomorphic to 16 copies of S 4 , one for each point of K 0 . Lemma (6.11) allows to compute the Betti numbers of S 4 : since S 4 is homotopy equivalent to a C * -bundle on P 1 , the Gysin sequence reads as (30) H 0 (S 4 ) ∼ = H 0 (P 1 ) ∼ = C; As the central map of (31) is the cup product with the Euler class of the bundle A, which is nontrivial, therefore it is non-zero and we have H 1 (S 4 ) = H 2 (S 4 ) = 0. By Poincaré duality, the E−polynomial of S 4 is E(S 4 ) = u 4 v 4 − u 2 v 2 By corollary 6.12 this completes the proof of proposition 6.10.
6.4. Unstable case. Consider the locus U of stable Higgs bundles (V, Φ) where V is an unstable vector bundle with trivial determinant. Then there exists a line bundle L of degree d > 0 that fits an exact sequence then the bundle L −2 K C has no non-zero global section because it has negative degree, hence L is Φ-invariant for any Higgs field Φ ∈ H 0 (End 0 (V ) ⊗ K C ). The only case we have to check is deg(L) = 1. Since the line bundle L −2 K C has degree 0, it has global sections if and only if it is trivial, i.e. is L = K 1 2 is one of the 16 roots of the canonical bundle K C . As a consequence, if there exists an unstable vector bundle V which is stable as a Higgs bundle, then it must be an extension of those bundles by their duals. Proposition 6.13. The locus U of stable Higgs bundles (V, Φ) with V unstable is isomorphic to 16 copies of C 3 , one for any root of the canonical bundle K C . As a consequence its cohomology with compact support is given by the generic Higgs field will be of the form Two Higgs fields define isomorphic Higgs bundles if and only if they are conjugate by an automorphism of the bundle, which will lie in C * × (H 0 (K C ), +) ⊂ SL(2, C). The action of C * on the Higgs field is precisely the one seen in the type (i) case. Therefore isomorphism classes of stable Higgs bundles are parametrized by the disjoint union of 16 copies of Consider the action of the automorphism group (C 2 , +): any element ζ ∈ H 0 (K C ) = C 2 acts on Φ as Such an action is linear and free on a ∈ H 0 (K C ) and whenever we fix a − ζc then the value of b + 2ζa − ζ 2 c is fixed as well. Therefore the quotient of H 0 (K C ) × H 0 (K 2 C ) by (C 2 , +) is C 3 . Non trivial case Non-trivial extensions of K 1 2 by K − 1 2 are parametrized by P(H 1 (K −1 )) = P 2 and fit in the exact sequence Tensor by K C and apply the functor Hom(V, −) restricted to traceless endomorphisms. Taking global sections one obtains Again, a Higgs bundle with V as underlying vector bundle is stable if and only if its Higgs field lies in the complement of the kernel of ρ. We shall prove that there are no stable bundles of this type, i.e. that ρ = 0.
Since H 0 (Hom(K 1 2 C , K 1 2 C )) ∼ = C this is equivalent to ask that the map ext is non zero, which is the case as we are considering nontrivial extensions. As a result there are no non-trivial extensions of K 6.5. Cohomology ofΣ SL \Ω SL andΩ SL .
Proof. Recall thatΩ consists of 16 copies of a nonsingular hypersurface G in P 4 . Therefore its cohomology is given by The E-polynomial ofΩ is First we notice that Σ SL = (Jac(C) × H 0 (K C )/Z 2 has the same cohomology as K, so Moreover, by lemma 5.6 we deduce P t (M SL Dol ) = 1 + 2t 2 + 23t 4 + 34t 6 .
We are now ready to prove the main theorem of the section.
Proof of theorem 6.1 . By (9), subtracting from E(M SL Dol ) the contributions E top coming from top cohomology of the fibres gives the intersection E−polynomial of M Dol : The intersection Betti numbers are obtained again by applying lemma 5.6. with the same fibres over the strata as in the SL(2, C). We stratifyM GL Dol as follows Observe that we as the fibres ofπ over both Ω GL and Σ GL \ Ω GL are irreducible, then the monodromy of the local system is trivial. Moreover since Ω GL is nonsingular and Σ GL have finite quotient singularities we have where the shifts (−1) and (−3) correspond to the Hodge structures Q(−1) of respectively H 2 (P 1 ) and H 6 (G). Taking hypercohomology with compact support in (7), we obtain the intersection cohomology groups and the splitting in the decomposition theorem becomes We compute the intersection E−polynomial and use lemma 5.6 to obtain intersection Betti numbers. Observe that and that by (9) and (10) ) − E(Ω GL × H 6 (G)); The rest of the section is devoted to computing each term of the equations above.
7.2. The stable case. We shall parametrize all stable Higgs pairs (V, Φ) where V is a stable vector bundle.
Proposition 7.3. Let N s be the locus of stable vector bundles on C. The locus of stable Higgs pairs (V, Φ) with V ∈ N s is isomorphic to the cotangent bundle T * N s and its E−polynomial is Proof. Suppose V ∈ N s . Since the stability of V ensures the stability of the Higgs pair (V, Φ) for all Φ, the locus of stable Higgs pairs with stable underlying vector bundle is isomorphic to the cotangent bundle T * N s . The moduli space N of semistable vector bundles on C is isomorphic to a P 3 -bundle over Jac(C). Namely, the fibre over a point ζ ∈ Jac(C) is a copy of the moduli space of semistable vector bundles of degree 0 and rank 2 with determinant ζ. From now we denote Jac(C) by J . A semistable vector bundle V is non stable if and only if it is of the form and therefore strictly semistable vector bundles are parametrized by the symmetric product J (2) . As a consequence N s is the complement of J (2) in N .
We compute E−polynomials of both N and J (2) , then E(N s ) = E(N ) − E(J (2) ). The cohomology of N has been computed by Kirwan in [25] and the E− polynomial is The cohomology of J (2) is the Z 2 invariant part of the cohomology of J × J . Alternatively one can use Macdonald formula [31] for symmetric product of surfaces. One has 7.3.1. Type (i). We shall determine stable Higgs bundles (V, Φ) with underlying vector bundle of type (i). Strictly semistable vector bundles are parametrized by J (2) . Let J 0 be the diagonal in J (2) fixed by the involution and let J 0 := J (2) − J 0 be its complement. The locus of stable Higgs bundles with underlying vector bundle of type (i) is a fibre bundle on J 0 . Proposition 7.4. Let N 1 be the locus of stable Higgs bundles with underlying vector bundle of type (i). Then N 1 is a (C 4 × C * )-bundle over J 0 and its E-polynomial is Proof. To compute the fibre we consider V = L ⊕ M with L, M ∈ J .
is stable if and only if both L and M are not preserved by Φ, that is b, c = 0. Since the automorphisms group of V is C * × C * , two Higgs pairs (V, Φ 1 ) and (V, Therefore, the stable Higgs pairs (V, Φ) with fixed underline vector bundle V are parametrized by (note that C * × C * acts with stabilizer C * ). Letting V vary, one obtains a (C 4 × C * )-bundle S 1 over J 0 . The E−polynomial is the product E(J 0 )E(C 4 )E(C * ). Since  Tensoring the sequence (35) by K C and applying the covariant functor Hom(V, −) restricted to traceless endomorphisms one obtains Pushing forward to J 0 one gets a long exact sequence (36) 0 ext Applying the functor p * Hom(−, MK C ) to (35) yields a long exact sequence The map ρ is the composition We prove that the second map is 0, i.e. there are no stable Higgs bundles of type (ii). Consider the fibre of (37) on a point (L, M ) ∈ J 0 : one has Since the extension map is non zero and H 1 ( In particular the map p * Hom(V, MK C ) → p * Hom(L, MK C ) is zero. 7.3.3. Type (iii). We now consider stable Higgs bundle with underlying vector bundle V = L ⊕ L with L ∈ J .
Proposition 7.6. Let N 3 be the locus of stable Higgs bundles with underlying vector bundle L ⊕ L with L ∈ J . Then the locus N 3 of stable Higgs pairs of type (iii) is a Zariski locally trivial N 3 -bundle over J and its E−polynomial is Proof. Consider V = L ⊕ L. In this case H 0 (End(V ) ⊗ K C ) ∼ = H 0 (K C ) ⊗ gl(2) ∼ = C 2 ⊗ gl(2) and the Higgs field is of the form with a, b, c, d ∈ H 0 (K C ). The bundle is not stable if and only if Φ is conjugate to an upper triangular matrix of elements of H 0 (K C ). The action of PGL(2, C) on H 0 (K C )⊗gl (2) is trivial on the first factor, so one can proceed as in the case of SL(2, C) looking for the couples of matrices (A, B) ∈ gl(2) ⊕ gl(2) that are not simultaneously triangulable. As before, A and B are simultaneously triangulable if det([A, B]) = 0. Writing where s = (x 1 − x 4 ) and t = y 1 − y 4 . Thanks to this substitution, one can interpret the locus of simultaneously triangulable matrices (A, B) ∈ gl(2) ⊕ gl(2) as a fibration over in the coordinates (s, x 2 , x 3 , t, y 2 , y 3 ) ∈ C 6 . The fibre over a point in Q is the 2-dimensional vector space in the coordinates (x 1 , x 4 , y 1 , y 4 ). Let H be the total space of this bundle over Q. One can summarize the above considerations in the following lemma. where the action of PGL(2, C) is the simultaneous conjugation on the matrices A and B as in (38).
Since H is a C 2 -bundle over Q, To obtain E(N 3 ) one needs to multiply E(N 3 ) by E(J ).
7.3.4. Type (iv). We now consider stable Higgs bundles of type (iv) and we prove the following result.
Proposition 7.8. Fix L ∈ J and let N 4 be the locus of stable Higgs bundles whose underlying vector bundle is a nontrivial extension of L by itself. Then the locus N 4 of stable Higgs bundles of type (iv) is a Zariski locally trivial N 4 -bundle over J and its E−polynomial is Proof. Let L ∈ J and let V be a nontrivial extension of L by itself: the isomorphism classes of such bundles are parametrized by (39) P(Ext 1 (L, L)) ∼ = P 1 .
Thus there exists a universal extension bundle on P 1 × C 0 → L → V → L → 0.
Let p : P 1 × C → P 1 be the projection map: as in the type (ii) case, one can tensor the above short exact sequence by K C , apply the covariant functor Hom(V, −) and pushforward to P 1 , getting the long exact sequence (40) 0 p * Hom(V, LK C ) p * (End(V) ⊗ K C ) p * Hom(V, LK C ) R 1 p * Hom(V, LK C ) R 1 p * (End(V) ⊗ K C ) R 1 p * Hom(V, LK C ) 0.
ext Starting again from (39), tensoring with K C , applying the contravariant functor Hom(−, L) and pushing forward to P 1 one obtains another long exact sequence (41) 0 p * K C p * Hom(V, LK C ) p * K C R 1 p * K C . ext As before, stable Higgs bundles are those whose Higgs field is in the complement of the kernel of the map p * End(V) ⊗ K C p * Hom(V, LK C ) p * K C .
Consider the sequence (41): p * K C has rank 2. Since R 1 p * K C has rank 1, being non zero, the map ext is surjective. Hence, the cokernel of p * Hom(V, K C ) → p * K C has rank 1 and consequently p * Hom(V, K C ) has rank 3. Looking at (40) one concludes that p * End(V )⊗K C is a vector bundle of rank 5. The group of automorphisms of a nontrivial extension of L by itself is the additive group (C, +) ⊂ GL(2, C), and an element t ∈ C acts on the Higgs field Φ by conjugation: Lemma 7.9. N 4 is a C 2 -bundle over a C 2 -bundle over a C * -bundle over P 1 .
Proof. Let A be the kernel of the extension map in (40) minus the zero section: A is a (C 2 − {0})bundle over P 1 . We can think of p * (End 0 (V) ⊗ K C ) − p * Hom(V, K C ) as a vector bundle of rank 3 over A. Similarly, the kernel of the extension map in (41) gives rise to a vector bundle A over A of rank 1 and the map p * Hom(V) → p * (End(V) ⊗ K C ) lifts to a (C, +)-equivariant map [p * (End(V) ⊗ K C ) − p * Hom(V, K C )] → A of vector bundles over A whose kernel is of rank 2. Observe that A is invariant under the automorphism action, while A and p * (End(V)⊗K C )−p * Hom(V, K C )] are not. In this way we have that p * (End(V)⊗ K C ) − p * Hom(V, K C )]/C is a C 2 -bundle on A which is a C 2 -bundle over A/C * , which is a C * -bundle over P 1 .
As all bundles are Zariski locally trivial one has: 7.4. Unstable case. Consider the locus N U of stable Higgs bundles (V, Φ) where V is an unstable vector bundle. Then there exists a line bundle L of degree d > 0 that fits an exact sequence 0 / / L / / V / / M / / 0 with M ∈ P ic d (C). If d > 1 the bundle L −1 M K C has no non-zero global section because it has negative degree, hence L is Φ-invariant for any Higgs field Φ ∈ H 0 (End(V ) ⊗ K C ). The only case to check is deg(L) = 1. The line bundle L −1 M K C has degree 0: it has global sections if and only if it is trivial, that is M = LK −1 C with L ∈ P ic 1 (C). As a consequence, if there exists an unstable vector bundle V which is stable as a Higgs bundle, then it must be an extension of the above form.
Proof. Trivial case Consider V = L ⊕ M with L ∈ P ic 1 (C) and M = LK −1 C . Then H 0 (End(V ) ⊗ K C ) ∼ = H 0 (K C ) ⊕ H 0 (K 2 C ) ⊕ H 0 (O) ⊕ H 0 (K C ) ∼ = C 2 ⊕ C 3 ⊕ C ⊕ C 2 Thus the generic Higgs field will be of the form Φ = a b c d with a, d ∈ H 0 (K C ), b ∈ H 0 (L 2 K C ), c ∈ H 0 (L −2 K C ). Two Higgs fields define isomorphic Higgs bundles if and only if they are conjugate by an automorphism of the bundle, which lies in C * × C * × (H 0 (K C ), +) ⊂ GL(2, C). The action of C * × C * on the Higgs field is the one seen in the type (i) case and it has stabilizer C * . Therefore isomorphism classes of stable Higgs bundles are parametrized by Thanks to the action of C * × C * we can suppose c = 1. Then we have to consider the action of ζ ∈ (H 0 (K C ), +): Such an action is linear and free on a ∈ H 0 (K C ) and we can fix a + ζ = 0. Therefore the quotient of H 0 (K C ) 2 × H 0 (K 2 C ) by (H 0 (K C ), +) is H 0 (K C ) 2 × H 0 (K 2 C ) ∼ = C 5 .

Non trivial case
Non-trivial extensions of L by M are parametrized by P(H 1 (L −1 M )) = P 2 and fit the exact sequence 7.5.2. Cohomology ofΣ GL \Ω GL .
By lemma 5.6, we get the Poincaré polynomial and the Hodge diamond.