2.1 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 $\mathbb {C}$ -algebra

$$ \begin{align*}R(X, L) := \bigoplus_{m \geq 0} H^0(X, L^m).\end{align*} $$

The affine cone over X with conormal bundle L is

$$ \begin{align*}C(X,L) := \operatorname{Spec} R(X, L).\end{align*} $$

Let $s_1, \ldots , s_N$ be a set of generators for $R(X,L)$ of degree $m_1, \ldots , m_N$ . Then there exists an embedding $C(X,L) \subseteq \mathbb {C}^N$ such that $C(X,L)$ is invariant with respect to the $\mathbb {G}_m$ -action

(7) $$ \begin{align} t \cdot (x_1, \ldots, x_N) = (t^{m_1}x_1, \ldots, t^{m_N}x_N). \end{align} $$

Conversely, any affine variety with a $\mathbb {G}_m$ -action and a fixed point which is attractive for $t \to 0$ is isomorphic to an affine cone; see, for instance, [Reference Demazure21, §3.5].

All of the singularities of this article are locally modelled on affine cones, whose coordinate rings are not necessarily generated in degree 1. For this reason, here we compute their intersection cohomology, thus generalising [Reference de Cataldo and Migliorini19, Example 2.2.1].

Proposition 2.1 Intersection cohomology of an affine cone

$$ \begin{align*} IH^d(C(X,L)) \simeq \begin{cases} IH^d_{\mathrm{prim}}(X) & \text{ for } d< n\\ 0 & \text{ for } d \geq n, \end{cases} \end{align*} $$

where $IH^d_{\mathrm {prim}}(X) := \ker (c_1(L) ^{n-d} \cup \colon IH^d(X) \to IH^{2n-d}(X))$ is the primitive intersection cohomology.

Proof. Denote by $C(X,L)^* := C(X,L) \setminus \{\text {vertex}\}$ the punctured affine cone. By [Reference Durfee23, Lemma 1] or [Reference Kirwan and Woolf45, Proposition 4.7.2], we can write

$$ \begin{align*} IH^d(C(X,L)) \simeq \begin{cases} IH^d(C(X,L)^*) & \text{ for } d< n\\ 0 & \text{ for } d\geq n. \end{cases} \end{align*} $$

Suppose now that $R(X,L)$ is generated in degree 1. Then the blow-up of the origin

$$ \begin{align*}p\colon BC(X,L) := \operatorname{Spec}_X \bigoplus_{m \geq 0} L^m \to C(X,L)\end{align*} $$

is the total space of the line bundle $L^*$ . By the hard Lefschetz theorem, the relative long exact sequence of the inclusion $C(X, L)^* \hookrightarrow BC(X,L)$ splits into the short exact sequences

(8) $$ \begin{align} 0 \to IH^{d-2}(X) \xrightarrow{c_1(L) \cup} IH^{d}(X) \to IH^{d}(C(X, L)^*) \to 0 \quad \text{ for } d < n. \end{align} $$

Therefore, we obtain that for $d < n$ ,

$$ \begin{align*}IH^{d}(C(X, L)) \simeq \mathrm{coker} (c_1(L) \cup\colon IH^{d-2}(X) \to IH^{d}(X)) \simeq IH^d_{\mathrm{prim}}(X).\end{align*} $$

If $R(X,L)$ is not generated in degree 1, then $BC(X,L)$ is the total space of a line bundle only up to a finite cover; see [Reference Orlik and Wagreich57, §1.2]. More precisely, consider the finite morphism $g\colon \mathbb {C}^N \to \mathbb {C}^N$ defined by

$$ \begin{align*}g(x_1, \ldots, x_N) = (x^{m_1}_1, \ldots, x^{m_N}_N).\end{align*} $$

Set $V'= g^{-1}C(X,L)$ , where $C(X,L)$ is embedded in $\mathbb {C}^N$ as in (7). We see that $C(X,L)$ is the quotient of $V'$ by the finite group $A= (\mathbb {Z}/m_1 \mathbb {Z}) \times \ldots \times (\mathbb {Z}/m_N \mathbb {Z})$ acting on $V'$ by coordinatewise multiplication.

$V'$ has a $\mathbb {G}_m$ -action defined by $t \cdot (x_1, \ldots , x_N) = (t x_1, \ldots , t x_N)$ and covering the $\mathbb {G}_m$ -action on $C(X,L)$ given by (7). Since the $\mathbb {G}_m$ -action on $V'$ has weight 1, $X'$ is the spectrum of a graded algebra generated in degree 1, say, $V' = C(X',L')$ for some projective variety $X'$ and ample line bundle $L'$ . In particular, there exists a commutative diagram

(9)

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 $IH^*(C(X',L'))^A \simeq IH^*(C(X, L))$ .

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.

2.2 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 $D^b(X, \mathbb {Q})$ be the bounded derived category of complexes of sheaves of $\mathbb {Q}$ -vector spaces with algebraically constructible cohomology. Denote the full abelian subcategory of perverse sheaves by $\mathrm {Perv}(X)$ and the perverse cohomology functors by ${}^{\mathfrak {p}}\mathcal {H}^i\colon D^b(S, \mathbb {Q}) \to \mathrm {Perv}(X)$ ; see [Reference Beilinson, Bernstein and Deligne5] or [Reference de Cataldo and Migliorini19].

Let $MHM_{\mathrm {alg}}(X)$ be the category of algebraic mixed Hodge modules with rational coefficients and $D^bMHM_{\mathrm {alg}}(X)$ its bounded derived category. Let ${}^{\mathfrak {p}}\mathcal {H}^i\colon D^bMHM_{\mathrm {alg}}(X) \to MHM_{\mathrm {alg}}(X)$ be the cohomology functors; see [Reference Saito62] or [Reference Schnell63].

The simple objects of $D^bMHM_{\mathrm {alg}}(X)$ (respectively $D^b(X, \mathbb {Q})$ ) are the intersection cohomology complexes $IC_X(L)$ , where L is a polarisable variation of pure Hodge structures (respectively a local system) on a Zariski-open subset of the smooth locus of X. We denote simply by $IC_X$ the complex $IC_X(\mathbb {Q}_{X \setminus \operatorname {Sing}(X)})$ . In particular, $IH^d(X) \simeq H^{d}(X, IC_X(\mathbb {Q}_{X \setminus \operatorname {Sing}(X)})[-\dim X])$ .

There is a forgetful functor $\mathrm {rat}\colon D^bMHM_{\mathrm {alg}}(X) \to D^b(X, \mathbb {Q})$ which commutes with ${}^{\mathfrak {p}}\mathcal {H}^i$ and pushforward $Rf_*$ and maps $MHM_{\mathrm {alg}}(X)$ in $\mathrm {Perv}(X)$ . We will make no notational distinction between $K \in D^bMHM_{\mathrm {alg}}(X)$ and $\mathrm {rat}(K)$ .

Now let $f\colon X \to Y$ be a proper morphism of varieties with defect of semismallness

$$ \begin{align*} r := \dim X \times_Y X - \dim X. \end{align*} $$

The decomposition theorem of Beilinson–Bernstein–Deligne–Gabber, or its mixed Hodge module version by Saito, says that there is an isomorphism in $D^bMHM_{\mathrm {alg}}(X)$ (respectively in $D^b(X, \mathbb {Q})$ )

$$ \begin{align*} Rf_*IC_X = \bigoplus^r_{i=-r} {}^{\mathfrak{p}}\mathcal{H}^i(Rf_*IC_X)[-i] = \bigoplus^r_{i=-r} \bigoplus_l IC_{\overline{Y}_{i,l}}(L_{i,l})[-i], \end{align*} $$

where $L_{i, l}$ are polarisable variations of pure Hodge structures (respectively local systems) on the strata of a stratification $Y = \bigsqcup _{l} Y_{i,l}$ ; see [Reference Beilinson, Bernstein and Deligne5] and [Reference Saito61].

The perverse (Leray) filtration is

$$ \begin{align*} P_k IH^d(X)= \mathrm{Im} \bigg\{ H^d(\bigoplus^{k-r}_{i=-r} {}^{\mathfrak{p}}\mathcal{H}^i(Rf_*IC_X)[-i-\dim X])\to IH^d(X)\bigg\}. \end{align*} $$

When Y is affine, de Cataldo and Migliorini provided a simple geometric characterisation of the perverse filtration; see [Reference de Cataldo and Migliorini20, Theorem 4.1.1]. Let $\Lambda ^s\subset Y$ be a general s-dimensional affine section of $Y\subset \mathbb {A}^N$ . Then

(10) $$ \begin{align} P_k IH^d(X) = \mathrm{Ker}\left\lbrace IH^d(X)\rightarrow IH^d(f^{-1}(\Lambda^{d-k-1})) \right \rbrace. \end{align} $$

This means that the cocycle $\eta \in IH^d(X)$ belongs to $P_kIH^d(X)$ if and only if its restriction to $f^{-1}(\Lambda ^{d-k-1})$ vanishes; that is, $\eta |_{f^{-1}(\Lambda ^{d-k-1})}=0$ .

2.3 Mixed Hodge structure of semi-projective varieties

In order to compute the intersection Poincaré polynomial of M, we observe that ${M}_{\mathrm {Dol}}$ and ${T}_{\mathrm {Dol}}$ are semi-projective.

Proof. Let $f\colon \widetilde {X} \to X$ be a $\mathbb {G}_m$ -equivariant resolution of singularities of X. Then $\widetilde {X}$ is smooth and semiprojective, and it has pure cohomology by [Reference Hausel and Rodriguez-Villegas34, Corollary 1.3.2]. Via the decomposition theorem, the mixed Hodge structure on $IH^*(X) \subset H^*(\tilde {X})$ is pure, too.

As X retracts onto the proper algebraic variety $\mathrm {Core}(X)$ , the weight filtration on $H^*(X)\simeq H^*(\mathrm {Core}(X))$ is concentrated in degree $[0, 2d]$ by [Reference Peters and Steenbrink60, Theorem 5.39]; that is, $W_d H^d(X) = H^d(X)$ .

The resolution f induces a surjective morphism $\mathrm {Core}(\widetilde {X}) \to \mathrm {Core}(X)$ of proper algebraic varieties. Hence, by [Reference Peters and Steenbrink60, Corollary 5.43] we have

$$ \begin{align*}W_{d-1}H^d(\mathrm{Core}(X)) = \ker \{f^*\colon H^d(\mathrm{Core}(X)) \to H^d(\mathrm{Core}(\tilde{X}))\},\end{align*} $$

and so $W_{d-1}H^d(X) = \ker \{ f^*\colon H^{d}(X) \to H^d(\widetilde {X})\}$ by Proposition 2.3. Finally, since $f^*$ factors as $H^d(X) \to IH^d(X) \hookrightarrow H^d(\widetilde {X})$ , we conclude that

$$ \begin{align*}W_{d-1}H^d(X) = \ker \{ H^{d}(X) \to IH^{d}(X)\}.\\[-34pt] \end{align*} $$

The multiplicative group $\mathbb {G}_m$ acts on ${M}_{\mathrm {Dol}}(C, \operatorname {SL}_n)$ by $\lambda \cdot (E, \phi )=(E, \lambda \phi )$ . The Hitchin fibration

(11) $$ \begin{align} \chi\colon {M}_{\mathrm{Dol}}(C, \operatorname{SL}_n) \to \bigoplus^{n}_{i=2} H^0(C, K_C^{\otimes i}) \end{align} $$

assigns to $(E, \phi )$ the characteristic polynomial of the Higgs field $\phi $ . By [Reference Simpson66, Theorem 6.11] the map $\chi $ is a proper $\mathbb {G}_m$ -equivariant map, where $\mathbb {G}_m$ acts linearly on $H^0(X, K_X^{\otimes i})$ with weight i. In particular, $\operatorname {Fix}({M}_{\mathrm {Dol}}(C, \operatorname {SL}_n))$ is contained in the nilpotent cone $\chi ^{-1}(0)$ . Therefore, ${M}_{\mathrm {Dol}}(C, \operatorname {SL}_n)$ is semiprojective.

The same argument works for $G=\operatorname {GL}_n, \operatorname {PGL}_n$ as well.

2.4 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 (nonproper) Dolbeault moduli spaces $M(C, \operatorname {GL}_n)$ and $M(C, \operatorname {SL}_n)$ 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 [Reference O’Grady56] or [Reference Choy and Kiem8] holds for $M(C, \operatorname {GL}_2)$ and $M(C, \operatorname {SL}_2)$ mutatis mutandis;

• • the same sequence of blow-ups which desingularises $M(S,v)$ in [Reference O’Grady56] resolves the singularities of $M(C, \operatorname {GL}_2)$ and $M(C, \operatorname {SL}_2)$ mutatis mutandis;

• • the description of the summands of the decomposition theorem in Theorem 1.2 holds for $M(C, \operatorname {GL}_2)$ , $M(C, \operatorname {SL}_2)$ , $M(S, v)$ and $K(S, v)$ with Mukai vector $v = 2 w \in H^*_{\mathrm {alg}}(S, \mathbb {Z})$ , where w is primitive and $w^2=2(g-1)$ .

We briefly recall the definition of Mukai moduli space. Fix an effective Mukai vectorFootnote ² $v \in H^{*}_{\mathrm {alg}}(S, \mathbb {Z})$ . Define $M(S, v)$ the moduli space of Gieseker H-semistable sheaves on S with Mukai vector v for a sufficiently general polarisation H (which we will typically omit in the notation); see [Reference Simpson65, §1]. Further, if S is an abelian variety with dual $\hat {S}$ , and $\dim M(S,v)\geq 6$ , then the Albanese morphism $\operatorname {alb}\colon M(S, v) \to S \times \hat {S}$ is isotrivial, and we set $K(S, v) := \operatorname {alb}^{-1}(0_S, \mathcal {O}_S)$ .

We start by stating the notion of stable isosingularity.

It is implicit in 3. that the stratifications above are analytically equisingular along each stratum; that is, the analytic type of the normal slices through $x \in X_i$ (respectively $y \in Y_i$ ) is independent of x (respectively y). Not all algebraic variety admits such a stratification; see [Reference Whitney68, Example 13.1]. However, the moduli spaces considered below will satisfy the following stronger condition of analytic normal triviality.

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 desingularisation of X, then the same sequence of blow-ups along the corresponding strata $Y_i$ gives a desingularisation 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.

Proof. Given a (quasi-projective) variety X equipped with the action of a reductive group G, let be the quotient map. Any fibre $\xi ^{-1}(y)$ , with $y \in Y$ , contains a closed G-orbit $T(y)$ . Denote the conjugacy class of a closed subgroup H of G by $(H)$ . Then $Y_{(H)}$ is the set of points $y \in Y$ such that the stabiliser of $x \in T(y)$ is in $(H)$ . The loci $Y_{(H)}$ are the strata of the stratification by orbit type of Y.

If Y is a Nakajima quiver variety,Footnote 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 [Reference Mayrand50, Proposition 4.2].

$M_{\mathrm {B}}(C, \operatorname {GL}_n)$ and $M(S,(0, nC, -nC^2/2))$ are $\operatorname {PGL}_N$ -quotients, and the quadraticity of the deformation spaces implies that they are locally modelled on Nakajima quiver varieties; see [Reference Bellamy and Schedler6, Theorem 2.5] and [Reference Arbarello and Saccà2, Proposition 6.1]. By construction, the stratifications by orbit type of $M_{\mathrm {B}}(C, \operatorname {GL}_n)$ 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

$$ \begin{align*}F = F^{l_1}_1\oplus \ldots \oplus F^{l_s}_s,\end{align*} $$

where $F_i$ are distinct stable factors. The automorphism group of F is

$$ \begin{align*}\prod^s_{i=1} \operatorname{GL}_{l_i} \subset \operatorname{GL}_n,\end{align*} $$

which can be identified up to constants with the stabiliser of a point in $T(F)$ under the $\operatorname {PGL}_N$ -action; see, for instance, [Reference Kaledin, Lehn and Sorger39, §2.5].Footnote ⁶

The poset of inclusions of the orbit type strata for both the Dolbeault and Mukai moduli spaces is isomorphic to the poset of inclusion of the stabilisers of $T(F)$ , and the analytic type of the normal slice through an orbit type strata is prescribed by the (abstract) isomorphism class of the stabiliser; see again [Reference Bellamy and Schedler6, Theorem 2.5] and [Reference Kaledin, Lehn and Sorger39, §2.7]. This gives 2 . and 3 . of Definition 2.6. The isosingularity follows from

$$ \begin{align*}\dim M_{\mathrm{B}}(C, \operatorname{GL}_n)=2(g-1)n^2+2=v^2+2= \dim M(S,(0, nC, -nC^2/2)).\\[-34pt] \end{align*} $$

There exists a clear geometric argument for Proposition 2.10, sketched below.

3.1 Singularities of M

Recall that M denotes indifferently the moduli spaces ${M}_{\mathrm {B}}(C,G)$ or ${M}_{\mathrm {Dol}}(C,G)$ with $G=\operatorname {GL}_2$ or $\operatorname {SL}_2$ . The stratification by orbit type of M (cf. Subsection 2.4) determines a filtration by closed subsets

$$ \begin{align*}M \supset \Sigma := \operatorname{Sing} M \supset \Omega := \operatorname{Sing} \Sigma.\end{align*} $$

In this section we characterise $\Sigma $ , $\Omega $ and their normal slices, mainly appealing to [Reference O’Grady56].

3.2 Geometry of the desingularisation

Inspired by [Reference Kirwan42], O’Grady exhibits a desingularisation of the Mukai moduli spaces $M(S,v)$ of semistable sheaves on a projective K3 surface S with Mukai vector $v=(2,0,-2c) \in H^*_{\mathrm {alg}}(S, \mathbb {Z})$ . By the stable isosingularity principle (cf. Subsection 2.4), the same sequence of blow-ups gives a desingularisation of M. In this section, we recall the geometry of the exceptional locus, and we compute the E-polynomials of its strata.

Proof. Since $\Omega (C, \operatorname {SL}_2)$ is a collection of $2^{2g}$ points, (1) obviously holds. Consider now the étale cover $\tau \colon M(C, \operatorname {SL}_n) \times M(C, \operatorname {GL}_1) \to M(C, \operatorname {GL}_n)$ trivialising (4). For some (or any) $x \in \Omega (C, \operatorname {SL}_n)$ we have

$$ \begin{align*}\pi_{T(C, \operatorname{SL}_n)}^{-1}(x) \times \Omega(C, \operatorname{GL}_n) \simeq \pi_{T(C, \operatorname{SL}_n)}^{-1}(x) \times M(C, \operatorname{GL}_1) \simeq \pi_{T(C, \operatorname{GL}_n)}^{-1}(\Omega(C, \operatorname{GL}_n)),\end{align*} $$

since $\tau $ is étale. Thus, (1) holds for $G=\operatorname {GL}_2$ , too.

(2), (3), (4), (5) follow instead from Proposition 3.4.(2), [Reference O’Grady56, (1.7.12) and (1.7.16)], [Reference O’Grady56, (3.5.1)] and [Reference O’Grady56, (3.5.1)] respectively; alternatively, see the proof of [Reference Choy and Kiem8, Proposition 3.2].

Proof. Let $\mathcal {U}_m$ be the universal bundle over $\operatorname {Gr}^{\omega }(m, V)$ , with $m=2,3$ , and $\operatorname {Hom}_k(W, \mathcal {U}_m)$ be the subbundle of $\operatorname {Hom}(W, \mathcal {U}_m)$ of rank $\leq k$ . The quotient space

is isomorphic to the space of quadrics $\mathbb {P}(S^2_k \mathcal {U}_m)$ of rank $\leq k$ . There are obvious forgetful maps

$$ \begin{align*} f_3\colon & \mathbb{P}\operatorname{Hom}(W, \mathcal{U}_3) \to \mathbb{P}\operatorname{Hom}^{\omega}(W,V)\\ f_2 \colon & \mathbb{P}\operatorname{Hom}_2(W, \mathcal{U}_2) \to \mathbb{P}\operatorname{Hom}^{\omega}_2(W,V), \end{align*} $$

which induces the following diagrams

A proof of the isomorphisms above is provided in [Reference O’Grady56, (3.1.1) and (3.5.1)]; alternatively, see [Reference Choy and Kiem8, Proposition 3.2]. This shows (1), (3), (4). To show (2), one can repeat the argument of [Reference Choy and Kiem8, Proposition 3.2.(2)] verbatim.

Proposition 3.9. $\operatorname {Gr}^{\omega }(m,V)$ and the fibres of $\Delta _S$ , $D_1$ , $D_3$ , $D_{13}$ and $\Omega _S$ over $\Omega $ have pure cohomology of Hodge–Tate type. In particular, they do not have odd cohomology. Their E-polynomials are

$$ \begin{align*} E(\operatorname{Gr}^{\omega}(2,V)) & = \frac{(1-q^{2g-2})(1-q^{2g})}{(1-q)(1-q^2)}\\ E(\operatorname{Gr}^{\omega}(3,V)) & = \frac{(1-q^{2g-4})(1-q^{2g-2})(1-q^{2g})}{(1-q)(1-q^2)(1-q^3)}\\ E(\Delta_S) & = \frac{(1-q^3)(1-q^{2g-2})(1-q^{2g})}{(1-q)^2(1-q^2)} \cdot E(\Omega)\\ E(D_1) & = \frac{(1-q^4)(1-q^{2g-4})(1-q^{2g-2})(1-q^{2g})}{(1-q)^3(1-q^2)} \cdot E(\Omega)\\ E(D_3) & = \frac{(1-q^{3})(1-q^{2g-3})(1-q^{2g-2})(1-q^{2g})}{(1-q)^3 (1-q^2)} \cdot E(\Omega)\\ E(D_{13}) & = \frac{(1-q^{3})(1-q^{2g-4})(1-q^{2g-2})(1-q^{2g})}{(1-q)^3 (1-q^2)} \cdot E(\Omega)\\ E(\Omega_S) & = E(D_1)-E(\Delta_S) \cdot \sum^{2g-6}_{i=0} q^{i+1} \\ & = \frac{(1-q^{2g-2})(1-q^{2g-1})(1-q^{2g})}{(1-q)^2(1-q^2)} \cdot E(\Omega). \end{align*} $$

3.3 The incidence variety $I_{2g-3}$

The incidence variety $I_{2g-3} \subset \mathbb {P}^{2g-3} \times \mathbb {P}^{2g-3}$ is the projectivisation of the vector bundle $\Omega ^1_{\mathbb {P}^{2g-3}}(1)$ over $\mathbb {P}^{2g-3}$ . Hence, we can write

(13) $$ \begin{align} H^*(I_{2g-3}) = \mathbb{Q}[a,b]/(a^{2g-2}, b^{2g-2}, \sum^{2g-3}_{i=0} (-1)^i a^{2g-3-i} b^{i}), \end{align} $$

where a and b have degree $2$ , and they are pullback of the first Chern classes of the tautological line bundle of $\mathbb {P}^{2g-3}$ via the two projections $I_{2g-3} \subset \mathbb {P}^{2g-3} \times \mathbb {P}^{2g-3} \to \mathbb {P}^{2g-3}$ . Note that $I_{2g-3}$ has no odd cohomology.

The involution which exchanges the factors of the product $\mathbb {P}^{2g-3} \times \mathbb {P}^{2g-3}$ 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 $\pm 1$ respectively)

$$ \begin{align*}H^*(I_{2g-3}) = H^*(I_{2g-3})^+ \oplus H^*(I_{2g-3})^-.\end{align*} $$

For $d=2k<4g-7$ , we have

$$ \begin{align*} H^d(I_{2g-3})^+ = H^d(\mathbb{P}^{2g-3}\times \mathbb{P}^{2g-3})^+ = \langle a^ib^j | i+j=d\rangle^+ = \langle a^ib^j + a^j b^i| i+j=d\rangle. \end{align*} $$

Therefore, we obtain that

$$ \begin{align*} \dim H^{2k}(I_{2g-3})=k+1, \,\, \dim H^{2k}(I_{2g-3})^+=\left\lceil \frac{k+1}{2} \right\rceil, \,\, \dim H^{2k}(I_{2g-3})^-=\left\lceil \frac{k}{2} \right\rceil. \end{align*} $$

Proposition 3.10. Setting $q := t^2=uv$ , the Poincaré polynomials (equivalently E-polynomials) of $I_{2g-3}$ of the invariant and variant parts of its cohomology are

(14) $$ \begin{align} P_t(I_{2g-3}) & = E(I_{2g-3}) = \frac{(1-q^{2g-2})(1-q^{2g-3})}{(1-q)^2} \end{align} $$

(15) $$ \begin{align} P_t(I_{2g-3})^+ & = E(I_{2g-3})^+ = \frac{(1-q^{2g-2})^2}{(1-q^2)(1-q)} \end{align} $$

(16) $$ \begin{align} P_t(I_{2g-3})^- & = E(I_{2g-3})^- = q \frac{(1-q^{2g-2})(1-q^{2g-4})}{(1-q^2)(1-q)}. \end{align} $$

Proof. $I_{2g-3}$ is a $\mathbb {P}^{2g-4}$ -bundle over $\mathbb {P}^{2g-3}$ , and this gives (14). We now estimate $P_t(I_{2g-3})^+ - P_t(I_{2g-3})^-$ . For $d=2k<4g-7$ , we have

$$ \begin{align*}\dim H^{2k}(I_{2g-3})^+ - \dim H^{2k}(I_{2g-3})^- = \left\lceil \frac{k+1}{2} \right\rceil- \left\lceil \frac{k}{2} \right\rceil = \begin{cases} 1 & \text{ for } k=2l,\\ 0 & \text{ for } k=2l+1. \end{cases}\end{align*} $$

Since the polarisation $\mathcal {O}(1,1)$ is $\iota $ -invariant, the hard Lefschetz theorem gives $\dim H^{d}(I_{2g-3})^{\pm } = \dim H^{8g-14 - d}(I_{2g-3})^{\pm }$ . Thus, we can write

(17) $$ \begin{align} P_t(I_{2g-3})^+ - P_t(I_{2g-3})^- = \sum^{g-2}_{l=0}q^{2l} + q^{2g-3} \sum^{g-2}_{l=0}q^{2l} = \frac{(1-q^{2g-2})(1+q^{2g-3})}{(1-q^2)}. \end{align} $$

Finally, substituting (14) and (17) in

$$ \begin{align*} P_t(I_{2g-3})^+ & = \frac{1}{2}(P_t(I_{2g-3}) + ( P_t(I_{2g-3})^+ - P_t(I_{2g-3})^- ))\\ P_t(I_{2g-3})^- & = P_t(I_{2g-3}) - P_t(I_{2g-3})^+, \end{align*} $$

we obtain (15) and (16).

3.4 Intersection cohomology of local models

The goal of this section is to compute the intersection cohomology of the normal slices $N_{\Sigma }$ and $N_{\Omega }$ . This is an important step to determine the summands of the decomposition theorem in Theorem 1.2.

Proposition 3.11. Let $N_{\Sigma }$ be a slice normal to $\Sigma $ at a point in $\Sigma \setminus \Omega $ . Then

(18) $$ \begin{align} IH^d(N_{\Sigma}) \simeq \begin{cases} H^{d}_{\mathrm{prim}}(I_{2g-3}) \simeq \mathbb{Q} & \text{ for } d=2k<4g-6,\\ 0 & \text{ otherwise.}\\ \end{cases} \end{align} $$

(19) $$ \begin{align} IH^{2k}(N_{\Sigma}) \subseteq \begin{cases} H^{2k}(I_{2g-3})^+ & \text{ for } k=2l,\\ H^{2k}(I_{2g-3})^- & \text{ for } k=2l+1. \end{cases} \end{align} $$

Proof. (18) follows from Proposition 2.1, since $N_{\Sigma }$ is locally isomorphic to an affine cone over the smooth variety $I_{2g-3}$ (with $\iota $ -invariant conormal bundle $\mathcal {O}(1,1)$ ) by Proposition 3.4.(1).

Since $\mathcal {O}(1,1)$ is $\iota $ -invariant, $H^d_{\mathrm {prim}}(I_{2g-3})$ is $\iota $ -invariant, too. Hence, (19) follows from the following dimensional argument:

$$ \begin{align*} 1 \geq \dim IH^{2k}(N_\Sigma) \geq \dim IH^{2k}(N_\Sigma)^+ & = \dim H^{2k}(I_{2g-3})^+ - \dim H^{2k-2}(I_{2g-3})^+ \nonumber \\ & = \left\lceil \frac{k+1}{2} \right\rceil- \left\lceil \frac{k}{2} \right\rceil = \begin{cases} 1 & \text{ for } k=2l,\\ 0 & \text{ for } k=2l+1. \end{cases} \\[-38pt] \end{align*} $$

Proposition 3.12. Let $N_{\Sigma _R \cap \Omega _R}$ be a slice normal to $\Sigma _R \cap \Omega _R$ in $\Omega _R$ . Then

(20) $$ \begin{align} IH^d(N_{\Sigma_R \cap \Omega_R}) \simeq \begin{cases} H^d_{\mathrm{prim}}(I^{\prime}_{2g-3}) \simeq \mathbb{Q} & \text{ for } d=4k<4g-6,\\ 0 & \text{ otherwise}. \end{cases} \end{align} $$

In particular,

(21) $$ \begin{align} \dim H^{4g-6+2i}(I^{\prime}_{2g-3}) - \dim IH^{4g-6+2i}(N_{\Sigma_R \cap \Omega_R}) = \left\lceil \frac{2g-3-|i|}{2}\right\rceil. \end{align} $$

Proposition 3.13. The intersection E-polynomial of $\Omega _R$ is

$$ \begin{align*}IE(\Omega_R) = \frac{(1-q^{4g-4})(1-q^{2g})}{(1-q)(1-q^2)}\cdot E(\Omega).\end{align*} $$

Proof. We apply the decomposition theorem to the restriction of $\pi _S$ to the strict transform $\Omega _S := \pi ^{-1}_{S, *} \Omega _R$ .

By Proposition 3.7.(3), the defect of semismallness of $\pi _S|_{\Omega _S}$ is

$$ \begin{align*} r(\pi_S|_{\Omega_S}) & := \dim \Omega_S \times_{\Omega_R} \Omega_S - \dim \Omega_R \\ & = 2 \dim I^{\prime}_{2g-3} + \dim \Sigma_ R \cap \Omega_R - \dim \Omega_R = 4g-8, \end{align*} $$

and $\Sigma _ R \cap \Omega _R$ is the only support of the decomposition theorem for $\pi _S|_{\Omega _S}$ . Note that $R^i \pi _{S, *} \mathbb {Q}_{\pi ^{-1}_S(\Sigma _ R \cap \Omega _R)}$ are trivial local systems over $\Sigma _ R \cap \Omega _R \simeq \mathbb {P}^{2g-1} \times \Omega $ , because of Proposition 3.7.(1) and the simple connectedness of $\mathbb {P}^{2g-1}$ . Hence, there exist integers $a(i)$ such that

$$ \begin{align*} R(\pi_{S}|_{\Omega_S})_{*} {\mathbb{Q}} [\dim {\Omega}_S] & = \bigoplus^{4g-8}_{i=-4g+8} {{}^{\mathfrak{p}}}{\mathcal{H}}^i((\pi_{S}|_{\Omega_S})_{*} {\mathbb{Q}}[\dim \Omega_S])[-i]\\ & = IC_{\Omega_R} \oplus {\bigoplus}^{2g-4}_{i=-2g+4} {\mathbb{Q}}^{a(i)}_{\Sigma_{R} \cap {\Omega}_R}[\dim{\Sigma}_ R \cap {\Omega}_R-2i](-2g+3-i). \end{align*} $$

At the stalk level, at $x \in \Sigma _R \cap \Omega _R$ , we obtain by (21)

$$ \begin{align*} a(i) = \dim H^{4g-6+2i}(I^{\prime}_{2g-3})-\dim IH^{4g-6+2i}(N_{\Sigma_{R} \cap \Omega_R}) = \left\lceil \frac{2g-3-|i|}{2}\right\rceil. \end{align*} $$

Together with Proposition 3.9, we get

$$ \begin{align*} IE(\Omega_R) &= E(\Omega_S) - E(\Omega) \cdot E(\mathbb{P}^{2g-1}) \cdot \sum^{2g-4}_{i=-2g+4}q^{2g-3+i} \left\lceil \frac{2g-3-|i|}{2}\right\rceil\\ & = \left( \frac{(1-q^{2g-2})(1-q^{2g-1})(1-q^{2g})}{(1-q)^2(1-q^2)} - \frac{q (1 - q^{2 g - 3}) (1 - q^{2 g - 2})(1-q^{2g})}{(1-q)^2 (1-q^2)} \right) \cdot E(\Omega)\\ & = \frac{(1-q^{4g-4})(1-q^{2g})}{(1-q)(1-q^2)} \cdot E(\Omega).\\[-40pt] \end{align*} $$

Proposition 3.14. Let $N_{\Omega }$ be a slice normal to $\Omega $ . Then $IH^*(N_{\Omega })$ is pure of Hodge–Tate type with intersection Poincaré polynomial (equivalently, intersection E-polynomials)

$$ \begin{align*}IP_t(N_{\Omega}) = IE(N_{\Omega}) = \frac{1-q^{2g}}{1-q^2}.\end{align*} $$

Proof. Since $IH^*(N_{\Omega }) \hookrightarrow IH^*(\pi _R ^{-1}(x))$ for some $x \in \Omega $ , $IH^*(N_{\Omega })$ is pure of Hodge–Tate type by Proposition 3.9.

Recall now that $N_{\Omega }$ is an affine cone over $\Omega _R$ by Proposition 3.2.(2). Hence, Proposition 2.1 implies that the intersection Poincaré polynomial $IP_t(N_{\Omega })$ is a polynomial in the variable $q = t^2$ of degree at most $3g-4$ , given by

$$ \begin{align*} IE(N_{\Omega})&=[IE(\pi_R ^{-1}(x))-q IE(\pi_R ^{-1}(x))]_{\leq 3g-4}= \frac{1}{E(\Omega)}[IE(\Omega_R)-q IE(\Omega_R)]_{\leq 3g-4} \\ & = \bigg[ \frac{(1-q^{4g-4})(1-q^{2g})}{(1-q^2)}\bigg]_{\leq 3g-4} = \frac{1-q^{2g}}{1-q^2}.\\[-40pt] \end{align*} $$

4.1 Applications of the decomposition theorem

Proof of Theorem 1.3. Taking cohomology with compact support, Theorem 1.2 gives

$$ \begin{align*} IE(M) & = E(T)- E(\Sigma_{\iota})^+ \cdot \left( \sum^{2g-3}_{i=-2g+3} \left\lceil \frac{2g-3-|i|}{2} \right\rceil q^{2g-3+i}\right)\\ &\quad - E(\Sigma_{\iota})^- \cdot \left(\sum^{2g-3}_{i=-2g+3} \left\lfloor \frac{2g-3-|i|}{2} \right\rfloor q^{2g-3+i} \right)\\ &\quad - E(\pi^{-1}_T(\Omega)) + E(\Omega) \cdot IE(N_{\Omega}) + E(\Omega) \cdot \sum^{2g-3}_{i=-2g+3}\left\lceil \frac{2g-3-|i|}{2} \right\rceil q^{2g-3+i} \\ & = E(M) - E(\Sigma) +E(D^{\circ}_{2})\\ &\quad - (E(\Sigma_{\iota})^+-E(\Omega))\cdot q \frac{(1 - q^{2 g - 3}) (1 - q^{2 g - 2})}{(1-q) (1-q^2)} \\ &\quad - E(\Sigma_{\iota})^- \cdot q^2 \frac{(1 - q^{2 g - 4}) (1 - q^{2 g - 3})}{(1-q) (1-q^2)} +E(\Omega) \cdot IE(N_{\Omega}). \end{align*} $$

Now Proposition 3.8.(2) gives

(22) $$ \begin{align} E(D^{\circ}_{2}) = E(I_{2g-3})^+ \cdot (E(\Sigma_{\iota})^+-E(\Omega)) + E(I_{2g-3})^- \cdot E(\Sigma_{\iota})^-. \end{align} $$

Therefore, we obtain

$$ \begin{align*} IE(M)& = E(M) - E(\Sigma) + (E(\Sigma_{\iota})^+-E(\Omega))\cdot \frac{1 - q^{2 g - 2}}{ 1-q^2} \\ &\quad + E(\Sigma_{\iota})^- \cdot q\frac{1 - q^{2 g - 4} }{1-q^2}+ E(\Omega) \cdot \frac{1-q^{2g}}{1-q^2}\\ & = E(M)+ (q^2E(\Sigma_{\iota})^+ + qE(\Sigma_{\iota})^-) \cdot \frac{1 - q^{2 g - 4}}{ 1-q^2} +E(\Omega) \cdot q^{2g-2}.\\[-42pt] \end{align*} $$

 □

The variant and anti-invariant E-polynomials of $(\mathbb {C}^*)^{2g}$ and $T^*\operatorname {Jac}(C)$ with respect to the involution $\iota $ defined in Definition 3.3 are

(23) $$ \begin{align} E((\mathbb{C}^*)^{2g})^+ & = \sum^{g}_{d=0} \dim (\Lambda^{2d}V) q^{2d}= \frac{1}{2}((1-q)^{2g}+(1+q)^{2g}), \end{align} $$

(24) $$ \begin{align} E((\mathbb{C}^*)^{2g})^- & = -\sum^{g-1}_{d=0} \dim (\Lambda^{2d+1}V) q^{2d+1}= \frac{1}{2}((1-q)^{2g}-(1+q)^{2g}), \end{align} $$

(25) $$ \begin{align} E(T^*\operatorname{Jac}(C))^+ & = \frac{1}{2}(uv)^g \left((1-u)^g(1-v)^g+(1+u)^g(1+v)^g\right), \end{align} $$

(26) $$ \begin{align} E(T^*\operatorname{Jac}(C))^- & = \frac{1}{2}(uv)^g \left((1-u)^g(1-v)^g-(1+u)^g(1+v)^g\right). \end{align} $$

Proof of Theorem 1.8. In view of

$$ \begin{align*}E(M)=E(M^{\mathrm{sm}})+E(\Sigma)=E(M^{\mathrm{sm}})+E(\Sigma_{\iota})^+,\end{align*} $$

we obtain Theorem 1.8 simply by substituting (25) and (26) in (2). □

Proof of Theorem 1.9. The purity of $IH^*({M}_{\mathrm {Dol}})$ (Proposition 2.4) and the Poincaré duality give

$$ \begin{align*} IP_t(M) = t^{2\dim M} IE({M}_{\mathrm{Dol}}; -t^{-1}, -t^{-1}).\end{align*} $$

Theorem 1.9 then follows from Theorem 1.8 and elementary algebraic manipulations. □

Proof of Theorem 1.12. By the additivity of the E-polynomial, we have

$$ \begin{align*} E(T)=E(M^{\mathrm{sm}})+E(D_{1})+E(D^{\circ}_{2})+E(D_{3})-E(D_{13}). \end{align*} $$

The formula (22), together with (23), (24), (25) and (26), yields

$$ \begin{align*} E(D^{\circ}_{2, \mathrm{B}}) & = \frac{1-q^{2g-2}}{2(1-q)} \big((1-q^{2g-3})(1-q)^{2g-1} + (1+q^{2g-3})(1+q)^{2g-1}\big)\\ &\quad -2^{2g}\frac{(1-q^{2g-2})^2}{(1-q^2)(1-q)},\\ E(D^{\circ}_{2, \mathrm{Dol}}) & = \frac{1}{2}(uv)^g \frac{1-(uv)^{2g-2}}{1-uv}\bigg( \frac{(1-u)^g(1-v)^g(1-(uv)^{2g-3})}{1-uv} \\ &\quad + \frac{(1+u)^g(1+v)^g(1+(uv)^{2g-3})}{1+uv}\bigg)-2^{2g} \frac{(1-(uv)^{2g-2})^2}{(1-(uv)^2)(1-uv)}, \end{align*} $$

and Proposition 3.9 gives

$$ \begin{align*}E(D_1)+E(D_3)-E(D_{13})=2^{2g}\frac{\left(1-q^{2g-2}\right) \left(1-q^{2g}\right) \left(1-q^4 -q^{2 g-3}-q^{2g-1}+2q^{2g}\right)}{\left(1-q\right)^3 \left(1-q^2\right)}.\\[-42pt] \end{align*} $$

 □

4.2 From $\operatorname {SL}_2$ to $\operatorname {PGL}_2$ or $\operatorname {GL}_2$

In order to compute $IE({M}_{\mathrm {Dol}}(C, G))$ or $IP_t(M(C, G))$ for $G=\operatorname {PGL}_2, \operatorname {GL}_2$ , one can repeat the arguments for $\operatorname {SL}_2$ and realise that in practise one can obtain the polynomials for $\operatorname {PGL}_2$ by replacing the coefficients $2^{2g}$ with $1$ in the corresponding polynomials for $\operatorname {SL}_2$ , as explained below. Further, one can use (5) and (6) to write the polynomials for $\operatorname {GL}_2$ from the $\operatorname {PGL}_2$ counterparts.

Definition 4.2. Let X be an algebraic variety endowed with an algebraic $\Gamma $ -action. The virtual Hodge realisation of $(X; \Gamma {\curvearrowright } X)$ is the element in the Groethendieck ring $K_0(\Gamma \mathrm {-mHs})$ defined by the formula

$$ \begin{align*}\chi_{\mathrm{Hdg}; \Gamma}(X)=\sum_k (-1)^k [H^k_c(T(C, \operatorname{SL}_2)); \rho_M \colon \Gamma \to \operatorname{Aut}(H^k_c(X))].\end{align*} $$

The morphism $\chi _{\mathrm {Hdg}; \Gamma }(\cdot )\colon K_0(\mathrm {Var}^{\Gamma }) \to K_0(\Gamma \mathrm {-mHs})$ is additive.

The same Hodge realisation was considered in [Reference Hausel and Thaddeus35, §4], when Hausel and Thaddeus defined E-polynomials with coefficient in the characters of the finite abelian group $\Gamma $ .

Now consider the $\Gamma $ -invariant stratification of ${T}_{\mathrm {Dol}}(C, \operatorname {SL}_2)$ whose strata are

1. 1. $S_0 \simeq \{(E, \phi ) \in {M}_{\mathrm {Dol}}(C, \operatorname {SL}_2) |\, E \text { is stable}\}$ ;

2. 2. $S_1 \simeq \{(E, \phi ) \in {M}_{\mathrm {Dol}}(C, \operatorname {SL}_2) |\, E \simeq L \oplus L^{-1}, \, L \in \operatorname {Jac}(C), \, L \neq L^{-1}\}$ ;

3. 3. $S_2 \simeq \{(E, \phi ) \in {M}_{\mathrm {Dol}}(C, \operatorname {SL}_2) |\, E \text { is a nontrivial extension of }L^{-1}\text { by }L \text { for } L\in \operatorname {Jac}(C)\\ \text { with } \, L \neq L^{-1}\}$ ;

4. 4. $S_3 \simeq \{(E, \phi ) \in {M}_{\mathrm {Dol}}(C, \operatorname {SL}_2) |\, E \simeq L \oplus L, \, L \in \operatorname {Jac}(C)\}$ ;

5. 5. $S_4 \simeq \{(E, \phi ) \in {M}_{\mathrm {Dol}}(C, \operatorname {SL}_2) |\, E \text { is a nontrivial extension of }L\text { by }L \text { for } L\in \operatorname {Jac}(C)\}$ ;

6. 6. $S_5 \simeq \{(E, \phi ) \in {M}_{\mathrm {Dol}}(C, \operatorname {SL}_2) |\, E \text { is unstable}\}$ ;

7. 7. $S_6 = D^{\circ }_2$ , $S_7 = D_3 \setminus D_{13}$ and $S_8 = D_1$ .

This is indeed a stratification of ${T}_{\mathrm {Dol}}(C, \operatorname {SL}_2)$ , since ${M}_{\mathrm {Dol}}(C, \operatorname {SL}_2)^{\text {sm}} = \bigsqcup ^5_{i=0} S_i \simeq {T}_{\mathrm {Dol}}(C, \operatorname {SL}_2) \setminus (D_1 \cup D_2 \cup D_3)$ by [Reference Hitchin37, Example 3.13] and Proposition 3.2. The additivity of the virtual realisation implies that

$$ \begin{align*}\chi_{\mathrm{Hdg}; \Gamma}({T}_{\mathrm{Dol}}(C, \operatorname{SL}_2))= \sum^8_{i=0} \chi_{\mathrm{Hdg}; \Gamma}(S_i).\end{align*} $$

By direct inspection (see [Reference Kiem and Yoo41, §3]) one can check that there exist algebraic varieties $Z_{ij}$ endowed with a $\Gamma $ -action such that

1. 1. the $\Gamma $ -module $H^k_c(Z_{ij})$ is isomorphic to the direct sum of copies of the trivial and of the regular representation of $\Gamma $ ; that is, there exist integers $l_{ijk}$ and $m_{ijk}$ such that there exists a $\Gamma $ -equivariant isomorphism

   $$ \begin{align*}H^k_c(Z_{ij}) \simeq V^{\oplus n_{ijk}}_{\text{tr}} \oplus V^{\oplus m_{ijk}}_{\text{reg}},\end{align*} $$
   where $\Gamma $ acts trivially on $V_{\text {tr}} \simeq \mathbb {Q}$ and via the regular representation on $V_{\text {reg}}\simeq \mathbb {Q}^{2^{2g}}$ . We call $V^{\oplus m_{ijk}}_{\text {reg}}$ the regular part of $H^k_c(Z_{ij})$ .

2. 2. we have

   $$ \begin{align*}\chi_{\mathrm{Hdg}; \Gamma}(S_i) = \sum_{j, k} \epsilon_{ij} (-1)^k [H^k_c(Z_{ij}); \rho_{ij}\colon \Gamma \to \operatorname{Aut}(H^k_c(Z_{ij}))],\end{align*} $$
   where $\epsilon _{ij}$ is $\pm 1$ , and $\rho _{ij}$ is a direct sum of copies of the trivial and/or of the regular representation.

Denote by $E_{\text {reg}}(Z_{ij})$ the E-polynomial of the regular part of $H^*_c(Z_{ij})$ , and let $E_{\text {tr}}(Z_{ij}):= E(Z_{ij}) - E_{\text {reg}}(Z_{ij})$ . Then we have

(27) $$ \begin{align} E({T}_{\mathrm{Dol}}(C, \operatorname{SL}_2)) & = \sum_{i,j}\epsilon_{ij} E_{\text{tr}}(Z_{ij}) + \sum_{ij}\epsilon_{ij} E_{\text{reg}}(Z_{ij}) \\ & = \sum_{ij}\epsilon_{ij} E_{\text{tr}}(Z_{ij}) + 2^{2g} \sum_{ij}\epsilon_{ij} E_{\text{reg}}(Z_{ij})^{\Gamma} \nonumber \\ E({T}_{\mathrm{Dol}}(C, \operatorname{SL}_2))^{\Gamma} & = \sum_{ij}\epsilon_{ij} E_{\text{tr}}(Z_{ij}) + \sum_{ij}\epsilon_{ij} E_{\text{reg}}(Z_{ij})^{\Gamma}. \nonumber \end{align} $$

Via the decomposition theorem, the same holds for $IE({M}_{\mathrm {Dol}}(C, \operatorname {SL}_2))$ , and so for $IP_t(M(C, \operatorname {SL}_2))$ , by the purity of $IH^*(M(C, \operatorname {SL}_2))$ , as explained in the proof of Theorem 1.9.

5.1 P=W conjecture for twisted character varieties

The computation of E-polynomials of character varieties was initiated in [Reference Hausel and Rodriguez-Villegas33] for twisted character varieties $M^{\mathrm {tw}}_{\mathrm {B}}=M^{\mathrm {tw}}_{\mathrm {B}}(C,G,d)$ ,

with $G=\operatorname {GL}_n, \operatorname {SL}_n$ or $\operatorname {PGL}_n$ and $\mathrm {gcd}(n,d)=1$ ; see also [Reference Mereb52].

As in the untwisted case, a nonabelian Hodge correspondence holds for $M^{\mathrm {tw}}_{\mathrm {B}}$ : there exists a diffeomorphism $\Psi \colon M^{\mathrm {tw}}_{\mathrm {Dol}} \to M^{\mathrm {tw}}_{\mathrm {B}}$ , from the Dolbeault moduli space $M^{\mathrm {tw}}_{\mathrm {Dol}}=M^{\mathrm {tw}}_{\mathrm {Dol}}(C,G,d)$ of semistable G-Higgs bundles over C of degree d; see [Reference Hausel and Thaddeus36]. However, contrary to the general untwisted character variety, $M^{\mathrm {tw}}_{\mathrm {B}}$ is smooth (a significant advantage!).

Surprisingly, Hausel and Rodriguez-Villegas [Reference Hausel and Rodriguez-Villegas33] in rank 2, and Mellit [Reference Mellit51] for $\operatorname {GL}_n$ , observed that the cohomology of $M^{\mathrm {tw}}_{\mathrm {B}}$ enjoys symmetries typical of smooth projective varieties, despite the fact that $M^{\mathrm {tw}}_{\mathrm {B}}$ is not projective. They called these symmetries curious hard Lefschetz theorem: there exists a class $\alpha \in H^2(M^{\mathrm {tw}}_{\mathrm {B}})$ which induces the isomorphism

$$ \begin{align*} \cup {\alpha^k}\colon \operatorname{Gr}^W_{\dim M_{\mathrm{B}}-2k}H^*(M^{\mathrm{tw}}_{\mathrm{B}})\xrightarrow{\simeq} \operatorname{Gr}^W_{\dim M_{\mathrm{B}}+2k}H^{*+2k}(M^{\mathrm{tw}}_{\mathrm{B}}). \end{align*} $$

Note that, as an immediate consequence of the curious hard Lefschetz theorem, the E-polynomial of $M^{\mathrm {tw}}_{\mathrm {B}}$ 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 [Reference de Cataldo, Hausel and Migliorini13]. This conjecture posits that the nonabelian Hodge correspondence exchanges two filtrations on the cohomology of $M^{\mathrm {tw}}_{\mathrm {Dol}}$ and $M^{\mathrm {tw}}_{\mathrm {B}}$ of very different origin, respectively the perverse Leray filtration (10) associated to the Hitchin fibration $\chi $ on $M^{\mathrm {tw}}_{\mathrm {Dol}}$ (the analogue of the map defined in (11)) and the weight filtration on $M^{\mathrm {tw}}_{\mathrm {B}}$ .

Conjecture 5.1 P=W conjecture for twisted moduli spaces

$$ \begin{align*}P_k H^*(M^{\mathrm{tw}}_{\mathrm{Dol}}(C,G,d)) = \Psi^* W_{2k}H^*(M^{\mathrm{tw}}_{\mathrm{B}}(C,G,d)).\end{align*} $$

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 $\chi $ on the Dolbeault side. The latter is an isomorphism between graded pieces of the perverse Leray filtration induced by cup product with a relative $\chi $ -ample class $\alpha \in H^2(M^{\mathrm {tw}}_{\mathrm {Dol}}(C,G,d))$ :

$$ \begin{align*} \cup {\alpha^k}\colon \operatorname{Gr}^P_{\dim {M}_{\mathrm{Dol}}/2-k}H^*(M^{\mathrm{tw}}_{\mathrm{Dol}})\xrightarrow{\simeq} \operatorname{Gr}^W_{\dim {M}_{\mathrm{Dol}}/2+2}H^{*+2k}(M^{\mathrm{tw}}_{\mathrm{Dol}}); \end{align*} $$

see, for instance, [Reference de Cataldo and Migliorini18, Theorem 2.1.1.(a)].

5.2 PI=WI and the intersection curious hard Lefschetz

In the untwisted (singular) case, curious hard Lefschetz fails in general; for example, [Reference Felisetti and Mauri26, Remark 7.6], and the E-polynomial of $M_{\mathrm {B}}(C,G)$ is not palindromic; see, for instance, [Reference Logares, Muñoz and Newstead46, Theorem 1.2], [Reference Martínez and Muñoz48, Theorem 2] or [Reference Baraglia and Hekmati4, Theorem 1.3]. In order to restore the symmetries, de Cataldo and Maulik suggested considering the intersection cohomology of $M_{\mathrm {B}}(C,G)$ , and in [Reference de Cataldo and Maulik15, Question 4.1.7] they conjectured the following.

Conjecture 5.2 PI=WI conjecture

$$ \begin{align*}P_k IH^*({M}_{\mathrm{Dol}}(C,G)) = \Psi^* W_{2k}IH^*(M_{\mathrm{B}}(C,G)).\end{align*} $$

As in the twisted case, the PI=WI conjecture and the relative hard Lefschetz theorem for $\chi $ would imply the intersection curious hard Lefschetz theorem.

Conjecture 5.3 intersection curious hard Lefschetz

There exists a class $\alpha \in H^2(M_{\mathrm {B}}(C,G))$ which induces the isomorphisms

$$ \begin{align*} \cup {\alpha^k}\colon \operatorname{Gr}^W_{\dim M_{\mathrm{B}}-2k}IH^*(M_{\mathrm{B}}(C,G))\xrightarrow{\simeq} \operatorname{Gr}^W_{\dim M_{\mathrm{B}}+2k}IH^{*+2k}(M_{\mathrm{B}}(C,G)). \end{align*} $$

In particular, the intersection E-polynomial of $M_{\mathrm {B}}(C,G)$ is palindromic.

In this article we provide some numerical evidence for Conjecture 5.3.

5.3 PI=WI for $\operatorname {SL}_2$ is equivalent to PI=WI for $\operatorname {GL}_2$

The P=W conjecture for $\operatorname {SL}_n$ implies the P=W conjectures for $\operatorname {PGL}_n$ and $\operatorname {GL}_n$ ; see [Reference Felisetti and Mauri26, §3.3]. The converse holds true in the twisted case for n prime by [Reference de Cataldo, Maulik and Shen17]. 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.

Theorem 5.5 [Reference de Cataldo, Maulik and Shen17]

Suppose that

1. 1. $ q^{(1-n^2)(2g-2)}IE_{\mathrm {var}}({M}_{\mathrm {Dol}}(C, \operatorname {SL}_n); q,q)=: q^{(1-n^2)(2g-2)}E(q)$ is palindromic;

2. 2. $IE_{\mathrm {var}}(M_{\mathrm {B}}(C, \operatorname {SL}_n); \sqrt {q},\sqrt {q})=q^{(2-n-n^2)(g-1)}E(q)$ .

Set $c_n := n(n-1)(g-1)$ . Then we have

$$ \begin{align*}IH^d_{\mathrm{var}}(M(C, \operatorname{SL}_n)) \simeq \operatorname{Gr}^P_{d-c_n} IH^d_{\mathrm{var}}({M}_{\mathrm{Dol}}(C, \operatorname{SL}_n)) \simeq \operatorname{Gr}^W_{2(d-c_n)} IH^d_{\mathrm{var}}(M_{\mathrm{B}}(C, \operatorname{SL}_n)).\end{align*} $$

Unfortunately, in the untwisted case the variant intersection E-polynomials are available only in rank 2; see Corollary 1.11.

5.4 Tautological classes

In [Reference Hausel and Thaddeus35] Hausel and Thaddeus proved that $H^*(M^{\mathrm {tw}}(C, \operatorname {SL}_2))^{\Gamma }$ is generated by tautological classes.Footnote ⁸ This is an essential ingredient of the proof of the P=W conjecture in the twisted case [Reference de Cataldo, Hausel and Migliorini13] and [Reference de Cataldo, Maulik and Shen16] 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 $B(C, \operatorname {SL}_2)$ be the (infinite-dimensional and contractible) space of $\operatorname {SL}_2$ -Higgs bundles on C of degree zero and $B^{ss}(C, \operatorname {SL}_2)$ be the corresponding locus of semistable Higgs bundles. Let $\mathcal {G}$ be the group of real gauge transformations with fixed determinant acting on this spaces by precomposition and $\mathcal {G}^{\mathbb {C}}$ its complexification.

We can identify the classifying space $B\mathcal {G} \simeq B(C, \operatorname {SL}_2)$ with the space of continuous maps $\operatorname {Map}(C, \operatorname {SU}_2)$ . The second Chern class of the tautological (flat) $\operatorname {SU}_2$ -bundle $\mathcal {T}$ on $C \times \operatorname {Map}(C, \operatorname {SU}_2)$ admits the Künneth decomposition

$$ \begin{align*} c_2(\mathcal{T})= \sigma \otimes \alpha + \sum^{2g}_{j=1} e_j \otimes \psi_j + 1 \otimes \beta, \end{align*} $$

where $\sigma \in H^2(C)$ is the fundamental cohomology class, and $e_1, \ldots , e_{2g}$ is a standard symplectic basis of $H^1(C)$ . Atiyah and Bott showed in [Reference Atiyah and Bott3] that the rational cohomology of $B\mathcal {G}$ is freely generated by the tautological classes $\alpha $ , $\psi _j$ and $\beta $ . That is, $H^*(B\mathcal {G})$ is the tensor product of the polynomial algebra on the classes $\alpha $ and $\beta $ of degree 2 and 4 with an exterior algebra on the classes $\psi _j$ of degree 3,

(28) $$ \begin{align} H^*(B\mathcal{G})\simeq \mathbb{Q}[\alpha, \beta] \otimes \Lambda (\psi_j). \end{align} $$

In particular, the Poincaré polynomial of the classifying space $B\mathcal {G}$ is

(29) $$ \begin{align} P_t(B\mathcal{G})=\frac{(t^3+1)^{2g}}{(t^2-1)(t^4-1)}. \end{align} $$

Now the nonabelian Hodge correspondence induces the following isomorphism in equivariant cohomology:

(30) $$ \begin{align} H^*_{\mathcal{G}}(B^{ss}(C, \operatorname{SL}_2))\simeq H^*_{\operatorname{SL}_2} (\operatorname{Hom}(\pi_1(C), \operatorname{SL}_2)); \end{align} $$

see [Reference Daskalopoulos, Wentworth and Wilkin12, Theorem 1.2]. Together with Kirwan surjectivity, [Reference Daskalopoulos, Weitsman, Wentworth and Wilkin10, Theorem 1.4]

(31) $$ \begin{align} H^{*}(B\mathcal{G})\simeq H^{*}_{\mathcal{G}}(B(C, \operatorname{SL}_2)) \twoheadrightarrow H^{*}_{\mathcal{G}}(B^{ss}(C, \operatorname{SL}_2))^{\Gamma}, \end{align} $$

this implies that the $\Gamma $ -invariant $\operatorname {SL}_2$ -equivariant cohomology of $\operatorname {Hom}(\pi _1(C), \operatorname {SL}_2)$ is generated by tautological classes.

Proof. Since $\Sigma $ has codimension $4g-6$ in M, we have

(32) $$ \begin{align}IH^{<4g-6}(M(C, \operatorname{SL}_2))\simeq H^{<4g-6}(M^{\mathrm{sm}}(C, \operatorname{SL}_2));\end{align} $$

see, for instance, [Reference Durfee23, Lemma 1]. In particular, $IH^{<4g-6}(M(C, \operatorname {SL}_2))$ has a natural structure of graded ring. The open subset of simple representations $\operatorname {Hom}^{s}(\pi _1(C), \operatorname {SL}_2)$ in $\operatorname {Hom}(\pi _1(C), \operatorname {SL}_2)$ is a $\operatorname {PGL}_2$ -principal bundle over the smooth locus $M^{\mathrm {sm}}_B(C, \operatorname {SL}_2)$ , and so

(33) $$ \begin{align} H^{*}(M^{\mathrm{sm}}(C, \operatorname{SL}_2)) \simeq H^*_{\operatorname{SL}_2} (\operatorname{Hom}^{s}(\pi_1(C), \operatorname{SL}_2)). \end{align} $$

We claim that the composition of (31), (30), (34), the inverse of (33) and (32),

$$ \begin{align*} H^{<4g-6}(B \mathcal{G}) & \xrightarrow{a} H^{<4g-6}_{\operatorname{SL}_2} (\operatorname{Hom}(\pi_1(C), \operatorname{SL}_2)) \\ & \xrightarrow{b} H^{<4g-6}_{\operatorname{SL}_2} (\operatorname{Hom}^s(\pi_1(C), \operatorname{SL}_2))\simeq IH^{< 4g-6}(M(C, \operatorname{SL}_2)), \end{align*} $$

is an isomorphism. Indeed, a is surjective by (31) (and Corollary 1.11) and actually bijective since by [Reference Daskalopoulos and Uhlenbeck11, Corollary 1.3] we have

$$ \begin{align*} P^{\operatorname{SL}_2}_t(\operatorname{Hom}(\pi_1(C), \operatorname{SL}_2)):= \sum_d \dim H^d_{\operatorname{SL}_2}(\operatorname{Hom}(\pi_1(C), \operatorname{SL}_2))t^d = P_t(B\mathcal{G}) + O(t^{4g-4}). \end{align*} $$

Further, b is injective by Lemma 5.9 and actually bijective due to (3) and (29). The free generation of $IH^{<4g-6}(M(C, \operatorname {SL}_2))$ now follows from (28). See [Reference Shende64] for the weight of the tautological classes. Finally, $\psi _i$ and $\beta $ are not cohomology classes by Corollary 1.10 and preceding lines.

Remark 5.8. Theorem 5.7 holds for $\operatorname {PGL}_2$ , as

$$ \begin{align*}IH^{< 4g-6}(M(C, \operatorname{SL}_2))=IH^{< 4g-6}(M(C, \operatorname{SL}_2))^{\Gamma} \simeq IH^{< 4g-6}(M(C, \operatorname{PGL}_2))\end{align*} $$

by Corollary 1.11, and so for $\operatorname {GL}_2$ , too. In the latter case, however, mind that there are additional generators $\epsilon _j$ which are pullback via the map (4) of a standard basis of $H^1(M(C,\operatorname {GL}_1))\simeq H^1(C)$ .

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 $M(C, \operatorname {SL}_n)$ . However, this surjectivity is an open problem for $n>2$ ; cf. [Reference Cliff, Nevins and Shen9].

We prove the lemma used in the proof of Theorem 5.7.

Lemma 5.9. The natural restriction map

(34) $$ \begin{align} H^d_{\operatorname{SL}_2} (\operatorname{Hom}(\pi_1(C), \operatorname{SL}_2)) \to H^d_{\operatorname{SL}_2} (\operatorname{Hom}^s(\pi_1(C), \operatorname{SL}_2)) \end{align} $$

is bijective for $d<4g-7$ and injective for $d=4g-7$ .

Proof. Set $c := \operatorname {codim}\operatorname {Sing} \operatorname {Hom}(\pi _1(C), \operatorname {SL}_2)=\dim M - \dim \Sigma + \dim \operatorname {Stab}_\Sigma = 4g-5$ , where $\operatorname {Stab}_\Sigma \simeq \mathbb {G}_m$ is the stabiliser of a closed orbit over $\Sigma $ . Since $\operatorname {Hom}(\pi _1(C), \operatorname {SL}_2)$ is a complete intersection (adapt [Reference Etingof24, Theorem 1.2] or [Reference Simpson66, Proposition 11.3]), there exists an isomorphism

(35) $$ \begin{align} H^d_{\operatorname{SL}_2} (\operatorname{Hom}(\pi_1(C), \operatorname{SL}_2)) \simeq H^d_{\operatorname{SL}_2} (\operatorname{Hom}^{\mathrm{sm}}(\pi_1(C), \operatorname{SL}_2)) \text{ for }d<c-1. \end{align} $$

Indeed, take an approximation $E_k$ of the universal $\operatorname {SL}_2$ -bundle $E\mathrm {SL}_2$ ; that is, a smooth variety $E_k$ with a free $\operatorname {SL}_2$ -action and such that $H^{< k}(X \times _{\operatorname {SL}_2}E_k)\simeq H^{< k}_{\operatorname {SL}_2}(X \times _{\operatorname {SL}_2}E\mathrm {SL}_2)=: H^{< k}_{\operatorname {SL}_2}(X);$ see [Reference Anderson1, Lemma 1.3]. By Luna slice theorem $X \times _{\operatorname {SL}_2}E_k$ is a local complete intersection, and the singular locus has again codimension c. Then (35) follows from [Reference Goresky and MacPherson31, p.199].

Further, the complement of $\operatorname {Hom}^s(\pi _1(C), \operatorname {SL}_2)$ in the smooth locus has codimension $2g-3$ ; see, for instance, [Reference González-Prieto29, §7.2] where the complement is denoted $\mathfrak {X}^{\rho }_{g}$ . Therefore, by the equivariant Thom isomorphism, the restriction map

$$ \begin{align*}H^d_{\operatorname{SL}_2} (\operatorname{Hom}^{\mathrm{sm}}(\pi_1(C), \operatorname{SL}_2)) \to H^d_{\operatorname{SL}_2} (\operatorname{Hom}^{s}(\pi_1(C), \operatorname{SL}_2))\end{align*} $$

is bijective for $d<4g-7$ and injective for $d=4g-7$ .

5.5 P=W vs PI=WI: nonpurity of $H^*(M_{\mathrm {B}})$

Despite the failure of curious hard Lefschetz, it still makes sense to conjecture $P=W$ phenomena for the ordinary cohomology of $M_{\mathrm {B}}$ .

Conjecture 5.10 P=W conjecture for untwisted character varieties

$$ \begin{align*}P_k H^*({M}_{\mathrm{Dol}}(C,G)) = \Psi^* W_{2k}H^*(M_{\mathrm{B}}(C,G)).\end{align*} $$

It was proved in [Reference Felisetti and Mauri26, Theorem 6.1] that the PI=WI conjecture for genus 2 and rank 2 implies the P=W conjecture simply by restriction, since $H^*(M_{\mathrm {B}}(C, \operatorname {SL}_2))$ injects into $IH^*(M_{\mathrm {B}}(C, \operatorname {SL}_2))$ or, equivalently, by the purity of $H^*(M_{\mathrm {B}}(C, \operatorname {SL}_2))$ ; see Proposition 2.4. In higher genus the situation is more subtle, as the following theorem shows.

5.6 P=W for resolution fails when no symplectic resolution exists

In [Reference Felisetti and Mauri26] 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 admit a symplectic resolution.

Conjecture 5.13 P=W conjecture for resolution

There exist resolutions of singularities $f_{\mathrm {Dol}}\colon \widetilde {M}_{\mathrm {Dol}}(C,G) \to {M}_{\mathrm {Dol}}(C,G)$ and $f_{\mathrm {B}}\colon \widetilde {M}_{\mathrm {B}}(C,G) \to {M}_{\mathrm {B}}(C,G)$ and a diffeomorphism $\widetilde {\Psi }\colon \widetilde {M}_{\mathrm {Dol}}(C,G) \to \widetilde {M}_{\mathrm {B}}(C,G)$ , such that the following square commutes:

(36)

and the lift $\widetilde {\Psi }^*$ of the nonabelian Hodge correspondence ${\Psi }^*$ satisfies the property

(37) $$ \begin{align} P_kH^*(\widetilde{M}_{\mathrm{Dol}}(C,G), \mathbb{Q}) = \widetilde{\Psi}^* W_{2k}H^*(\widetilde{M}_{\mathrm{B}}(C,G), \mathbb{Q}). \end{align} $$

In [Reference Felisetti and Mauri26, Theorem 3.4] Camilla Felisetti and the author proved that resolutions of singularities satisfying (36) do exist; for instance, the Kirwan–O’Grady desingularisations are such.

In Theorem 5.14 we show, however, that if $M(C,\operatorname {GL}_n)$ does not admit a symplectic resolution, no resolution of $M(C,\operatorname {GL}_n)$ satisfies (37), despite the palindromicity of the E-polynomial of $T_B$ ; see Theorem 1.12. A fortiori, the same negative result holds for $G=\operatorname {SL}_n$ .

This means that the hypotheses of [Reference Felisetti25, Main Theorem, 3] were optimal for $G=\operatorname {GL}_n, \operatorname {SL}_n$ : the proof of Theorem 5.14 suggests that the semismallness of the desingularisation 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 [Reference de Cataldo, Hausel and Migliorini14, §4.4].

Proof. Let $f\colon \widetilde {M} \to M(C, \operatorname {GL}_n)$ be a resolution of singularities of $M(C, \operatorname {GL}_n)$ as in (36) and E be an f-exceptional divisor whose image is contained in the singular locus $\Sigma := \operatorname {Sing} M(C, \operatorname {GL}_n)$ . Recall that $\chi \colon {M}_{\mathrm {Dol}}(C, \operatorname {GL}_n) \to \Lambda := \bigoplus ^n_{i=1} H^0(C, K^{\otimes i}_C)$ is the Hitchin fibration (11).

The locus $\chi (\Sigma )$ consists of reducible characteristic polynomials, and it has codimension

$$ \begin{align*} \frac{1}{2}(\dim M(C, \operatorname{GL}_n) - \max\{\dim M(C, \operatorname{GL}_{n_1})+\dim M(C, \operatorname{GL}_{n_2}): n=n_1+n_2\}) \geq 2. \end{align*} $$

The last inequality follows, for instance, from [Reference Bellamy and Schedler6, Lemma 2.2.(2)]. In particular, the general affine line in $\Lambda $ avoids $\chi \circ f(E) \subseteq \chi (\Sigma )$ . Then by (10) the Poincaré dual of E belongs to $P_0 H^2(\widetilde {M})$ . However, since $\widetilde {M}_{\mathrm {B}}$ is smooth, $H^2(\widetilde {M}_B)$ has weight not smaller than 2. This contradicts (37).