Intersection cohomology of rank two character varieties of surface groups

For $G = \mathrm{GL}_2, \mathrm{SL}_2, \mathrm{PGL}_2$ we compute the intersection E-polynomials and the intersection Poincar\'e polynomials of the $G$-character variety of a compact Riemann surface $C$ and of the moduli space of $G$-Higgs bundles on $C$ of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.


Introduction
Let C be a compact Riemann surface of genus g ≥ 2, and G be a complex reductive algebraic group. The G-character variety of C, or Betti moduli space, is the affine GIT quotient M B (C, G) := Hom(π 1 (C), G) G It parametrises isomorphism classes of semi-simple representations of the fundamental group of C with value in G. Remarkably, the space M B (C, G) is homeomorphic to the Dolbeault moduli space M Dol (C, G), parametrising instead isomorphism classes of semistable principal G-Higgs bundles on C of degree zero; see [66]. For example, a GL n -Higgs bundle is a pair (E, φ) with E vector bundle of rank n and φ ∈ H 0 (C, End(E) ⊗ K). Such a pair is an SL n -Higgs bundle, if in addition the determinant of E is trivial and the trace of φ vanishes.
Since the pioneering work of Hitchin [37], the (non-algebraic) homeomorphism between M B (C, G) and M Dol (C, G) have been exploited to study the topology of character varieties. The main result of this paper is the computation of some geometric invariants of M Dol (C, G) and M B (C, G), namely the intersection E-polynomials and the intersection Poincaré polynomials of M Dol (C, G) and M B (C, G) for G = GL 2 , SL 2 , PGL 2 ; see §1.2.
The motivation for this work stems from the newly stated P=W conjectures [26, Conjecture 1.2, 1.4, 1.5] for singular character varieties; see also [15,Question 4.1.7] and the seminal paper [13]. In fact, the explicit knowledge of intersection E-polynomials and intersection Poincaré polynomials is an essential ingredient in the proof of the P=W conjectures in rank 2 and genus 2 in [26,Main Theorem].
Here, as an applications of our computations, we collect in Theorem 1.1 several results related to the P=W conjectures in rank 2 and arbitrary genus. For brevity, we simply write M B for M B (C, G) and M Dol for M Dol (C, G) when G = GL 2 or SL 2 , and we suppress subscripts B or Dol when we refer indifferently to the Betti or the Dolbeault side. B. the PI=WI conj. for SL 2 is equivalent to the PI=WI conj. for GL 2 ; C. IH <4g−6 (M ) is freely generated by tautological classes; D. the mixed Hodge structure on H * (M Dol ) is not pure for g > 3; E. the P=W conjecture for any resolution of M (C, GL n ) fails, when M (C, GL n ) does not admit a symplectic resolution 1 .
In §5 we recall the content of the P=W conjectures, and we give a proof of Theorem 1.1. Here we briefly explain the relevance of the previous statements in view of the P=W conjectures.
A. Theorem 1.1.A (Theorem 5.4) provides a numerical evidence for the PI=WI conjecture. Indeed, the PI=WI conjecture implies the palindromicity of IE(M B ).
B. Theorem 1.1.B (Corollary 5.6) is a useful reduction statement. It says that it is enough to prove the PI=WI conjecture only for a portion of the intersection cohomology, namely its Γ-invariant part; see (5) and (6).
C. The known proofs of the P=W conjecture for twisted character varieties [13] and [16] (cf also §5.1) rely on the generation by tautological classes of the Γ-invariant part of H * (M ). This is unknown for the intersection cohomology of the singular moduli spaces. Theorem 1.1.C (Theorem 5.7) provides a partial answer, i.e. the tautological generation of the intersection cohomology in low degree.
Our strategy to compute the intersection E-polynomials of M is to use the Kirwan-O'Grady desingularization π T : T → M ( §3. 2), and determine all the summands of the decomposition theorem for π T ; cf [44,Remark 2.28]. This is a subtle task that we can complete thanks to a tight control of the geometry of π T .
where • Σ is the singular locus of M ; • Ω is the singular locus of Σ; • i Σ • : Σ • := Σ \ Ω → Σ is the natural inclusion; • L is the rank-one local system on Σ • corresponding to a quasi-étale double cover q : Σ ι → Σ branched along Ω (see Definition 3.3); • b(−3g + 3 + j) is the coefficient of the monomial q j in the polynomial Note in addition that the same decomposition holds for the Mukai moduli space of semistable sheaves on K3 or abelian surfaces with Mukai vector v = 2w ∈ H * alg (S, Z), where w is primitive and w 2 = 2(g − 1), thus suggesting other potential applications of Theorem 1.2. This is indeed a consequence of the stable isosingularity principle (Theorem 2.11), which roughly says that Betti, Dolbeault and Mukai moduli spaces have the same type of singularities, in the sense of Definition 2.6.
It is conceivable that the computation of the intersection E-polynomials in rank 3 can be pursued with no substantial conceptual difference. In higher rank however this seems hard. In fact, closed formulas may be cumbersome and less enlightening. Notwithstanding, we believe that the rank 2 case can inspire the investigation of the higher rank case, especially in relation to the P=W conjectures [26] and the Hausel-Thaddeus topological mirror symmetry conjecture for singular character varieties [32, Remark 3.30].

Notation
The intersection cohomology of a complex variety X with middle perversity and rational coefficients is denoted by IH * (X). Ordinary singular cohomology with rational coefficients is denoted by H * (X). The subscript c stands for compactly supported intersection or ordinary cohomology, respectively IH * c (X) and H * c (X). Recall that they all carry mixed Hodge structures.
The Poincaré polynomial, the intersection Poincaré polynomial, the intersection Euler characteristic, the E-polynomial and the intersection E-polynomial are defined by We will often write q := uv.
The action of a finite group Γ on X induces the splitting where H d (X) Γ is fixed by the action of Γ, and H d var (X) is the variant part, i.e. the unique Γ-invariant complement of H d (X) Γ in H d (X). Analogous splittings hold for the Γ-modules H d c (X), IH d (X) and IH d c (X). The label Γ or var , written after the polynomials above, imposes to replace ordinary (or intersection) cohomology with its Γ-invariant or Γ-variant part respectively, e.g. IE(X) Γ = r,s,d (−1) d dim(Gr W r+s IH d c (X, C) Γ ) r,s u r v s . If ι : X → X is an involution, we simply use the superscript + or − to denote the ι-invariant and ι-variant part, e.g. P (X) We always denote by C a complex projective curve of genus g ≥ 2, unless differently stated. For notational convenience, we simply write M B for M B (C, G) and M Dol for M Dol (C, G) when G = GL 2 or SL 2 , and we suppress subscripts B or Dol when we refer indifferently to the Betti or the Dolbeault side. We adopt the same convention for the strata Σ B (C, G), Σ Dol (C, G), Ω B (C, G), Ω Dol (C, G).

Computations
As an application of the decomposition theorem (Theorem 1.2), we can express IE(M ) as a function of the E-polynomials of M , Σ ι and Ω; see Proposition 3.2 for the definition of these strata, and §4.1 for the proofs of the following expressions.
Remark 1.7. The intersection E-polynomial of M B (C, SL 2 ) is a polynomial in q 2 . This fails for twisted SL 2 -character varieties (cf [52, (2)]; see also §5.1 for the definition of the twist), but it holds true for twisted PGL 2 -character varieties, since their cohomology is generated by classes of weight 4; see [34,Proposition 4.1.8]. The E-polynomial of M B (C, SL 2 ) is a polynomial in q 2 too; see [48,Theorem 2] or [4,Theorem 1.3].
An explicit formula for the E-polynomial of the smooth locus of M Dol (C, SL 2 ) was computed in [41,Theorem 3.7]. Together with Theorem 1.8, this gives the intersection Poincaré polynomial of M (C, SL 2 ).
At this point it is worthy to recall how the (intersection) cohomology of M (C, G) with G = GL n , SL n and PGL n compare, and how to extend the previous results for the SL 2 case to PGL 2 and GL 2 .
The morphism alb : M (C, GL n ) → M (C, GL 1 ) sends a representation or a Higgs bundle to its associated determinant representation or Higgs bundle. It is an étale locally trivial fibration with monodromy group Γ ≃ (Z/2Z) 2g and fibre isomorphic to M (C, SL n ). The quotient of M (C, SL n ) for the residual action of Γ is M (C, PGL n ). Hence, there exist morphisms of mixed Hodge structures Analogous splittings hold for IH * , H * c , IH * c . A proof of these facts can be found for instance in [26, §3.2].
The analogues of Theorems 1.8 and 1.9 for PGL 2 can be obtained by substituting all the occurrence of the coefficient 2 2g with 1 in the formulas of the theorems; see Remark 4.3. According to (5), the corresponding polynomials for GL 2 are the product of the E-polynomials or the (intersection) Poincaré polynomial for PGL 2 with E(T * Jac(C)) = (uv) g (1 − u) g (1 − v) g or P t (Jac(C)) = (t + 1) 2g respectively.
Here, as a corollary of Theorems 1.4, 1.9 and (6), we study the portion of IH * (M (C, SL 2 )) on which Γ acts non-trivially. The following should be considered the untwisted analogue of [35,Proposition 8.2] in rank 2. This suggests that intersection cohomology may be the right cohomology theory to formulate a topological mirror symmetry conjecture for M (C, SL n ) and M (C, PGL n ); see [32,Remark 3.30] and [49].
As a byproduct, we also obtain the E-polynomials and the Poincaré polynomial of the Kirwan-O'Grady desingularization π T : T → M . We write explicitly the E-polynomials of M (C, SL 2 ), and leave the straightforward computations of the other polynomials to the reader (cf §4.2 and §2.3).
In particular, the E-polynomial E(T B ) is palindromic.

Outline and relation with other work
• In §2 we collect some preliminary results: the intersection cohomology of an affine cone, the decomposition theorem, some properties of the mixed Hodge structures of singular semi-projective varieties (cf [34]), and the stable isosingularity principle (implicitly used in [41, p.834]). Analogous degeneration techniques employed to establish the stable isosingularity principle have been discussed in [16] and [17, §4] in relation to twisted character varieties.
• In §3 we describe the singularities of M and the geometry of the Kirwan-O'Grady desingularization π T : T → M . This part highly relies on [56] and [41]. The computation of the Poincaré polynomial of the incidence variety I 2g−3 in [41] contains a mistake, and we fix it in §3.3.
• In §3.4 we use several times the decomposition theorem to determine the intersection cohomology of the normal slice to strata of a Whitney stratification of M .
The singularities of the Betti and Dolbeault moduli spaces are locally modelled on Nakajima quiver varieties which usually do not admit a symplectic resolution. Although a lot is known about the intersection cohomology of quiver varieties with symplectic resolutions (see for instance [54]), the local computations in §3.4 seem new.
• In §4 we complete the proof of Theorem 1.2. Then in §4.1 and §4.2 we argue how to compute the intersection E-polynomials and intersection Poincaré polynomial of M , and we prove the results of §1.2. The Epolynomial of M are known thanks to [48], [4] and [41], while the ordinary Poincaré polynomial of M appears in [12]. Despite the active research in the field and the stimuli from the PI=WI conjecture, there are few previous works exhibiting explicit computations of the intersection cohomology of Dolbeault and Betti moduli spaces; see [25] and [26].
In [40] Kiem studied the intersection cohomology of character varieties with coefficients in a compact Lie group. However, the methods in [40] do not extend to the complex reductive case, since for a general complex reductive group G the representation space Hom(π 1 (C), G) is not smooth, and the quotient map Hom(π 1 (C), G) → Hom(π 1 (C), G) G is not placid in the sense of [30].
We mention another remarkable precedent. The Dolbeault moduli space is a partial compactification of the cotangent bundle of the moduli space of stable vector bundles. The intersection cohomology of the moduli space of semistable vector bundles was determined in [43] for rank two (or equivalently in [40]), and in [53]

Intersection cohomology of affine cones
Let X be a complex projective variety of dimension n − 1 with an ample line bundle L. The graded ring associated to L is the graded C-algebra The affine cone over X with conormal bundle L is Let s 1 , . . . , s N be a set of generators for R(X, L) of degree m 1 , . . . , m N . Then there exists an embedding C(X, L) ⊆ C N such that C(X, L) is invariant with respect to the G m -action Conversely, any affine variety with a G m -action and a fixed point which is attractive for t → 0 is isomorphic to an affine cone; see for instance [21, §3.5].
All the singularities of this paper are locally modelled on affine cones, whose coordinate rings are not necessary generated in degree one. For this reason here we compute their intersection cohomology, thus generalising [19, Example 2.2.1].
Proof. Denote by C(X, L) * := C(X, L) \ {vertex} the punctured affine cone. By [23,Lemma 1] or [45,Proposition 4.7.2], we can write Suppose now that R(X, L) is generated in degree one. Then the blow-up of the origin is the total space of the line bundle L * . By the hard Lefschetz theorem, the relative long exact sequence of the inclusion C(X, L) * ֒→ BC(X, L) splits into the short exact sequences Therefore, we obtain that for d < n If R(X, L) is not generated in degree one, then BC(X, L) is the total space of a line bundle only up to a finite cover; see [57, §1.2]. More precisely, consider the finite morphism g : (7). We see that C(X, L) is the quotient of V ′ by the finite group A = (Z/m 1 Z) × . . . × (Z/m N Z) acting on V ′ by coordinatewise multiplication.
V ′ has a G m -action defined by t · (x 1 , . . . , x N ) = (tx 1 , . . . , tx N ), and covering the G m -action on C(X, L) given by (7). Since the G m -action on V ′ has weight one, X ′ is the spectrum of a graded algebra generated in degree one, say V ′ = C(X ′ , L ′ ) for some projective variety X ′ and ample line bundle L ′ . In particular, there exists a commutative diagram where p and p ′ are blow-ups of the vertices of the cones, i and i ′ are the embedding of the exceptional divisors, and the vertical arrows are quotients with respect to (the lift of) the action of A. Thus we have IH * (C(X ′ , L ′ )) A ≃ IH * (C(X, L)).
The discussion above shows that the sequences (8) are exact for C(X ′ , L ′ ), and it is A-equivariant by the commutativity of (9). Taking invariants, we show then that (8) holds for C(X, L) unconditionally.

The perverse Leray filtration
In this section we briefly recall the statement of the decomposition theorem and the definition of the perverse filtration.
For a complex algebraic variety X let D b (X, Q) be the bounded derived category of complexes of sheaves of Q-vector spaces with algebraically constructible cohomology. Denote the full abelian subcategory of perverse sheaves by Perv(X) and the perverse cohomology functors by p H i : D b (S, Q) → Perv(X); see [5] or [19].
Let M HM alg (X) be the category of algebraic mixed Hodge modules with rational coefficients, and D b M HM alg (X) its bounded derived category. Let p H i : D b M HM alg (X) → M HM alg (X) be the cohomology functors; see [62] or [63].
The simple objects of D b M HM alg (X) (resp. D b (X, Q)) are the intersection cohomology complexes IC X (L), where L is a polarizable variation of pure Hodge structures (resp. a local system) on a Zariski-open subset of the smooth locus of X. We denote simply by IC X the complex IC X (Q X\Sing(X) ). In particular, There is a forgetful functor rat : with p H i and pushforward Rf * , and maps M HM alg (X) in Perv(X). We will make no notational distinction between K ∈ D b M HM alg (X) and rat(K).
The decomposition theorem of Beilinson-Bernstein-Deligne-Gabber, or its mixed Hodge module version by Saito, says that there is an isomorphism in where L i,l are polarizable variations of pure Hodge structures (resp. local systems) on the strata of a stratification Y = l Y i,l ; see [5] and [61].
The perverse (Leray) filtration is When Y is affine, de Cataldo and Migliorini provided a simple geometric characterization of the perverse filtration; see [20,Theorem 4 This means that the cocycle η ∈ IH d (X) belongs to P k IH d (X) if and only if its restriction to

Mixed Hodge structure of semi-projective varieties
In order to compute the intersection Poincaré polynomial of M , we observe that M Dol and T Dol are semi-projective.
A semiprojective variety is a complex quasi-projective variety X with a G m -action such that: • the fixed point set Fix(X) is proper; • for every m ∈ X the limit lim λ→0 λ · m exists.
The core of X, denoted Core(X), is the (proper) union of the repelling sets of Proposition 2.3. Let X be a semi-projective variety. Then the inclusion Core(X) ⊂ X is a homotopy equivalence.
Proof. The flow R + × X → X, induced by the G m -action, defines a deformation retract of X onto a neighbourhood of Core(X), which in turn is homotopy equivalent to Core(X).
Proposition 2.4. If X is a semiprojective variety, then 1. the mixed Hodge structure on IH * (X) is pure; Proof. Let f : X → X be a G m -equivariant resolution of singularities of X. Then X is smooth and semiprojective, and it has pure cohomology by [34,Corollary 1.3.2]. Via the decomposition theorem, the mixed Hodge structure on IH * (X) ⊂ H * (X) is pure too.
As X retracts onto the proper algebraic variety Core(X), the weight filtration The resolution f induces a surjective morphism Core( X) → Core(X) of proper algebraic varieties. Hence, by [60, Corollary 5.43] we have The multiplicative group G m acts on M Dol (C, SL n ) by λ · (E, φ) = (E, λφ). The Hitchin fibration assigns to (E, φ) the characteristic polynomial of the Higgs field φ. By [66, The- The same argument works for G = GL n , PGL n as well.

Stable isosingularity principle
Let S be a smooth projective K3 surface or an abelian surface. In this section we establish a stable isosingularity principle for the (non-proper) Dolbeault moduli spaces M (C, GL n ) and M (C, SL n ) and the (proper) Mukai moduli spaces M (S, v) and K(S, v). This means that these moduli spaces have the same analytic singularities, up to multiplication by a polydisk. The upshot is that: • the description of the local model of the singularities of M (S, v) in [56] or [8] holds for M (C, GL 2 ) and M (C, SL 2 ) mutatis mutandis; • the same sequence of blow-ups which desingularises M (S, v) in [56] resolves the singularities of M (C, GL 2 ) and M (C, SL 2 ) mutatis mutandis; • the description of the summands of the decomposition theorem in Theorem 1.2 holds for M (C, GL 2 ), M (C, SL 2 ), M (S, v) and K(S, v) with Mukai vector v = 2w ∈ H * alg (S, Z), where w is primitive and w 2 = 2(g − 1). We briefly recall the definition of Mukai moduli space. Fix an effective Mukai vector 2 v ∈ H * alg (S, Z). Define M (S, v) the moduli space of Gieseker H-semistable sheaves on S with Mukai vector v for a sufficiently general polarization H (which we will typically omit in the notation); see [65, §1]. Further, if S is an abelian variety with dualŜ, and dim M (S, v) ≥ 6, then the Albanese morphism alb : M (S, v) → S ×Ŝ is isotrivial, and we set K(S, v) := alb −1 (0 S , O S ).
Remark 2.5 (Donagi-Ein-Lazarsfeld degeneration). Mukai moduli spaces should be thought as locally trivial deformations of Dolbeault moduli spaces as follows. Fix an ample curve C ⊂ S of genus g ≥ 2. Donagi, Ein and Lazarsfeld shows in [22] that there exists a flat family π W : W → P 1 such that Unless g = 2 and n = 2, M (S, (0, nC, −nC 2 /2)) and M Dol (C, GL n ) have Qfactorial terminal symplectic singularities; see [ [55,Theorem 17] the morphism π W is locally analytically trivial 3 . Roughly this means that the two moduli spaces have the same singularities. We make this statement precise in Proposition 2.10.
We start by stating the notion of stable isosingularity.
Definition 2.6. The varieties X and Y are stably isosingular if there exist complex Whitney stratifications 4 by (smooth non-necessarily connected) locally Zariski-closed subsets X i and Y i such that 2. the posets of closed subsets {X i } and {Y i } ordered by inclusion are equal; 3. the normal slices through X i and Y i are locally analytically isomorphic.
If dim X = dim Y , then we say that X and Y are isosingular.
It is implicit in 3. that the stratifications above are analytically equisingular along each stratum, i.e. the analytic type of the normal slices through x ∈ X i (resp. y ∈ Y i ) is independent of x (resp. y). Not all algebraic variety admits such a stratification; see [68,Example 13.1]. However, the moduli spaces considered below will satisfy the following stronger condition of analytic normal triviality.
Definition 2.7. A Whitney stratification X = i X i is analytically trivial in the normal direction to each strata, if for any x ∈ X i there exists a normal slice N x through X i at x, and a neighbourhood of x in X which is locally Note that if X and Y are stably isosingular via Whitney stratifications which are analytically trivial in the normal direction, and a sequence of blow-ups along (the strict transforms of) some X i gives a desingularization of X, then the same sequence of blow-ups along the corresponding strata Y i gives a desingularization of Y . In addition, if X and Y are isosingular, then an analytic neighbourhood of any point of X is isomorphic to an analytic neighbourhood of some point in Y .
Example 2.8 (Analytically trivial fibrations). Let f : X → Y be an analytic locally trivial fibration, and suppose that F := f −1 (y), with y ∈ Y , admits an analytically equisingular Whitney stratification. Then X and F are stably isosingular. Indeed, by the local triviality, any Whitney stratification of f −1 (y) can be lifted to a Whitney stratification on X with the same normal slices.
In particular, if W is a smooth algebraic variety, F and F × W are stably isosingular. If Y is a Nakajima quiver variety 5 , then the stratification by orbit type is a complex Whitney stratification, which is analytically trivial in the normal direction to each stratum, due to [50,Proposition 4.2].
M B (C, GL n ) and M (S, (0, nC, −nC 2 /2)) are PGL N -quotients, and the quadraticity of the deformation spaces imply that they are locally modelled on Nakajima quiver varieties; see [6, Theorem 2.5] and [2, Proposition 6.1]. By construction, the stratifications by orbit type of M B (C, GL n ) and M (S, (0, nC, −nC 2 /2)) are locally isomorphic to stratification by orbit type of quiver varieties, and so they are complex Whitney stratifications, analytically trivial in the normal direction to each stratum.
A singular point of either moduli space is a polystable objects There exists a clear geometric argument for Proposition 2.10, sketched below.
Sketch of the proof of Proposition 2.10 via a degeneration argument. Via the Donagi-Ein-Lazarsfeld degeneration one can actually prove that M Dol (C, GL n ) is isosingular to a neighbourhood of a nilpotent cone of M (S, (0, nC, −nC 2 /2)) as defined in [22, §2]. This is an analytic open set of the Mukai moduli space that intersects all the orbit type strata, if C generates the Picard group of S. In order to extend the result to the whole Mukai moduli space, it is sufficient to invoke the analytic triviality in the normal direction of the stratification by orbit type of M (S, (0, nC, −nC 2 /2)), which follows from the quadraticity of the deformation spaces Lemma 2.9. • Now let S ′ be a K3 or an abelian surface such that C embeds in S ′ and generates its Picard group. Then M (S ′ , (0, nC, −nC 2 /2)) and M (S, v) (resp. K(S ′ , (0, nC, −nC 2 /2)) and K(S, v)) are isosingular by [59, Theorem 1.17], independently on the complex structure of S.
• M B (C, GL n ) and M (S ′ , (0, nC, −nC 2 /2)) are isosingular by Proposition 2.10. 6 Note that this is the only place where we use the assumption that S has Picard number one. Otherwise, if [C] can be decomposed in the sum of effective classes, then the rank of the automorphism group of F ∈ M (S, (0, nC, −nC 2 /2)) may be greater than n, and the stratification by orbit type of M (S, (0, nC, −nC 2 /2)) would have more strata than that of M B (C, GLn). Observe however that a general result is achieved in Theorem 2.11.
Further, the morphism alb is locally given by the linear projection onto the last factor of (12) by Lemma 2.9. The same argument works for M Dol (C, SL n ) too. As in Example 2.8, we conclude that M Dol (C, GL n ) and M Dol (C, SL n ) (resp. M (S, v) and K(S, v)) are stably isosingular. In this section we characterise Σ, Ω and their normal slices, maily appealing to [56].  where Z/2Z acts on (C * ) 2g by v → −v. Set Σ ι,B (C, SL 2 ) := (C * ) 2g .

The largest open subset of
There exists a rank-one local system L on Σ • := Σ \ Ω such that Proposition 3.4 (Normal slices). 1. A slice N Σ normal to Σ at a point in Σ\ Ω is locally analytically isomorphic to an affine cone over the incidence variety 2. Let (W, q) be a vector space of dimension 3 endowed with a quadratic form q of maximal rank, and (V, ω) be a symplectic vector space of dimension 2g. Let Hom ω (W, V ) be the cone of linear maps from W to V whose image is isotropic. Note that the group SO(W ) acts on Hom ω (W, V ) by pre-composition.
Then a normal slice N Ω through Ω is isomorphic to an affine cone over P Hom ω (W, V ) SO(W ).
Proof. The local models have been described in [56,

Geometry of the desingularization
Inspired by [42], O'Grady exhibits a desingularization of the Mukai moduli spaces M (S, v) of semistable sheaves on a projective K3 surface S with Mukai vector v = (2, 0, −2c) ∈ H * alg (S, Z). By the stable isosingularity principle (cf §2.4) the same sequence of blow-ups gives a desingularization of M . In this section, we recall the geometry of the exceptional locus, and we compute the E-polynomials of its strata.

Proposition 3.5 (Kirwan-O'Grady desingularization). Let
• π R : R → M be the blow-up of M along Ω; • π S : S → R be the blow-up of R along Σ R := π −1 R, * Σ; • π T : T → S be the blow-up of S along its singular locus.
Then the composition π := π T • π S • π R : T → M is a log resolution of M 7 .
• D 1 , D 2 and D 3 are (the strict transform of) the exceptional divisors in T of the blow-ups π R , π S and π T respectively.
• D ij := D i ∩ D j and D 123 := D 1 ∩ D 2 ∩ D 3 are (smooth closed) strata of the exceptional locus of π.
isotropic with respect to the symplectic form ω. 7 In genus 2 the (unique) symplectic resolution of M can be obtained by contracting a P 2 -bundle in S. See [26,Proposition 8.6] where π R and π S are denoted η and ζ respectively. In particular, the resolution π is not symplectic.
•P 5 is the blow-up of P 5 ≃ P(S 2 (W )) (space of quadratic forms on W ) along P 2 (locus of quadratic form of rank 1).
•Q is the blow-up of Q ⊂ P(S 2 (W )) (space of degenerate quadratic forms on W ) along P 2 (locus of quadratic form of rank 1).
3. Let I ′ 2g−3 be the quotient of I 2g−3 by the involution which exchanges the coordinates x i and y i . A slice normal to Σ R ∩ Ω R ≃ (P Hom ω 1 (W, V ) SO(W )) × Ω ≃ P 2g−1 × Ω in Ω R is locally analytically isomorphic to an affine cone over I ′ 2g−3 . 4. The singular locus ∆ S of S is the strict transform of (P Hom ω 2 (W, V ) SO(W )) × Ω ⊆ Ω R via π R , which is isomorphic to a P 2 -bundle over Gr ω (2, V ) × Ω.

5.
A slice normal to ∆ S in S is locally analytically isomorphic to the quotient C 2g−3 / ± 1.

Let I 2g−3 be the (Zariski locally trivial)
extending the involution ι on Σ • defined in Definition 3.3.
Proof. Let U m be the universal bundle over Gr ω (m, V ), with m = 2, 3, and Hom k (W, U m ) be the subbundle of Hom(W, U m ) of rank ≤ k. The quotient space P Hom k (W, U m ) SO(W ) is isomorphic to the space of quadrics P(S 2 k U m ) of rank ≤ k. There are obvious forgetful maps which induces the following diagrams f2 SO(W ) A proof of the isomorphisms above is provided in [ Proof. Note that Gr ω (m, V ) is a smooth projective variety whose cohomology is of Hodge-Tate type; see for instance [8,

The incidence variety I 2g−3
The incidence variety I 2g−3 ⊂ P 2g−3 × P 2g−3 is the projectivization of the vector bundle Ω 1 P 2g−3 (1) over P 2g−3 . Hence, we can write where a and b have degree 2, and they are pullback of the first Chern classes of the tautological line bundle of P 2g−3 via the two projections I 2g−3 ⊂ P 2g−3 × P 2g−3 → P 2g−3 . Note that I 2g−3 has no odd cohomology.
The involution which exchanges the factors of the product P 2g−3 × P 2g−3 leaves I 2g−3 invariant, and in cohomology exchanges the classes a and b. Consider the decomposition into eigenspaces for the involution (relative to eigenvalues ±1 respectively) Therefore, we obtain that Proposition 3.10. Setting q := t 2 = uv, the Poincaré polynomials (equivalently E-polynomials) of I 2g−3 , of the invariant and variant part of its cohomology are Proof. I 2g−3 is a P 2g−4 -bundle over P 2g−3 , and this gives (14). We now estimate (17) Finally, substituting (14) and (17) in we obtain (15) and (16).

Intersection cohomology of local models
The goal of this section is to compute the intersection cohomology of the normal slices N Σ and N Ω . This is an important step to determine the summands of the decomposition theorem in Theorem 1.2.
Hence, (19) follows from the following dimensional argument Proposition 3.12. Let N ΣR∩ΩR be a slice normal to Σ R ∩ Ω R in Ω R . Then In particular, Proof. Proposition 3.13. The intersection E-polynomial of Ω R is Proof. We apply the decomposition theorem to the restriction of π S to the strict transform Ω S := π −1 S, * Ω R . By Proposition 3.7.(3) the defect of semismallness of π S | ΩS is and Σ R ∩ Ω R is the only support of the decomposition theorem for π S | ΩS . Note that R i π S, * Q π −1 S (ΣR∩ΩR) are trivial local systems over Σ R ∩ Ω R ≃ P 2g−1 × Ω, because of Proposition 3.7.(1) and the simple connectedness of P 2g−1 . Hence, there exist integers a(i) such that At the stalk level, at x ∈ Σ R ∩ Ω R , we obtain by (21) Together with Proposition 3.9, we get Proposition 3.14. Let N Ω be a slice normal to Ω. Then IH * (N Ω ) is pure of Hodge-Tate type with intersection Poincaré polynomial (equivalently intersection E-polynomials) Proof. Since IH * (N Ω ) ֒→ IH * (π −1 R (x)) for some x ∈ Ω, IH * (N Ω ) is pure of Hodge-Tate type by Proposition 3.9.
Recall now that N Ω is an affine cone over Ω R by Proposition 3.2.(2). Hence, Proposition 2.1 implies that the intersection Poincaré polynomial IP t (N Ω ) is a polynomial in the variable q = t 2 of degree at most 3g − 4, given by

Decomposition theorem
Proof of Theorem 1.2. Let π • T be the restriction of π T over M • := M \ Ω. By Proposition 3.2.(1) the defect of semismallness of π • T is and Σ • is the only support of the decomposition theorem for π • T . Hence, there exists a splitting for some semisimple local systems L i supported on Σ • . Restricting to By Proposition 3.8. (2) there exists a commutative square where the horizontal arrows are étale double covers, p 2 is a Zariski locally trivial fibration with fibre I 2g−3 , and (C * ) 2g,• is the complement in (C * ) 2g of the locus fixed by the involution v → −v. Taking cohomology, we write where L is the rank-one local system defined in Definition 3.3. Together with Proposition 3.11, we obtain This splitting holds on M • , and we now extend it through Ω. Note that the defect of semismallness of π T is Since Σ is a rational homology manifold, Localising at x ∈ Ω, we obtain

Applications of the decomposition theorem
Proof of Theorem 1.3. Taking cohomology with compact support, Theorem 1.2 gives Now Proposition 3.8. (2) gives Therefore, we obtain The variant and anti-invariant E-polynomials of (C * ) 2g and T * Jac(C) with respect to the involution ι defined in Definition 3.3 are Proof of Theorem 1.12. By the additivity of the E-polynomial, we have The formula (22), together with (23), (24), (25) and (26), yields and Proposition 3.9 gives In order to compute IE(M Dol (C, G)) or IP t (M (C, G)) for G = PGL 2 , GL 2 , one can repeat the arguments for SL 2 , and realise that in practise one can obtain the polynomials for PGL 2 by replacing the coefficients 2 2g with 1 in the corresponding polynomials for SL 2 , as explained below. Further, one can use (5) and (6) to write the polynomials for GL 2 from the PGL 2 counterparts.
Definition 4.1. Var Γ is the category of algebraic varieties endowed with a Γaction, with Γ-equivariant morphisms as morphisms.
Γ−mHs is the abelian category whose objects are Γ-modules over Q endowed with a mixed Hodge structure, and whose morphisms are Γ-equivariant morphisms of mixed Hodge structures.
Definition 4.2. Let X be an algebraic variety endowed with an algebraic Γaction. The virtual Hodge realization of (X; Γ X) is the element in the Groethendieck ring K 0 (Γ−mHs) defined by the formula where ǫ ij is ±1, and ρ ij is a direct sum of copies of the trivial and/or of the regular representation.
Denote by E reg (Z ij ) the E-polynomial of the regular part of H * c (Z ij ), and let Via the decomposition theorem, the same holds for IE(M Dol (C, SL 2 )), and so for IP t (M (C, SL 2 )), by the purity of IH * (M (C, SL 2 )), as explained in the proof of Theorem 1.9.
Remark 4.3. Since in our case the varieties Z ij are completely explicit, we can check that all the coefficients 2 2g in Theorem 1.8 comes from the E-polynomial of the regular part. So we obtain IE(M Dol (C, SL 2 )) Γ by replacing 2 2g by 1.
Remark 4.4. By the vanishing of the odd part of IH * var (M (C, SL 2 )) (cf Corollary 1.11), every non-trivial (Gr W r+s IH d var (M (C, SL 2 )) r,s will contribute with non-negative coefficient to IE var (M (C, SL 2 )). Therefore, there is no cancellation, and the Γ-modules IH * (M (C, SL 2 )) and H * (T (C, SL 2 )) are direct sum of copies of the trivial and of the regular representation of Γ by (27). Comparing with [12], one can check that the same holds for H * (M (C, SL 2 )).

P=W conjecture for twisted character varieties
The computation of E-polynomials of character varieties have been initiated in [33], for twisted character varieties M tw with G = GL n , SL n or PGL n and gcd(n, d) = 1; see also [52].
As in the untwisted case, a non-abelian Hodge correspondence holds for M tw B : there exists a diffeomorphism Ψ : M tw Dol → M tw B , from the Dolbeault moduli space M tw Dol = M tw Dol (C, G, d) of semistable G-Higgs bundles over C of degree d; see [36]. However, contrary to the general untwisted character variety, M tw B is smooth (a significant advantage!).
Surprisingly, Hausel and Rodriguez-Villegas [33] in rank 2, and Mellit [51] for GL n , observed that the cohomology of M tw B enjoys symmetries typical of smooth projective varieties, despite the fact that M tw B is not projective. They called these symmetries curious hard Lefschetz theorem: there exists a class α ∈ H 2 (M tw B ) which induces the isomorphism Note that, as an immediate consequence of the curious hard Lefschetz theorem, the E-polynomial of M tw B is palindromic. In the attempt to explain the curious hard Lefschetz theorem, de Cataldo, Hausel and Migliorini conjectured the P=W conjecture, and they verified it for rank 2; see [13]. This conjecture posits that the non-abelian Hodge correspondence exchanges two filtrations on the cohomology of M tw Dol and M tw B of very different origin, respectively the perverse Leray filtration (10) associated to the Hitchin fibration χ on M tw Dol (the analogue of the map defined in (11)) and the weight filtration on M tw B .
This suggests that the symmetries of the mixed Hodge structure of the cohomology of twisted character varieties, noted by Hausel and Rodriguez-Villegas, should be understood as a manifestation of the standard relative hard Lefschetz symmetries for the proper map χ on the Dolbeault side. The latter is an isomorphism between graded pieces of the perverse Leray filtration induced by cup product with a relative χ-ample class α ∈ H 2 (M tw Dol (C, G, d)):
As in the twisted case, the PI=WI conjecture and the relative hard Lefschetz theorem for χ would imply the intersection curious hard Lefschetz theorem.

Conjecture 5.3 (intersection curious hard Lefschetz).
There exists a class α ∈ H 2 (M B (C, G)) which induces the isomorphisms In particular, the intersection E-polynomial of M B (C, G) is palindromic.
In this paper we provide some numerical evidence for Conjecture 5.3.

PI=WI for SL 2 is equivalent to PI=WI for GL 2
The P=W conjectures for SL n implies the P=W conjectures for PGL n and GL n ; see [26, §3.3]. The converse holds true in the twisted case for n prime by [17]. By (5) and (6), this reduction boils down to prove the P=W conjecture for the variant cohomology. The proof in Theorem 5.5 does not rely on the smoothness of twisted character varieties, and it extends to the singular case verbatim. Set c n := n(n − 1)(g − 1). Then we have Unfortunately, in the untwisted case the variant intersection E-polynomials are available only in rank 2; see Corollary 1.11.
Corollary 5.6. The PI=WI conjecture for M (C, SL 2 ) is equivalent to the PI=WI conjecture for M (C, GL 2 ).

Tautological classes
In [35] Hausel and Thaddeus proved that H * (M tw (C, SL 2 )) Γ is generated by tautological classes 8 . This is an essential ingredient of the proof of the P=W conjecture in the twisted case [13] and [16], and a missing desirable piece of information in the untwisted case. Here we provide a partial result: we show that tautological classes do generate the low-degree intersection cohomology of M .
Let B(C, SL 2 ) be the (infinite dimensional and contractible) space of SL 2 -Higgs bundles on C of degree zero, and B ss (C, SL 2 ) be the corresponding locus of semistable Higgs bundles. Let G be the group of real gauge transformations with fixed determinant acting on this spaces by precomposition, and G C its complexification.
We can identify the classifying space BG ≃ B(C, SL 2 ) with the space of continuous maps Map(C, SU 2 ). The second Chern class of the tautological (flat) SU 2 -bundle T on C × Map(C, SU 2 ) admits the Künneth decomposition where σ ∈ H 2 (C) is the fundamental cohomology class, and e 1 , . . . , e 2g is a standard symplectic basis of H 1 (C). Atiyah and Bott showed in [3] that the rational cohomology of BG is freely generated by the tautological classes α, ψ j and β. That is, H * (BG) is the tensor product of the polynomial algebra on the classes α and β of degree 2 and 4 with an exterior algebra on the classes ψ j of degree 3, In particular, the Poincaré polynomial of the classifying space BG is .
As a final remark, note that the proof of Theorem 5.7 shows the more general statement that Kirwan surjectivity implies the tautological generation of the low-degree intersection cohomology for M (C, SL n ). However, this surjectivity is an open problem for n > 2; cf [9].
We prove the lemma used in the proof of Theorem 5.7.
Indeed, take an approximation E k of the universal SL 2 -bundle ESL 2 , i.e. a smooth variety E k with a free SL 2 -action and such that H <k (X × SL2 E k ) ≃ H <k SL2 (X × SL 2 ESL 2 ) =: H <k SL2 (X); see [1,Lemma 1.3]. By Luna slice theorem X × SL 2 E k is a local complete intersection, and the singular locus has again codimension c. Then (35) follows from [31, p.199].

P=W vs PI=WI: non-purity of H * (M B )
Despite the failure of curious hard Lefschetz, it still makes sense to conjecture P = W phenomena for the ordinary cohomology of M B .
It was proved in [26, Theorem 6.1] that the PI=WI conjecture for genus 2 and rank 2 implies the P=W conjecture simply by restriction, since H * (M B (C, SL 2 )) injects into IH * (M B (C, SL 2 )), or equivalently by the purity of H * (M B (C, SL 2 )); see Proposition 2.4. In higher genus the situation is more subtle, as the following theorem shows.
Theorem 5.11. Let C be a curve of genus g > 3. Then the natural map H * (M ) → IH * (M ) is not injective. Equivalently, M Dol has no pure cohomology.
Proof. Otherwise the polynomial IP t (M ) − P t (M ) would have only positive coefficients, but this is not the case by Corollary 1.10 and preceding lines.

P=W for resolution fails when no symplectic resolution exists
In [26] Camilla Felisetti and the author proposed a strong version of PI=WI conjecture, called P=W for resolution, and proved it for character varieties which admits a symplectic resolution. This means that the hypotheses of [25,Main Theorem,3] were optimal for G = GL n , SL n : the proof of Theorem 5.14 suggests that the semismallness of the desingularization may be a necessary requirement for the P=W conjecture for resolutions to hold for a G-character variety with G arbitrary reductive group. This is compatible with the expectation of [14, §4.4].
Theorem 5.14. Let M (C, GL n ) be an untwisted GL n -character variety with no symplectic resolution, i.e. for g, n > 1 and (g, n) = (2, 2). Then the P=W conjecture for any resolution of M (C, GL n ) does not hold.
Proof. Let f : M → M (C, GL n ) be a resolution of singularities of M (C, GL n ) as in (36), and E be an f -exceptional divisor whose image is contained in the singular locus Σ := Sing M (C, GL n ). Recall that χ : M Dol (C, GL n ) → Λ := n i=1 H 0 (C, K ⊗i C ) is the Hitchin fibration (11).