Perversity equals weight for Painlev\'e spaces

We provide further evidence to the $P=W$ conjecture of de Cataldo, Hausel and Migliorini, by checking it in the Painlev\'e cases. Namely, we compare the perverse Leray filtration induced by the Hitchin map on the cohomology spaces of the Dolbeault moduli space and the weight filtration on the cohomology spaces of the irregular character variety corresponding to each of the Painlev\'e $I-VI$ systems. We find that up to a shift the two filtrations agree.


Introduction and statement of main result
Throughout the paper we let X denote one of the symbols I, II, III(D6), III(D7), III(D8), IV, V deg , V, V I so that the index P X refers to Painlevé X. In [25], irregular Betti moduli spaces (also called wild character varieties following [6]) M P X B are defined and shown to be C-analytically isomorphic under the Riemann-Hilbert correspondence to irregular de Rham spaces M P X dR . (At a higher level of generality, moduli spaces of untwisted irregular connections of arbitrary rank on a compact Riemann surface of arbitrary genus were constructed in [5] as algebraic symplectic manifolds, and the irregular Riemann-Hilbert correspondence for the moduli spaces was proven in [7], building on the categorical correspondence of Malgrange [23,Chapitre 4].) It follows from [4, Theorem 1] that for every X in the so-called untwisted cases II, III(D6), IV, V, V I smooth complex analytic moduli spaces M P X Dol exist and are diffeomorphic under non-abelian Hodge theory to the corresponding M P X dR . A combination of these results implies that in the untwisted cases M P X B and M P X Dol are diffeomorphic; such a diffeomorphism is expected to exist in the remaining (twisted) cases too. In [18], [19] we gave an explicit description of the spaces M P X Dol and of their (irregular) Hitchin map in terms of elliptic pencils. In [25], an explicit description of the spaces M P X B is provided as affine cubic surfaces. Deligne [14] constructs a weight filtration W on the complex cohomology spaces of an affine algebraic variety. In particular, the cohomology spaces of M P X B carry a mixed Hodge structure. On the other hand, the Hitchin map endows the complex cohomology spaces of M P X Dol with a perverse Leray filtration P [3]. Following [15, page 2], we set Remarkably, in the rank 2 case without (regular or irregular) singularities equality between these two polynomials for Dolbeault and Betti spaces corresponding to each other under non-abelian Hodge theory and Riemann-Hilbert correspondence was proven in [12,Theorem 1.0.1], and conjectured to be the case in general (the P = W conjecture). The perverse filtration for some logarithmic Hitchin systems was studied by Z. Zhang [29], where he showed multiplicativity of the filtration with respect to wedge product on Hilbert schemes of smooth projective surfaces fibered over a curve, and thereby computed their perverse polynomials. More generally, W. Chuang, D. Diaconescu, R. Donagi and T. Pantev conjectured a formula for the perverse Hodge polynomial of moduli spaces of meromorphic Higgs bundles with one irregular singularity [11]. On the Betti side, T. Hausel, M. Mereb and M. Wong investigated the weight filtration on the cohomology of character varieties of punctured curves with one irregular singularity, and extended the P = W conjecture to this case [15,Problem 0.1.4]. The purpose of this paper is to give an affirmative answer to this conjecture in the Painlevé cases. Notice that not all the cases we study fall into the class studied in [15], because some of them admit two irregular singularities, some of which with twisted formal type.
Theorem 1. For every X the perverse Leray and weight filtrations on the cohomology of the Dolbeault and Betti spaces mapped to each other by non-abelian Hodge theory and Riemann-Hilbert correspondence agree up to a shift by 1: P H P X (q, t) = q −1 W H P X (q, t). (1) Moreover, in the untwiseted cases we will prove that the classes generating the exotic pieces of the P and W filtrations match up under non-abelian Hodge theory. Our proof goes through establishing a conjecture of C. Simpson [26,Conjecture 11.1] in these special cases. It may be considered as "a homotopy version in the highest graded part" of the P = W conjecture. Theorem 2. There exists a smooth compactification M P X B of M P X B by a simple normal crossing divisor D such that the body |N P X | of the nerve complex N P X of D is homotopy equivalent to S 1 . Moreover, for some sufficiently large compact set K ⊂ M P X B , there exists a homotopy commutative square Here, h denotes the Hitchin map, D × ⊂ Y is a neighbourhood of ∞ in the Hitchin base, and the top row is the diffeomorphism coming from non-abelian Hodge theory.
For details, see Section 3. An analogous statement for two 2-dimensional and a 4-dimensional logarithmic Dolbeault moduli spaces has been proven by A. Komyo [22].
Acknowledgements: During the preparation of this document, the author benefited of discussions with P. Boalch, T. Hausel, M-H. Saito, C. Simpson and A. Stipsicz, and was supported by the Lendület Low Dimensional Topology grant of the Hungarian Academy of Sciences and by the grants K120697 and KKP126683 of NKFIH.

Perverse Leray filtration
We first deal with the left-hand side of (1). We study moduli spaces M P X Dol parameterizing parabolically (semi-)stable Higgs bundles of rank 2 over CP 1 with irregular singularities at ≤ 4 points of total pole order equal to 4, and having specific local forms near these punctures. We assume that the parabolic weights are general, so that stability is equivalent to semi-stability. Moreover, we fix the degree of the underlying vector bundle to be odd. For the local forms of the Higgs field • in case X = I, see: [ However, for the purposes of this paper, we need a certain completion of the Dolbeault moduli spaces studied for instance in [18,19,20]. Namely, in case the residue Res p (θ) of the Higgs field at some logarithmic point p is assumed to have two equal eigenvalues with non-trivial nilpotent part, then we consider the moduli space M P X Dol of corresponding Higgs bundles completed with all Higgs bundles having the same eigenvalues of its residue but with trivial nilpotent part, equipped with a quasi-parabolic structure of full flag type at these points. In order to simplify notation, we will continue to denote our completed moduli space by the symbol M P X Dol . The reason we consider this completion is that the Hitchin fibers of the non-completed moduli spaces may be non-compact, as endomorphisms with nontrivial nilpotent part may converge to ones with trivial nilpotent part. Importantly for our purposes, we have: Lemma 1. The completed moduli space is a smooth complex manifold, and the irregular Hitchin map h : is proper.
Proof. The proof of the first statement follows from [4,Theorem 5.4]. Indeed, let us consider an endomorphism A ∈ gl(E| p ) of the fiber of a given rank 2 smooth vector bundle E at p. Let the decomposition of A into semi-simple and nilpotent part be and assume that A nil = 0 (so necessarily A s is a multiple of identity). Finally, let stand for the parabolic subalgebra containing A nil and π : p → l its Levi quotient. It follows from [4,Theorem 5.4] that the moduli space parameterizing irregular Higgs bundles (E, θ) endowed with a compatible parabolic structure, with fixed underlying smooth vector bundle E and such that π(Res p (θ)) = π(A) is a smooth complex manifold. Now, given that π(A s ) = π(A) we get that the completed Dolbeault moduli space is smooth. Properness follows from [9,24] for the moduli space of Higgs bundles with some poles of total order n over any compact Riemann surface C, without any condition on the polar parts and residues at these points. In the case C = CP 1 and a divisor of total multiplicity 4, the base C 8 of the Hitchin map for this system contains the image of those Higgs bundles having prescribed polar parts and residues as an affine open subspace A ∼ = C. Namely, A is specified by the jet of the characteristic coefficients at the punctures of given order (see [18,19,20], or in greater generality [2,Theorems 5,6]). The preimage h −1 (a) of any a ∈ A is the set of all Higgs bundles having characteristic polynomial corresponding to a. By [19,Lemma 10.1], at any logarithmic singularity p the characteristic polynomial of the residue of the Higgs field is prescribed by a, but its adjoint orbit is not. Conversely, if a sequence of Higgs bundles (E n , θ n ) n≥1 in h −1 (a) converge to some Higgs bundle (E 0 , θ 0 ) then the residue of θ 0 at p has the same characteristic polynomial as the residues of θ n at p. Replacing a finite number of points in the fiber by projective lines (corresponding to choices of a parabolic line ℓ ⊂ E| p ) does not modify properness. This implies properness for the completed moduli problem.
is an irregular parabolic Higgs bundle such that with A nil = 0 then the compatible quasi-parabolic line ℓ ⊂ E| p is uniquely determined by the requirement A nil ∈ p. On the other hand, if (E 1 , θ 1 ) is an irregular parabolic Higgs bundle such that The irregular Hitchin map (2) endows H * (M P X Dol , Q) with a finite decreasing perverse filtration P • through the perverse Leray spectral sequence. As usual, we set Gr P k = P k /P k+1 . Proposition 1. We have and all the other graded pieces of H * for P vanish. In particular, we have b 2 (M P X Dol ) = 1 + d P X and P H P X (q, t) = q −1 + d P X q −2 t 2 + q −3 t 2 . Furthermore, we have d P X = 10 − χ(F P X ∞ ), where F P X ∞ is the fiber at infinity of M P X Dol listed in Table 1. Table 1. Fiber at infinity and perverse Hodge polynomial of M P X

Dol
The specific forms of P H P X (q, t) can then easily be determined using Proposition 1 and the fibers F P X ∞ , and for convenience they are included in Table 1.
Proof. As M P X Dol is a non-compact oriented 4-manifold, by Poincaré duality we have b 4 (M P X Dol ) = 0. We use the geometric characterization of the perverse filtration provided in [13,Theorem 4.1.1] in terms of the flag filtration F . Namely, let Rh * denote the right derived direct image functor in the derived category of constructible sheaves and R l h * be the l'th right derived direct image sheaf. Let H denote hypercohomology of a complex of sheaves and H stand for cohomology of a single sheaf. Let Y −1 ∈ Y be a generic point and Q M be the constant sheaf with fibers Q on M P X Dol . Here and throughout this section, for ease of notation we drop the subscript Dol and the superscript P X of M P X Dol whenever this latter is in subscript. It is known that there exists a spectral sequence L E k,l r called the (ordinary) Leray spectral sequence degenerating at the second page . We then have the equality It immediately follows that dim Q Gr P −1 H 0 (M P X Dol , Q) = 1 and all other graded pieces of H 0 vanish. Moreover, it also follows that and the non-trivial associated graded pieces of P are Let us turn to the computation of these graded pieces in general. We know from [18, for some non-reduced curve F P X ∞ , moreover there exists an elliptic fibratioñ so that the following diagram commutes In particular, we haveh −1 (∞) = F P X ∞ . The type of the curves F P X ∞ is determined by X and is listed in Table 1. If the residues of the Higgs field at the simple poles are assumed to have distinct eigenvalues then exactly the same results hold in the cases II, IV, V deg , V, V I too. Lemma 2. In cases X = II, IV, V deg , V , assume that there is a simple pole of the Higgs field such that the residue has equal eigenvalues. Then, there exists an embedding M P X Dol ֒→ E(1) of the completed Dolbeault moduli spaces so that for the non-reduced curve F P X ∞ listed in Table 1, and the diagram (6) commutes.
Proof. The proof is similar to the P V I case. We consider the pencil of spectral curves associated to Higgs bundles with poles of the given local forms. According to [28,Theorem 1.1], the moduli space arises as a certain relative compactified Picard-scheme of this pencil. In order to determine the relative compactified Picardscheme, one first needs to blow up the base locus of the pencil of spectral curves; in general this process involves blowing up infinitesimally close points. The common phenomenon in the cases when the residue of the Higgs field at a simple pole p 1 has equal eigenvalues is that one exceptional divisor E of the blow-up process (with self-intersection number equal to (−2)) maps to p 1 under the ruling and becomes a component of one of the fibers X t in the fibration. In the cases X = II, IV this is precisely proven in [19,Lemma 4.5]. The same proof goes along for the other types too, because both the assumptions and the assertion is local at the fiber of the ruling over p 1 . Let us denote by Z t the singular curve in the pencil of spectral curves whose proper transform contains E, so that X t is the proper transform of Z t . It follows from [27, Section 6] that X t is one of the Kodaira types The corresponding spectral curves Z t are listed before [19, Lemma 10.1], except in case X t is of type I 4 . The case I 4 may only occur in cases X = V deg , V , under the assumption that there exists two simple poles p 1 , p 2 of the Higgs field such that for i ∈ {1, 2} Res pi (θ) has two equal eigenvalues. In this case two non-neighbouring components of X t get mapped to p 1 , p 2 respectively under the ruling and Z t consists of two rational curves (sections of the Hirzebruch surface of degree 2) intersecting each other transversely in two points, one on the fiber over p 1 and another one on the fiber over p 2 . Indeed, Z t may have at most two components because it is a 2 : 1 ramified covering of the base curve CP 1 , so two components of X t must be exceptional divisors of the blow-up process; one of these two components must come from blow-ups at p 1 and the other one from blow-ups at p 2 , for otherwise the dual graph could not be a cycle. By [19, Lemma 10.1], Higgs bundles whose residue at p 1 (and p 2 in case IV ) has non-trivial nilpotent part correspond to locally free spectral sheaves over Z t at p 1 (respectively, p 2 ). For such curves Z t , [19, Lemma 10.2] determines the families of locally free spectral sheaves giving rise to parabolically stable Higgs bundles. On the other hand, any torsion-free sheaf on Z t is the direct image of a locally free sheaf on a partial normalization. Let us separate cases according to the type of X t .
(1) If X t is of type I 2 then Z t is a nodal rational curve with a single node on the fiber over p 1 ; there exists a unique torsion-free but not locally free sheaf of given degree on Z t . This gives rise to a unique Higgs bundle whose residue has the required eigenvalue of multiplicity 2 and trivial nilpotent part. This object is irreducible hence stable. On the other hand, the choice of quasi-parabolic structure at p 1 compatible with this unique Higgs bundle is an arbitrary element of CP 1 . This gives us that the Grothendieck class of the Hitchin fiber of the completed moduli space over the point t is As the unique Kodaira fiber in this class is I 2 , we deduce from Lemma 1 that the Hitchin fiber of the completed moduli space over the point t is of this type, i.e. the same type as X t .
(2) If X t is of type I 3 then Z t is composed of two sections of the ruling intersecting each other transversely in two distinct points, one of them lying on the fiber over p 1 . As shown in [19, Lemma 10.2. (2)], Higgs bundles with spectral curve Z t and residue having non-trivial nilpotent part form a family parameterized by a variety in the class As in the previous point, there exists a single torsion-free but not locally free sheaf giving rise to a Higgs bundle with spectral curve Z t such that Res p1 (θ) has trivial nilpotent part. Again, the quasi-parabolic structure at p 1 compatible with this unique Higgs bundle is parameterized by CP 1 , so the class of the Hitchin fiber of the completed moduli space over the corresponding point t is The only Kodaira fiber in this class is I 3 , hence the Hitchin fiber of the completed moduli space over t is I 3 .
(3) For X t is of type I 4 , as we already mentioned, Z t is a union of two sections of the ruling that intersect each other transversely in two points: one on the fiber over each of p 1 , p 2 . Therefore, the analysis is quite similar to the case of I 3 treated above: Higgs bundles with spectral curve Z t such that both Res p1 (θ), Res p2 (θ) have non-trivial nilpotent part are parameterized by a variety in class 2[C × ].
(The class of the point [pt] that appears in the case I 3 is missing here because it corresponds to a torsion-free but non-locally free sheaf at p 2 which would give rise to a Higgs bundle with trivial nilpotent part at p 2 .) Now, there exists a single sheaf of given degree that is locally free at p 1 and torsion-free but non-locally free at p 2 . For the Higgs bundle obtained as the direct image of this sheaf, compatible quasi-parabolic structures at p 1 are parameterized by CP 1 . The same observations clearly apply with p 1 , p 2 interchanged too. Finally, notice that stability excludes that the spectral sheaf be torsion-free but non-locally free at both p 1 , p 2 : this would mean that the spectral sheaf comes from the normalization of Z t , so the corresponding Higgs bundle would be decomposable. In sum, the class of the Hitchin fiber of the completed moduli space over t is As the only Kodaira fiber in this class is I 4 , we infer that the Hitchin fiber of the completed moduli space over t is of type I 4 . (4) If X t is of type III then Z t is a cuspidal rational curve with a single cusp on the fiber over p 1 . Stability of any Higgs bundle with spectral sheaf supported on Z t again follows from irreducibility. By virtue of [19,Lemma 7.2], locally free sheaves of given degree on Z t are parameterized by C × and there exists a single non-locally free torsion free sheaf of given degree on Z t . This latter gives rise to a unique Higgs bundle in the extended moduli space with residue having trivial nilpotent part. Again, compatible quasi-parabolic structures at p 1 provide a further CP 1 of parameters, so that the class of the Hitchin fiber of the completed moduli space over t is As the unique Kodaira fiber in this class is III, we see that the Hitchin fiber of the completed moduli space over t is of type III. As the only Kodaira fiber in this class is IV , the corresponding Hithin fiber is of type IV too.

Remark 2.
In the Lemma we found that the completed Hitchin system has the same type of singular fibers as the associated fibration of spectral curves. A similar statement is shown in [1, Corollary 6.7], based on the analysis [17] of Fourier-Mukai transform for sheaves on various singular elliptic curves. Our result is more general than the one of [1] in that it also treats the ramified Dolbeault moduli spaces and consequently more types of singular fibers enter into the picture, and we also consider the dependence of our result on parabolic weights. As Gl(2, C) is Langlands-selfdual, the above relative self-duality result can be considered as an irregular version of Mirror Symmetry of Hitchin systems [16]. Now, consider the smooth de Rham complex According to de Rham's theorem, the cohomology spaces of this complex of sheaves are isomorphic to the singular cohomology spaces H n (M P X Dol , C). The morphism h defines the following finite decreasing filtration on Ω • M : . This filtration gives rise to the Leray spectral sequence The first page L E 1 of this spectral sequence is defined by taking cohomology of the relative smooth de Rham differentials so we have The morphisms d 1 on these groups are induced by the smooth de Rham differential By definition, these morphisms are called Gauß-Manin connections It is known that ∇ l GM is a smooth integrable connection over the base set Y reg ⊂ Y parameterizing smooth fibers of h. The holomorphic vector bundle induced by the (0, 1)-part of ∇ l GM extends over the finite set Y \ Y reg , and the meromorphic connection induced by the (1, 0)-part of ∇ l GM on the extended holomorphic vector bundle has regular singularities at Y \ Y reg . For l ∈ {0, 2}, the complex vector bundles R l h * C M ⊗ C Ω k Y are of rank 1 over Y , and the monodromy of ∇ l GM is trivial, so that ∇ 0 GM = d = ∇ 2 GM . For l = 1, the vector bundles underlying the local systems To compute L E 2 , we need to take the cohomology groups of the terms of L E 1 with respect to ∇ l GM . For l ∈ {0, 2} the cohomology groups of ∇ l GM = d compute the singular C-valued cohomology spaces of Y = C. For l = 1, we have ) ∨ , and this latter vanishes because there exist no compactly supported invariant sections of the dual integrable connection (∇ 1 GM ) t on the non-compact space Y . We get that L E k,l 2 is of the form k = 2 0 0 0 It is known that the Leray spectral sequence degenerates at this term. In particular, the dimension b 1 (M) of L E 0,1 2 is equal to the first Betti number b 1 (M P X Dol ). Lemma 3. We have b 1 (M P X Dol ) = 0. Proof. Let N denote a tubular neighbourhood of F P X ∞ in E(1) and consider the covering Part of the associated Mayer-Vietoris cohomology long exact sequence reads as We know that H 1 (E(1), Q) vanishes because it has the structure of a CW-complex only admitting even-dimensional cells. On the other hand, M P X Dol ∩ N is homotopy equivalent to an S 1 -bundle π : M → F P X ∞ .
(11) According to Table 1, for each X the Dynkin diagram of F P X ∞ is simply connected. It follows from the fibration homotopy long exact sequence that there exists a morphism

then a generator is given by the dual of the class
[c] ∈ H 1 (M P X Dol ∩ N, Q) of a fiber S 1 of (11). To prove the assertion it is clearly sufficient to show that the connecting morphism δ between cohomology groups is a monomorphism, or dually, that the connecting morphism ∂ : H 2 (E(1), Q) → H 1 (M P X Dol ∩ N, Q) on singular homology is an epimorphism. Let us now recall the definition of ∂. Assume given a singular 2-cycle C in E(1), that decomposes as where A and B are singular 2-chains in M P X Dol and N respectively. (Such a decomposition always exists using barycentric decomposition.) We then let Given this definition, we need to show that there exists a singular 2-cycle C and a decomposition (12) such that ∂(A) = c. Now, the intersection form of E(1) is non-degenerate, specifically it turns H 2 (E(1), Q) into the lattice Let us denote by H the generator of the component (1) determined by the hyperplane class of CP 2 . On the other hand, the intersection form of N is obviously isomorphic to the negative semi-definite lattice associated to an extended Dynkin diagram of type listed in Table 1, with 1-dimensional radical. Therefore, the intersection lattice of F P X ∞ can not be in the orthogonal complement of H (which is negative definite), and we have Let C = H. It then follows that C ∩ N is homologous to a positive multiple B of a disc with boundary c. Up to modifying C by a boundary, we may assume that C ∩ N = B. Setting A = C − B, the singular 2-chain A then clearly lies in M P X Dol . This finishes the proof of the Lemma.
Let us set (where we reintroduced the superscript P X of M in order to emphasize the way in which the right-hand side depends on the specific irregular type that we work with). Let us denote Euler-characteristic of a CW-complex by χ.
Lemma 4. We have d P X = 10 − χ(F P X ∞ ). Proof. We see from degeneration of the Leray spectral sequence at L E 2 that By Lemma 3 and additivity of χ with respect to stratifications we deduce b 0 (M P X Dol ) + b 2 (M P X Dol ) + χ(F P X ∞ ) = χ(E(1)) = 12. The assertion follows because M P X Dol is connected. Lemma 5. The inclusion Y −1 ֒→ Y induces a non-trivial morphism on H 2 : Proof. Dually, we need to show that Y −1 ֒→ Y induces a non-trivial morphism on second singular homology

that a generic fiber of h is not a boundary in M P X
Dol . This follows immediately from the known fact that the generic fiber ofh is not a boundary in E(1).

This Lemma and (3), (4) coupled with (13) finish the proof of the Proposition.
Remark 3. It would be certainly possible to prove Proposition 1 merely using the definition of the perverse Leray filtration, i.e. by applying appropriate shifts to the sheaves R l h * Q M or, equivalently, applying the Decomposition Theorem [3] to Rh * Q M [2]. This direct proof would be actually quite similar to the argument presented above.

Weight filtration
We now turn our attention to the right hand side of (1). Observe first that according to [25], for all X the space M P X B is a smooth affine cubic surface defined by a polynomial for an affine quadric Q P X . Each of these quadrics depends on some subset (possibly empty) of complex parameters s 0 , s 1 , s 2 , s 3 , α, β. For a generic choice of these parameters, that we will assume from now on, the obtained affine cubic surfaces are smooth. Moreover, in case the cubics do not depend on any parameter, the affine cubic surfaces are always smooth. Denote by the homogenization of f P X as a homogeneous cubic polynomial and consider the projective surface which is a compactification of M P X B . In general, M P X B is not smooth: it has some isolated singularities over x 0 = 0. Let us set where µ stands for the Milnor number of an isolated surface singularity and the summation ranges over all singular points of M P X B . Proposition 2. The non-trivial graded pieces of H * for W are The singularities of M P X B and the weight polynomial of M P X B in the various cases are summarized in Table 2.
Proof. We use the definition given in [14] of the weight filtration on the mixed Hodge structure on the cohomology of an affine variety in terms of a smooth projective compactification. The form (14) of f P X implies that the compactifying divisor is defined by the equation where each L i is a complex projective line such that each two of them L i , L j for i < j intersect each other transversely in a point p ij . Said differently, the nerve complex of D consists of the edges (and vertices) of a triangle A 2 . In particular, the body of this complex is homeomorphic to a circle S 1 . As we will see in Subsections 3.1-3.9, all singularities of M P X B are located at some of the points p ij and are of type A k for k = µ(p ij ). We obtain a smooth compactification M P X consisting of reduced projective lines. We know that D P X contains the proper transform of each component of (18). More precisely, the nerve complex N P X of D P X arises from the graph A (1) 2 of (18) by replacing the edge corresponding to an intersection point p ij by a diagram A µ(pij ) . On the other hand, the generic plane section of M P X B is a cubic curve, therfore the nerve complex N P X must appear on Kodaira's list [21]. From this we see that the nerve complex of D P X is a cycle of length N P X + 3 As customary, we will denote by N P X 0 and N P X 1 the set of 0-and 1-dimensional cells of N P X , respectively.
We are now ready to determine the Betti numbers of M P X B . Lemma 6. We have Proof. The assertion for b 0 is obvious, and then immediately follows by Poincaré duality for b 4 too.
In case N P X = 0, i.e. M P X B is a smooth projective cubic surface, it is known that M P X B is given by a blow-up of CP 2 in six different points, and so carries the structure of a CW-complex with only even-dimensional cells, with 7 two-dimensional cells.
The non-smooth surfaces M P X B clearly belong to the 20-dimensional family of projective cubic surfaces. The points parameterizing smooth cubics form a dense set in C 20 with respect to the analytic topology. We will see in Subsections 3.1-3.9 that the spaces M P X B only admit singularities of type A k . It is known that a smoothing of a projective surface with ADE singularities coincides up to diffeomorphism with a minimal resolution thereof. In our case, a smoothing is a smooth cubic surface.
The smooth case treated in the previous paragraph therefore implies the general statement. Now, [14,Théorème (3.2.5)] implies that there exists a spectral sequence Let us list a few properties (either obvious or directly following from [14, Théorème (3.2.5)]) related to this spectral sequence.
(3) The filtration W N is induced by n ≤ N on the above diagram, and its shifted filtration W [k] defines the mixed Hodge structure on H k (M P X B , C). (4) Up to identifying H 2 (L, C) with H 0 (L, C) via the Lefschetz operator, the map δ is the differential of the simplicial complex N P X . In particular, as the body of N P X is homeomorphic to S 1 , we have Proof. Assuming that δ 4 vanish, the term H 4 M P X B , C in the spectral sequence could not be annihilated at any further page by any other term, so we would get H 4 M P X B , C = 0. This, however, would contradict that M P X B is an oriented, non-compact 4-manifold. It is also easy to derive the result directly using the explicit description of δ 4 as wedge product by the Thom class Φ L of the tubular neighbourhood N L of L in M P X B , as in Lemma 8 below. Indeed, the image of the class of a generator [ω L ] of H 2 (L, C) is then represented in N L by the compactly supported 4-form The normal bundle of L in M P X B is orientable, and the above 4-form is cohomologous to a positive multiple of a volume form of N L . This implies the assertion.
The Lemma and (20) now imply that in the top row of W E 2 the only nonvanishing term will be the upper-left entry, and it is of dimension 1. be the inclusion map. Notice that as Φ L is vertically of compact support, its class can be extended by 0 to define a class The restriction of δ 2 to the component H 0 (L, C) then maps Now, according to Proposition 6.24 [8], we have where P D V stands for Poincaré duality in V and [L] is the cohomology class defined by integration on L. Therefore, for any we have As Poincaré duality is perfect, the assertion is equivalent to showing that the classes [L] for L ∈ N P X 0 are linearly independent in H 2 M P X B , C . For this purpose, we fix a generic line ℓ in the projective plane x 0 = 0, and let CP 2 t , t ∈ CP 1 denote the pencil of projective planes in CP 3 passing through ℓ. We may assume that t = ∞ corresponds to the plane x 0 = 0. For each t ∈ CP 1 , the curve is an elliptic curve. The line ℓ intersects for each k ∈ {1, 2, 3} the line L k in a single point p k , which is (by genericity of ℓ) different from all the intersection points p ij . The elliptic pencil (21) with center B; E is then an elliptic surface over CP 1 , in particular it is diffeomorphic to (5). The exceptional divisors ω −1 (p k ) are sections of Y , in particular they do not belong to the fiber E ∞ over ∞. Let us denote byL the proper transform of L with respect to ω. We may then write for some m L,k ∈ {0, 1}. The quotient We have already determined the type of E ∞ in (19), in particular its intersection form is negative semi-definite, with non-trivial radical. The only possible vanishing linear combination of these classes would then be one in the radical of (19), generated by (1, . . . , 1). However, one sees immediately that the intersection number of and any [ω −1 (p k )] is equal to 1, hence (23) is a non-zero class. Alternatively, the same argument as at the end of the proof of Lemma 3 shows that the class (23) is non-zero: the hyperplane class [H] must intersect it positively because the orthogonal complement of [H] in H 2 (E, C) is negative definite while the lattice of (19) has non-trivial radical.
Taking into account that the sequence degenerates at W E 2 and that the weight with respect to the filtration W [k] defining the mixed Hodge structure on H * (M P X B , C) is defined by from Lemma 7 we derive that the only non-vanishing graded pieces of the weight filtration on cohomology read as: Recalling (19) that N P X is a cycle of length N P X + 3, Lemmas 6 and 8 finish the proof.

Proofs of the theorems
Proof of Theorem 1 in the untwisted cases. As explained in the Introduction, for each X ∈ {II, III(D6), IV, V, V I} the Dolbeault and Betti spaces are known to be diffeomorphic, in particular we have . Propositions 1 and 2 imply that the graded pieces for the respective filtrations P and W have only two non-trivial terms: p ∈ {−2, −3} for P and −k − n ∈ {−2, −4} for W , and that the graded pieces for the −3 and −4 weights are of the same dimension. Then necessarily the graded pieces for the only remaining weights must be of the same dimension too, and we get the assertion.
Let [C] be a generator of Gr W −4 H 2 (M P X B , C). We need to show that the diffeomorphism M P X B → M P X Dol , maps [C] to a generator of Gr P −3 H 2 (M P X Dol , Q). For this purpose, we describe a representative of [C]. Throughout this proof, we work dually with homology classes. We use the notations of Proposition 2 for the dual nerve complex of the compactifying divisor of M P X B . Let us furthermore introduce the notation N P X 1 = {p 1 , . . . , p N P X +3 } for the intersection points of the divisor at infinity. Then, for each 1 ≤ j ≤ N P X +3 we introduce a cycle C j such that The cycle C j is defined as follows: let z 1 , z 2 be local coordinates defining the two divisors crossing at p j , then set for some sufficiently small 0 < ε << 1; topologically, C j (ε) is a 2-torus. We choose ε sufficiently small so that C j (ε) ∩ C j ′ (ε) = ∅ unless j = j ′ . In homology the classes C j (ε) are clearly independent of the choice of ε, hence we may omit to include ε in their notation. It follows that given any compact subset K ⊂ M P X B , the generator [C] can be represented in To match the graded pieces of degree w = −4 and p = −3 of the two filtrations W and P respectively, it is therefore sufficient to show that if a class [C] ∈ H 2 (M P X Dol , Q) can be represented in the complement of any prescribed compact K ⊂ M P X Dol , then [C] is a multiple of the class of the generic Hitchin fiber . Indeed, the class [HF ] is non-zero because it intersects any section of the Hitchin fibration non-trivially, and by (3) it is a generator of Gr P −3 H 2 (M P X Dol , Q) (see also Lemma 5). Let us denote by N P X Dol a closed tubular neighbourhood of F P X ∞ ⊂ E(1); N P X Dol is a plumbed 4-manifold over F P X ∞ . Let us now pick a compact K ⊂ M P X Dol so that M P X Dol \ K ⊂ N P X Dol , and assume that C is a representative of a class Then in particular C represents a non-zero class in the punctured tubular neighbourhood: The same also holds for [HF ]: a Hitchin fiber over a point sufficiently far away from 0 ∈ Y lies in the complement of K, hence gives rise to a class The open smooth 4-manifold M P X Dol ∩N P X Dol has as deformation retract the unit circle bundle ∂N P X Dol of N P X Dol , which is a smooth oriented 3-manifold. By Poincaré-duality, we have H 2 (M P X Dol ∩ N P X Dol , R) ∼ = H 1 (M P X Dol ∩ N P X Dol , R). As the dual graph of every F P X ∞ is a tree, one readily sees that the fundamental group of M P X Dol ∩ N P X Dol is generated by a simple loop ∂D around 0 in a punctured disc fiber D × . Therefore, we have for some q ∈ Q × . This finishes the proof.
Proof of Theorem 1, general case. We merely need to compute N P X and compare the perverse and weight polynomials explicitly for each X. As a consistency check and for sake of completeness, we treat the untwisted cases too. We will determine N P X using the explicit form of the quadratic terms Q P X provided in [25]. Before turning to the study of the various cases, let us address a result that will be needed in some of the cases. Namely, assume that M P X B has a singularity at [0 : 0 : 0 : 1]. Plugging x 3 = 1 into F P X we get with f i homogeneous of order i. If f 2 is a non-degenerate quadratic form, then the Hessian of F P X at the singular point is non-degenerate, and so the singularity is of type A 1 . Up to exchanging x 0 and x 2 , we have the following. The study of the specific cases, based on Lemma 9, is contained in Subsections 3.1-3.9 below. We note that in Subsections 3.1, 3.2, 3.4, 3.5 and 3.8 we rederive the weight polynomials obtained in Section 6 of [15] using different methods.
Proof of Theorem 2. The statement that the nerve complex of the boundary is homotopic to S 1 immediately follows from (19). Let us denote the components of the compactifying divisor D P X of M P X of these components. We may assume that for any j and for any pairwise distinct j, j ′ , j ′′ we have Set so that T P X

B
is an open tubular neighbourhood of D P X in M P X B ; T P X B is a plumbed 4-manifold. A simple computation shows that the group π 1 (M P X B ∩ T P X B ) ∼ = Z 2 is generated by two classes: one class [α] coming from the fundamental group of D P X and the class of the boundary of a fiber ∂D of T P X B . Now, let ρ 1 , . . . , ρ N P X +3 be a partition of unity subordinate to the cover (28), and consider the map The image of φ is contained in the standard simplex ∆ N P X +2 of dimension N P X +2 because the family ρ j forms a partition of unity. Moreover, it follows from (27) that Im(φ) ⊂ ∆ N P X +2 1 , the 1-skeleton of ∆ N P X +2 . Let us denote the vertices of ∆ N P X +2 listed in the standard order by P 0 , . . . , P N P X +2 and by P j P j ′ the edge connecting P j and P j ′ . Then, (26) implies that Im(φ) = P 0 P 1 ∪ P 1 P 2 ∪ · · · ∪ P N P X +2 P 0 , which is homotopy equivalent to S 1 . We claim that the loop α in T P X B ∩ M P X B coming from the fundamental group of D P X maps to a generator of the fundamental group of S 1 under φ. To be specific, for any j let us consider a pointp j close to p j ∈ N P X 1 , so thatp j ∈ ∂T j ∩ ∂T j+1 , where j + 1 is understood modulo N P X + 3 and p j ∈ T j ∩ T j+1 . Let α j be any path in ∂T j+1 connectingp j top j+1 . Then φ maps the path α = α 1 · · · α N P X +3 to a generator of the fundamental group of S 1 . Now, it is easy to see that the intersection number of α with the class (24) in the smooth oriented 3-manifold ∂T P X B (given the suitable orientation) is Indeed, for each j the curve α j intersects C j (ε) and C j+1 (ε) positively. Moreover, the intersection number of [C] with the boundary ∂D of a fiber of T P X B is easily seen to vanish, so that [α] is proportional to the Poincaré dual of [C] in ∂T P X B : for some r ∈ Q × . On the other hand, a simple loop β around ∞ ∈ Y in the Hitchin base can be lifted to a pathβ : [0, 1] → M P X Dol . We may choose any path γ connectingβ(1) toβ(0) in h −1 (β(0)). Then,βγ is a loop in M P X Dol ∩ N P X Dol such that in ∂N P X Dol we have [βγ] ∩ [HF ] = 1.
By virtue of (25) it follows from this and Poincaré duality that 3.1. Case X = V I. In this case the quadric is of the form This is the generic quadric, so it is smooth at infinity, and W H P V I (q, t) = 1 + 4q −1 t 2 + q −2 t 2 .
3.2. Case X = V . In this case the quadric is of the form 0 . An easy computation gives that the only singular point of M P V B over x 0 = 0 is [0 : 0 : 0 : 1]. We consider the affine chart x 3 = 0 and normalize x 3 = 1. Then, we have f 2 = x 1 x 2 − s 3 x 2 0 , which is a non-degenerate quadratic form because s 3 = 0. We infer that this singular point is of type A 1 , in particular its Milnor number is 1, hence W H P V (q, t) = 1 + 3q −1 t 2 + q −2 t 2 .
3.3. Case X = V deg . In this case the quadric is of the form The same analysis as in Subsection 3.2 shows that [0 : 0 : 0 : 1] is the only singular point. This time, however, we have which is degenerate. On the other hand, we have , in particular f 3 (1, 0, 0) = 1. Lemma 9 shows that the singularity is of type 3.4. Case X = IV . In this case the quadric is of the form In the first point, the second-order homogeneous term of F P IV in affine co-ordinates (x 0 , x 1 , x 3 ) is given by x 1 x 3 − s 2 2 x 2 0 , which is non-degenerate because s 2 = 0, so this singular point is of type A 1 . In the second point, the second-order homogeneous term of F P IV in affine co-ordinates (x 0 , x 1 , x 2 ) is given by f 2 = x 1 x 2 − s 2 2 x 2 0 , which shows that this singular point is again of type A 1 . We infer that M P IV B has two singular points, each of Milnor number 1, and W H P IV (q, t) = 1 + 2q −1 t 2 + q −2 t 2 . This time we have f 3 = x 0 x 2 1 + x 0 x 2 2 + (1 + αβ)x 2 0 x 1 (α + β)x 2 0 x 2 + αβx 3 0 . Now, we again see that f 3 (1, 0, 0) = αβ = 0, so Lemma 9 implies that we have an A 2 -singularity, thus W H P III(D6) (q, t) = 1 + 2q −1 t 2 + q −2 t 2 .
3.6. Case X = III(D7). This case is obtained from degeneration of Subsection 3.5 by setting the parameter β of Q P III(D6) (corresponding to the eigenvalue of the formal monodromy at one of the the irregular singular points) equal to 0. In this case (up to exchanging the variables x 1 , x 2 ) the quadric is of the form Q P III(D7) = x 2 1 + x 2 2 + αx 1 + x 2 with α ∈ C × . The only singular point of M P III(D7) B is [0 : 0 : 0 : 1], with homogeneous terms . This time f 2 is degenerate and we have f 3 (1, 0, 0) = 0, so the singularity is neither of type A 1 nor of type A 2 . Plugging x 1 = 0 in f 3 gives f 3 (x 0 , 0, x 2 ) = x 0 x 2 2 + x 2 0 x 2 . As this form has non-trivial linear term in x 2 at x 0 = 1, we get that k 1 = 1.
3.7. Case X = III(D8). This case is obtained from further degeneration of Subsection 3.6 by setting the parameter α of Q P III(D7) (corresponding to the eigenvalue of the formal monodromy at the only remaining unramified irregular singularity) equal to 0 too. We find the quadric 1 Q P III(D8) = x 2 1 + x 2 2 + x 2 .
1 Notice that this differs from the result x 1 x 2 x 3 +x 2 1 −x 2 2 −1 obtained in [25, 3.6]. We are grateful to Masa-Hiko Saito for pointing out that in this case the monodromy data has the extra symmetry x i → −x i for i ∈ {1, 2}. Indeed, the two-fold Weyl group S 2 × S 2 acts on the monodromy data by passing to opposite Borel subgroups at the two irregular singular points, and only the diagonal S 2 leaves invariant the constraints on the parameters. Now, introducing the invariant co-ordinates y 1 = x 2 1 , y 2 = x 2 2 , y 3 = x 1 x 2 and eliminating y 2 we are led to the formula y 1 y 3 x 3 + y 2 1 − y 2 3 − y 1 , which in turn transforms into (29) after some obvious changes of co-ordinates.
3.8. Case X = II. In this case the quadric is of the form Q P II = −x 1 − αx 2 − x 3 + α + 1 with α ∈ C × . We have As in the corresponding affine co-ordinates the degree two terms are respectively given by 0 , x 1 x 3 − αx 2 0 , x 1 x 2 − x 2 0 , and α = 0, we see that all these points are of type A 1 . As a conclusion, we get W H P II (q, t) = 1 + q −1 t 2 + q −2 t 2 .
3.9. Case X = I. In this case the quadric is of the form Q P I = x 1 + x 2 + 1.
There are three singular points of M At the first two of these, the degree two terms in the corresponding affine co-ordinates respectively read as 0 , x 1 x 3 + x 2 0 , so these singularities are of type A 1 . At [0 : 0 : 0 : 1] however, we have 0 . As f 3 (1, 0, 0) = 1 = 0, by virtue of Lemma 9 this singularity is of type A 2 . In total we have three singular points, with Milnor numbers 1, 1, 2 respectively, therefore W H P I (q, t) = 1 + q −2 t 2 .