COHOMOLOGICAL χ -INDEPENDENCE FOR HIGGS BUNDLES AND GOPAKUMAR–VAFA INVARIANTS

. The aim of this paper is two-fold: Firstly, we prove Toda’s χ -independence conjecture for Gopakumar–Vafa invariants of arbitrary local curves. Secondly, following Davison’s work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank r and Euler characteristic χ which are not necessary coprime

variety, i.e., the quotient variety by the conjugate GL r action.Assume for a while that r and m are coprime so that the moduli spaces in (1.1) are smooth.The homeomorphism (1.1) induces an isomorphism (1.2) between the singular cohomology groups.However, since (1.1) is only a diffeomorphism, the isomorphism (1.2) is not an isomorphism of mixed Hodge structures.Indeed, the mixed Hodge structure on H * (M Dol (r, m)) is pure, while that on H * (M B (r, m)) is not pure.Instead, the cohomology group H * (M Dol (r, m)) has the so-called perverse filtration induced by the Hitchin morphism h : M Dol (r, m) → B.
For character varieties, M B (r, m) and M B (r, m ′ ) are Galois conjugate to each other, for all m, m ′ ∈ Z with gcd(r, m) = gcd(r, m ′ ) = 1.In particular, we have an isomorphism of mixed Hodge structures.According to the P=W conjecture, the perverse filtration on H * (M Dol (r, m)) should be independent of m ∈ Z, as long as we have gcd(r, m) = 1.We prove this statement using the cohomological Donaldson-Thomas theory.
Theorem 1.1 (Example 5.18).Let r, m, m ′ be integers such that r > 0 and gcd(r, m) = gcd(r, m ′ ) = 1 hold.Then there exists an isomorphism preserving the Hodge structure and the perverse filtration.
This kind of statement is called a χ-independence phenomenon, as an invariant of the moduli space of Higgs bundles depends only on the rank r and independent of the choice of the Euler characteristic χ.Note that the above result for the perverse filtration was obtained by [dCMSZ] independently, via a completely different method.Now assume that (r, m) is not coprime.In this case, the moduli spaces M Dol (r, m) and M B (r, m) are singular.Hence it is not clear which cohomology theory is a right choice to obtain a P=W type statement.There are two candidates for this: (1) Intersection cohomology ( [dCM, FM, Mau, FSY]).
One advantage of using the intersection cohomology is that it is mathematically defined whereas the BPS cohomology is defined in the physical language.Instead of this, BPS cohomology has its own advantage: whereas the χ-independence phenomena for the intersection cohomology is only expected when we have gcd(r, m) = gcd(r, m ′ ), the χ-independence for the BPS cohomology is expected to hold without any assumption.Further, the BPS cohomology groups in both sides are expected to carry a Lie algebra structure (see [Davb]) and the non-abelian Hodge correspondence (1.1) is expected to induce an isomorphism of these Lie algebras [SS20, Conjecture 1.5].This suggests that we would have a representation theoretic approach to the original P=W conjecture.Following Davison's idea [Davc], we propose a definition of the BPS cohomology for the Dolbeault moduli space H * BPS (M Dol (r, m)) as a cohomology of a pure Hodge module BPS r,m on M Dol (r, m) defined using the cohomological Donaldson-Thomas theory (or refined BPS state counting) for Tot C (O C ⊕ ω C ).We have a split injection IC M Dol (r,m) ֒→ BPS r,m which is an isomorphism when gcd(r, m) = 1, but not necessarily so for general (r, m).We prove the following χ-independence for the BPS cohomology, which is a non-coprime generalization of Theorem 1.1: Theorem 1.2 (Corollary 5.15).Let r, m, m ′ be integers such that r > 0. Then there exists an isomorphism H * BPS (M Dol (r, m)) ∼ = H * BPS (M Dol (r, m ′ )) preserving the Hodge structure and the perverse filtration.
Remark 1.3.When we have gcd(r, m) = gcd(r, m ′ ), the Betti moduli spaces M B (r, m) and M B (r, m ′ ) are Galois conjugate.Therefore we expect that there exists an isomorphism (1.4) H * BPS (M B (r, m)) ∼ = H * BPS (M B (r, m ′ )) preserving the mixed Hodge structure, though we do not discuss the definition of the BPS cohomology for the Betti moduli spaces in this paper.Therefore Theorem 1.2 gives an evidence of the P=W conjecture for the BPS cohomology.Conversely, P=W conjecture and Theorem 1.2 suggest that the isomorphism (1.4) holds without the assumption gcd(r, m) = gcd(r, m ′ ), which is of independent interest.Remark 1.4.Recently, Davison-Hennecart-Schlegel-Mejia [DHM22] established a theorem relating the BPS cohomology and the intersection cohomology for the moduli space of Higgs bundles and for the character varieties.Their work imply the equivalence of two versions of the P=W conjectures via the BPS cohomology and via the intersection cohomology, and that the χindependence of the intersection cohomology of the Dolbeault moduli space follows from Theorem 1.2 as long as gcd(r, m) = gcd(r, m ′ ) holds.
We also establish the cohomological integrality theorem for Higgs bundles, which claims the decomposition of the Borel-Moore homology of the moduli stack of Higgs bundles H BM * (M Dol (r, m)) into tensor products of the BPS cohomology (see Theorem 5.16 for the precise statement).A similar statement was proved for quivers with potentials in [DM20, Theorem A] and for preprojective algebras in [Davc, Theorem D].As explained in [DM, §6.7], a plethystic computation and the cohomological integrality theorem imply that the Euler characteristic of the BPS cohomology is equal to the genus zero BPS invariant introduced by Joyce-Song [JS12, §6.2].In particular, cohomological integrality theorem strengthens the integrality conjecture for the genus zero BPS invariants [JS12, Conjecture 6.12].
Combining the cohomological integrality theorem and the χ-independence theorem (Theorem 1.2), we obtain the following χ-independence result for the Borel-Moore homology of the moduli stack M Dol (r, m) of Higgs bundles: 1.1.2.Gopakumar-Vafa (BPS) invariants.More generally, we investigate the χ-independence phenomena for curve counting theory on a class of Calabi-Yau (CY) 3-folds called local curves.By definition, a local curve is a CY 3-fold of the form Tot C (N ), where C is a smooth projective curve and N is a rank 2 vector bundle on C such that det(N ) ∼ = ω C .To explain our result, we recall some basic background of curve counting theory for CY 3-folds.
There are several ways to count curves in a CY 3-fold X, and one of them is the Gromov-Witten (GW) theory: For an integer g ≥ 0 and a homology class β ∈ H 2 (X, Z), denote by M g,β (X) the moduli space of stable maps f : C → X with C nodal curves of arithmetic genus g and f * [C] = β.Then the GW invariant is defined as where [M g,β (X)] vir denotes the virtual fundamental cycle.Due to the existence of stacky points in the moduli space M g,β (X), the GW invariant GW g,β is in general a rational number.
Based on string theory, Gopakumar-Vafa [GV] conjectured the existence of integer valued invariants n g,β ∈ Z for g ≥ 0 and β ∈ H 2 (X, Z), satisfying the equation We call the invariants n g,β the Gopakumar-Vafa (GV) invariants (also known as the BPS invariants).
Building on the previous works by Hosono-Saito-Takahashi [HST01] and Kiem-Li [KL], Maulik-Toda [MT18] and Toda [Todb] proposed the mathematical definition of the GV invariants.Following the original idea of Gopakumar-Vafa, they consider the moduli space M X (β, m) of slope semistable one-dimensional sheaves E on X satisfying which sends a sheaf to its fundamental cycle.Maulik-Toda [MT18] and Toda [Todb] defined the generalized GV invariants by the formula , where ϕ M X (β,m) is a certain perverse sheaf on M X (β, m), see Sections 2.2 and 2.3 for more detail.
As the GV invariants are conjecturally equivalent to the GW invariants by the formula (1.5), the GV invariants should be independent of the additional choice of the Euler characteristic m ∈ Z: ).The generalized GV invariants are independent of the choice of m ∈ Z, i.e., we have We call the above conjecture as χ-independence conjecture for GV invariants.In this paper, we prove it for local curves in full generality: Theorem 1.7 (Theorem 3.1).Conjecture 1.6 holds for X = Tot C (N ).

Results on local curves.
The key ingredient in our arguments is the main result of the companion paper by the first author and Masuda [KM21] on the construction of a global d-critical chart for the moduli space M X := M X (β, m) of one-dimensional semistable sheaves on a local curve X = Tot C (N ), i.e., the description of the moduli space M X as the critical locus inside a certain smooth space: Take an exact sequence where L 1 , L 2 are line bundles with deg(L 2 ) sufficiently large.We denote by Y := Tot C (L 2 ).Then it is shown in [KM21, Theorem 5.6] that there exists a function f : M Y → A 1 on the good moduli space of one-dimensional semistable sheaves on Y such that we have an isomorphism where p Y : M Y → M Y is the natural map from the moduli stack of onedimensional semistable sheaves to its good moduli space.In this situation, the perverse sheaf appeared in the definition (1.6) of the generalized GV invariants coincides with the vanishing cycle sheaf: and the proof of Theorem 1.7 is reduced to proving the corresponding statement for the intersection complex IC M Y .The latter is proved in the recent paper by Maulik-Shen [MSa], hence we obtain Theorem 1.7.
1.2.2.Results on Higgs bundles.We define the BPS sheaf BPS r,m on the moduli space M Dol (r, m) using the vanishing cycle complex ϕ M X for X = Tot C (O C ⊕ ω C ).Then the argument as above also implies Theorem 1.2.The cohomological integrality theorem for Higgs bundles (Theorem 5.16) is obtained by extending the argument for quivers with potentials [DM20, Theorem A] using the global critical locus description of M X and applying the first author's dimensional reduction theorem [Kin,Theorem 4.14] which relates the vanishing cycle cohomology for M X and the Borel-Moore homology for M Dol .
(1) Mellit [Mel20], Groechenig-Wyss-Ziegler [GWZ20], and Yu [Yu] proved that the E-polynomial of M Dol (r, m) is independent of m ∈ Z when gcd(r, m) = 1.These results were proved via the reduction to the positive characteristics.We extended the result to the non-coprime case and further lifted the equality to an isomorphism of Hodge structures via the completely different methods.
(2) Recently, de Cataldo-Maulik-Shen-Zhang [dCMSZ] used a positive characteristic method to prove that the isomorphism (1.3) preserves the perverse filtration induced by the non-abelian Hodge theorem.At present, we do not know whether our cohomological χ-independence results are compatible with the Galois conjugate.
(3) Toda [Todb] proved Conjecture 1.6 for primitive classes β ∈ H 2 (X, Z) (assuming a technical conjecture on orientation data).For nonprimitive classes, Maulik-Shen [MSa] proved it for local toric del Pezzo surfaces and recently [Yua] removed the toric assumption from their result.Maulik-Shen [MSa] also proved the conjecture for local curves of the form Our Theorem 1.7 proves Conjecture 1.6 for arbitrary local curves.In particular, the result for X = Tot C (N ) with indecomposable N is completely new.1.4.Structure of the paper.The paper is organized as follows.In Section 2, we recall Joyce's theory on d-critical structures.Then we recall the definition of the GV invariants, and introduce the notion of local curves and twisted Higgs bundles.
In Section 3, we prove Theorem 1.7.In Section 4, we prove the cohomological integrality theorem for D-twisted Higgs bundles where deg(D) > 2g − 2, which plays an important role in the proofs of Theorems 1.7 and Corollary 1.5.Finally in Section 5, we discuss applications to Higgs bundles.We prove Theorem 1.2 and the cohomological integrality theorem for Higgs bundles (Theorem 5.16).
In Appendix A, we give a brief overview of the shifted symplectic geometry and prove some technical lemmas that we use in this paper.
In Appendix B, we prove a version of the support lemma of the vanishing cycle complexes which is needed to define the BPS sheaf.
Acknowledgement.The authors would like to thank Professors Ben Davison, Yukinobu Toda and Junliang Shen for fruitful discussions and for carefully reading the previous version of this article.The first author would like to thank Naruki Masuda for the collaboration on the companion paper [KM21].The second author would like to thank Professors Arend Bayer and Jim Bryan for related discussions.
T.K. was supported by WINGS-FMSP program at the Graduate School of Mathematical Science, the University of Tokyo and JSPS KAKENHI Grant number JP21J21118.N.K. was supported by ERC Consolidator grant Wall-CrossAG, no.819864.

Notation and Convention.
In this paper, we work over the complex number field C. We use the following notations: • We let S denote the ∞-category of spaces (see [Lur09, Definition 1.2.16.1]).• We basically write stacks in Fraktur (e.g.X, Y, . ..), and derived schemes, derived stacks and morphisms between derived stacks in bold (e.g.X, X, f , . ..).We will write X = t 0 (X), X = t 0 (X), f = t 0 (f ) and so on.• A derived Artin stack X is said to be quasi-smooth if the cotangent complex L X has Tor-amplitude [−1, 1].• All derived/underived Artin stacks are assumed to have quasi-compact and separated diagonals and locally finitely presented over the complex number field.As the fiber product of finite type separated schemes over such a stack is again of finite type and separated, we can consider the category of mixed Hodge modules on such stacks (see §4.1 for the detail).• For a separated complex analytic space X, we let D b c (X) denote the bounded derived category of complexes of sheaves in Q-vector spaces on X with constructible cohomology.
• For a complex analytic stack X, we let D b c (X) denote the bounded derived category of sheaves in Q-vector spaces on X lis-an with constructible cohomology.Here X lis-an denote the lisse-analytic topos of X (see [Sun17, §3.2.3]).
• If there is no confusion, we use expressions such as f * and f ! for the derived functors Rf * and Rf ! .

Preliminaries
In this section, we collect some basic notions that we use in this paper.Firstly we recall Joyce's theory of d-critical locus in §2.1.Then we review the construction of vanishing cycle complexes associated with d-critical stacks in §2.2.In §2.3 we review Maulik-Toda's construction [MT18] of Gopakumar-Vafa invariants based on vanishing cycle complexes.In §2.4 we collect some basic facts on local curves and recall Maulik-Shen's cohomological χ-independence theorem [MSa].
2.1.D-critical structures.In [Joy15], Joyce introduced the notion of dcritical structures which are classical models of derived critical loci.We now briefly recall it.
Let X be a complex analytic space.Joyce [Joy15, Theorem 2.1] introduced a sheaf S X on X with the following property: for an open subset R ⊂ X and an embedding i : R ֒→ U to a complex manifold U , there exists a short exact sequence where I R,U is the ideal sheaf of R in U .One can show that the natural map glues to define a morphism S X → O X .We define a subsheaf S 0 X ⊂ S X by the kernel of the map If X is the critical locus Crit(f ) of a holomorphic function f on a complex manifold U such that f | X red = 0, then f + I 2 X,U defines an element of S 0 X .
Definition 2.1.Let X be a complex analytic space.A section s ∈ Γ(X, S 0 X ) is called a d-critical structure if for each point x ∈ X, there exists an open neighborhood R ⊂ X, an embedding i : R ֒→ U into a complex manifold, and a holomorphic function f on U with the property A complex analytic space equipped with a d-critical structure is called a d-critical analytic space.
The sheaf S 0 X has the following functorial property: for a given morphism of complex analytic spaces q : X 1 → X 2 , there exist natural morphisms Now assume that q is smooth surjective and take a section s ∈ Γ(X, S 0 X ).Then it is shown in [Joy15, Proposition 2.8] that q ⋆ s is a d-critical structure if and only if s is a d-critical structure.Now let X be a complex analytic stack.Then it is shown in [Joy15, Corollary 2.52] that there exists a sheaf S 0 X on the lisse-analytic site of X with the following property: • For a smooth morphism t : T → X, there exists a natural isomorphism η between complex analytic spaces smooth over X, the natural map q −1 (S 0 X | T 2 ) → S 0 X | T 1 is identified with q ⋆ .For a smooth morphism t : T → X from a scheme and a section s ∈ Γ(X, S 0 X ), we write t ⋆ s := η t (s| T ) ∈ Γ(T, S 0 T ).
Definition 2.2.For a complex analytic stack X, a section s ∈ Γ(X, S 0 X ) is called a d-critical structure if for any smooth surjective morphism t : T → X, the element t ⋆ s is a d-critical structure on T .A d-critical stack is a complex analytic stack X equipped with a d-critical structure.
For a complex analytic stack X equipped with a d-critical structure s, Joyce [Joy15, §2.4,§2.8] defines a line bundle K vir X,s on X red called the virtual canonical bundle of (X, s).If there is no confusion, we simply write K vir X = K vir X,s .We now recall some of its basic properties.Firstly assume X is a complex analytic space and write X = X.Take a d-critical chart R = (R, U, f, i) of (X, s).Then there exists a natural isomorphism Let q : X 1 → X 2 be a smooth morphism and s 2 be a d-critical structure on X 2 .Write s 1 = q ⋆ s 2 .Then it is shown in [Joy15, Proposition 2.30] that there exists a natural isomorphism with the following property: if we are given d-critical charts the following diagram of line bundles on R red 1 commutes: Here the bottom horizontal arrow is defined by the natural isomorphism . Now we treat the stacky case.Let X be a complex analytic space and t : T → X be a smooth morphism from an analytic space.Then there exists a natural isomorphism between complex analytic spaces smooth over X, the following diagram commutes: (2.1) For a d-critical stack (X, s), an orientation is a choice of a line bundle L on X red and an isomorphism o : L ⊗2 ∼ = K vir X,s .For a smooth morphism t : T → X, we define an orientation If we are given a smooth morphism q : (t 1 : T 1 → X) → (t 2 : T 2 → X) between analytic spaces smooth over X, there exists a natural isomorphism

2.2.
Vanishing cycle complexes on d-critical stacks.In this subsection, we recall some basic properties of the vanishing cycle functors and the vanishing cycle complexes associated with oriented d-critical stacks.
Let U be a complex manifold and f be a holomorphic function on U .Write U 0 = f −1 (0).Then the vanishing cycle functor . is defined by the following formula Let q : V → U be a holomorphic map between complex manifolds.Write V 0 := (f • q) −1 (0) and we let q 0 : V 0 → U 0 be the restriction of q.By the definition of the vanishing cycle functor, we have the following base change morphisms The first morphism is an isomorphism if q is proper and the latter morphism is an isomorphism if q is smooth.These are direct consequences of the proper/smooth base change theorem.Now let U be a smooth complex analytic stack and f be a holomorphic function on U. Write U 0 := f −1 (0).For a perverse sheaf P ∈ Perv(X), we define the perverse sheaf as follows: Take a smooth surjective morphism q : U → U. We let pr i : U × U U → U denote the i-th projection and pr i,0 : (f . This isomorphism satisfies the cocycle condition, hence the shifted perverse sheaf ϕ f •q (q * P) descends to a perverse sheaf ϕ f (P) ∈ Perv(U 0 ).One can show that the construction does not depend on the choice of the smooth morphism q.Now we recall the vanishing cycle complex associated with an oriented d-critical stack constructed in [BBBBJ15, Theorem 4.8].
First we treat the non-stacky case.Let (X, s, o) be an oriented d-critical analytic space.Then it is shown in [BBD + 15, Theorem 6.9] that there is a natural perverse sheaf ϕ X,s,o ∈ Perv(X) called the vanishing cycle complex associated with (X, s, o).We sometimes omit s and o and write Example 2.3.Let U be a complex manifold and f : U → A 1 be a holomorphic function such that f | Crit(f ) red = 0. Write X = Crit(f ) and equip it with the canonical d-critical structure s and the canonical orientation Let q : X 1 → X 2 be a smooth morphism and equip X 2 with a d-critical structure s 2 and an orientation o 2 .Write s 1 = q ⋆ s 2 and o 1 = q ⋆ o 2 .Then there exists a natural isomorphism of perverse sheaves with the following property: If we are given d-critical charts where the right vertical arrow is defined using the natural isomorphisms with the following property: If we are given a smooth morphism t : T → X from a complex analytic space, there exists a natural isomorphism Furthermore, if we are given a smooth morphism q : (t 1 : T 1 → X) → (t 2 : T 2 → X) between schemes smooth over X, the following diagram commutes: (2.4) Let U be a smooth Artin stack and f : U → A 1 be a regular function on it.Then it is shown in Example A.7 that Crit(f ) carries a natural (−1)-shifted symplectic structure hence there exists a natural d-critical structure s on its classical truncation X := Crit(f ).We will see in Lemma A.10 that the d-critical analytic stack (X, s) admits a canonical orientation o : There exists a natural isomorphism of perverse sheaves: We postpone the proof to §A.2.
Remark 2.5.The argument in [BBBBJ15, Theorem 4.8] shows that the perverse sheaf ϕ X,s,o naturally extends to a mixed Hodge module ϕ mhm X,s,o and to a monodromic mixed Hodge module ϕ mmhm X,s,o .Proposition 2.4 extends to an isomorphism of monodromic mixed Hodge modules with the same proof.We refer the reader to §4.2 for a brief discussion on monodromic mixed Hodge modules.
Definition 2.6.Let E be a pure one-dimensional coherent sheaf with compact support on X.
(1) We define the H-slope to be where holds. For The stack M H (v) admits the good moduli space p : M H (v) → M H (v), and we have the Hilbert-Chow morphism sending a sheaf E to its fundamental one cycle.Here, Chow β (X) denotes the Chow variety of compactly supported effective one cycles with homology class β (see [Kol96] for the definition.Note that it is denoted as Chow ′ (X) in [Kol96]).We denote by π M the composition Recall from Example A.2 that the stack M H (v) is the classical truncation of a (−1)-shifted derived Artin stack.In particular, the stack M H (v) carries a natural d-critical structure and (A.2) implies that there exists a natural isomorphism denotes the projection and E denotes the universal sheaf on M H (v) × X.In order to define the well-defined notion of Gopakumar-Vafa invariants, Maulik-Toda [MT18] and Toda [Todb] proposed the following conjecture on the virtual canonical bundle of the stack . Suppose that Conjecture 2.7 holds.Then we can take an orientation of , which we call a Calabi-Yau (CY) orientation.As we have seen in §2.2, we have the associated perverse sheaf We then define the perverse sheaf on the good moduli space as where we denote by along the open embedding U ⊂ Chow β (X), respectively.Note that we denote by H i (−) the i-th perverse cohomology.
(1) By [Todb, Lemma 2.14], the Laurent polynomial (2.7) is independent of the choice of a CY orientation on M H (v)| U .
(2) The definition of the perverse sheaf in (2.6) is motivated from the notion of BPS sheaves for quivers with super-potentials introduced by Davison-Meinhardt [DM20].See [Todb, Section 2.8] for the detailed discussion.
At this moment, the above conjecture is known to hold in the following cases: • X = Tot S (ω S ), where S is a smooth projective surface, and γ is primitive [Todb].
, where S is a del Pezzo surface, and γ is arbitrary [MSa, Yua].
Note that we can drop the subscript H in the notation since for m = 1, the moduli space is independent of the choice of an ample divisor H. Furthermore, for m = 1, we know that the perverse sheaf φ M H (v) is Verdier self-dual.Hence there exist integers n g,γ ∈ Z for g ≥ 0 such that the equation holds.Following Maulik-Toda [MT18], we call the integers n g,γ as the GV invariants of X.

2.4.
Local curves and twisted Higgs bundles.In this section, we introduce a class of Calabi-Yau threefolds which we call local curves.Then we review the results on the twisted Higgs bundles due to Maulik-Shen [MSa].
2.4.1.Spectral correspondence for local curves.Let C be a smooth projective curve and N be a rank two vector bundle on C with det N ∼ = ω C .Then the total space X := Tot C (N ) of the bundle N gives an example of quasiprojective Calabi-Yau threefolds, which we call a local curve.Denote by p : X → C the projection.
In this section, we recall the spectral-type correspondence for coherent sheaves on local curves.See e.g.[Sim94] for the details.
Lemma 2.12.Giving a compactly supported pure one-dimensional coherent sheaf on X is equivalent to giving a pair (E, φ) of a locally free sheaf E on C and a morphism φ ∈ Hom(E, E ⊗ N ) satisfying φ ∧ φ = 0.
We call a pair (E, φ) in the above lemma as an N -Higgs bundle.We can define the slope semistability for N -Higgs bundles as in Definition 2.6: Definition 2.13.Let (E, φ) be an N -Higgs bundle.
Lemma 2.14.Take an ample divisor H on C. Let E be a pure onedimensional coherent sheaf on X and (E, φ) be the corresponding N -Higgs bundle.
Then the sheaf E is µ p * H -(semi)stable if and only if the N -Higgs bundle X (r, m) be the good moduli space of M ss X (r, m).By the above lemma, C-valued points of M ss X (r, m) correspond to µ-semistable N -Higgs bundles.
The moduli space M ss X (r, m) admits a Hitchin type morphism: Define a Hitchin base as and a Hitchin morphism as follows: Remark 2.15.We can construct a bijection between the sets of closed points of im(h X ) and im(π M ) by sending a point in im(h X ) to its spectral curve, where π M : M ss X (r, m) red → Chow r[C] (X) denotes the Hilbert-Chow morphism defined as in (2.5).Moreover, by the properness of the morphisms h X and π M , the spaces im(h X ) and im(π M ) are homeomorphic.
As a result, the GV invariants do not change if we replace the Hilbert-Chow morphism with the Hitchin morphism.Hence we use the Hitchin morphism for the GV theory of local curves in this paper.2.4.2.Twisted Higgs bundles.Let L be a line bundle on a smooth projective curve C. Denote by Y := Tot C (L) the total space of L.
An L-Higgs bundle is a pair (E, θ) consisting of a locally free sheaf E on C and a homomorphism θ ∈ Hom(E, E ⊗ L).For the canonical divisor L = K C , the notion of K C -Higgs bundles agrees with the usual notion of Higgs bundles.
As in Definition 2.13, we can define the notion of µ-semistability for L-Higgs bundles.We denote by M ss Y (r, m) the moduli stack of µ-semistable L-Higgs bundles (E, θ) with rk(E) = r, χ(E) = m, and M ss Y (r, m) its good moduli space.Similarly to (2.8), we have a Hitchin morphism (2.9) Given an element a ∈ B Y , we denote by C a ⊂ Y its spectral curve.Define an open dense subset U ⊂ B Y as and let g : C → U be the universal spectral curve.The following result plays a key role in this paper: Then we have an isomorphism where d denotes the genus of the fibers of g : C → U .
In particular, we have isomorphisms for all m, m ′ ∈ Z.

Cohomological χ-independence for local curves
Let C be a smooth projective curve of genus g and N be a rank two vector bundle on C with det N ∼ = ω C .We put X := Tot C (N ).The goal of this section is to prove the following theorem: (i) There exists a function f on M Y such that the projection from X to Y induces an equivalence of (−1)-shifted symplectic derived Artin stacks (ii) Let (E, φ) be an L 2 -Higgs bundle.Then we have an equality where is the class corresponding to the short exact sequence (3.1).
We now want to describe the moduli stack of semistable N -Higgs bundle as a global critical locus.We begin with the following easy lemma: Lemma 3.3.Let C be a smooth projective curve and N be a rank two vector bundle on C. Then we can take the short exact sequence (3.1) so that deg(L 2 ) > 2g(C) − 2 holds.More generally, we can take L 2 so that its degree is arbitrarily large.
Proof.Let O C (1) be an ample line bundle on C. Then there exists an integer l 0 > 0 such that for every integer l ≥ l 0 , the bundle N ∨ (l) is globally generated.Then a general element s ∈ Hom(N, O C (l)) ∼ = H 0 (C, N ∨ (l)) is surjective.Putting L 2 := O C (l) and L 1 := Ker(s), we get the desired exact sequence as in (3.1).
Lemma 3.4.Take integers r, m ∈ Z with r > 0. Then there exists an integer k(r) > 2g(C) − 2 depending only on r, such that, for any short exact sequence (3.1) with deg(L 2 ) ≥ k(r), the following statement holds: For every µ-semistable N -Higgs bundle (E, φ) ∈ M ss X (r, m), the L 2 -Higgs bundle Proof.Let (E, φ) ∈ M ss X (r, m) be a µ-semistable N -Higgs bundle.Suppose that the L 2 -Higgs bundle (E, s • φ) is not µ-semistable.We claim that there exists a saturated subsheaf F ⊂ E such that µ(F ) > µ(E) and Hom(F, E/F ⊗ L 1 ) = 0. Indeed, let F ⊂ E be the maximal destabilizing subsheaf of (E, s • φ).This means that we have µ(F ) > µ(E) and is zero.On the other hand, by the µ-semistability of (E, φ), we have φ(F ) F ⊗ N , i.e., the composition As a result, we obtain the following diagram hence we have Hom(F, E/F ⊗ L 1 ) = 0.By Lemma 3.5 below, we can replace an exact sequence (3.1) so that Hom(F, E/F ⊗ L 1 ) = 0 for all µ-semistable N -Higgs bundles (E, φ) ∈ M ss X (r, m) and all saturated subsheaves F ⊂ E with µ(F ) > µ(E).Hence the above argument shows that (E, s • φ) remains µ-semistable for such a choice of the exact sequence (3.1).
Lemma 3.5.Take integers r, m ∈ Z with r > 0. Then the following sets are bounded: HN(S) := {A : A is a Harder-Narasimhan factor of G ∈ S} .
Moreover, there exists an integer k ′ (r) ∈ Z, depending only on r, such that for all line bundles L 1 with deg(L 1 ) ≤ k ′ (r) and for all F, E/F ∈ S, we have the vanishing Hom(F, E/F ⊗ L 1 ) = 0.
Proof.The boundedness of the sets S, HN(S) follows from the boundedness of M ss X (r, m) and Grothendieck's boundedness lemma (cf.[HL97, Lemma 1.7.9]).
In particular, there exist integers a, b such that the inequalities µ min (F ) ≥ a and µ max (F/E) ≤ b hold for all F, E/F ∈ S. By setting k ′ (r, m) := a−b−1, we obtain the inequality for all line bundles L 1 with deg(L 1 ) ≤ k ′ (r, m) and all F, E/F ∈ S.
Then the claim follows from Theorem 3.2 (ii), the fact that the derived moduli stack M ss Y (r, m) is a smooth (classical) stack, and that the classical truncation of the derived critical locus of a function on a smooth stack coincides with the classical critical locus.
For the vanishing cycle sheaves on the good moduli spaces, we have the following result: Proposition 3.7.Let r, m, m ′ ∈ Z be integers with r > 0. Let g : B Y → A 1 be a function as in Proposition 3.6.Then we have an isomorphism , denote the Hitchin morphisms (2.9) on the good moduli spaces.
Proof.Since the Hitchin morphism h Y : M ss Y (r, m) → B Y is proper, the result follows from Theorem 2.16 together with the commutativity of the vanishing cycle functors and proper push forwards.
In the following subsections, we will show that the complexes in (3.5) compute the generalized GV invariants for the local curve X = Tot C (N ).

CY property for local curves.
In this subsection, we fix integers r, m ∈ Z with r > 0, and an exact sequence (3.1).We assume that the line bundle L 2 satisfies the following conditions: • L 2 is globally generated.
Recall that we denote as X := Tot C (N ) and Y := Tot C (L 2 ).By Proposition 3.6, the moduli stack M ss X (r, m) is written as the global critical locus inside M ss Y (r, m).
Proposition 3.8.The canonical bundle K M ss Y (r,m) of the stack M ss Y (r, m) is trivial, and hence so is the virtual canonical bundle K vir M ss X (r,m) of the stack M ss X (r, m).In particular, the stack M ss X (r, m) is CY at any point γ ∈ B X , i.e., Conjecture 2.7 holds for M ss X (r, m).
Proof.A similar argument can be found in [Todb, Theorem 7.1].Take a morphism T → M ss Y (r, m) from a scheme T .Let E ∈ Coh(Y × T ) be the corresponding family of µ-semistable one-dimensional sheaves on Y .We consider the following diagram: We need to construct an isomorphism which is functorial in T .We have the following exact sequence Applying the functor RHom π T (−, E) to the exact sequence (3.6), we obtain the exact triangle where we put F := p T * E. By taking the determinants, we get (3.7) On the other hand, we have an exact sequence where Z ∈ |L 2 | is a finite set of points.Applying the functor RHom q T (F, F⊠ (−)) and taking the determinants, we get (3.8) where we put F Z := F| Z×T .Combining the equations (3.7) and (3.8), we obtain the desired isomorphism where we denote by r T : Z × T → T the projection.For the third isomorphism, we put Z = {p 1 , . . ., p k } and F p i := F| {p i }×T .
The triviality of the virtual canonical bundle K vir M ss X (r,m) now follows from Lemma A.10.

3.3.
Proof of Theorem 3.1.In this subsection, we finish the proof of Theorem 3.1.
Consider the following commutative diagram: where the morphism b : B X → B Y is induced by the surjection Sym k (N ) → L ⊗k 2 for k = 1, . . ., r. Lemma 3.9.The morphism Proof.It is enough to show that the morphism in (3.10) is proper and affine.The composition is proper as so are h Y and ι.Furthermore, the morphism h X : M ss X (r, m) → im(h X ) is proper and surjective.Hence the morphism (3.10) is proper.
On the other hand, by the properness of the Hitchin morphism h X , the inclusion im(h X ) ֒→ B X is closed.As the morphism b : B X → B Y is just the projection of affine spaces, it is also affine.We conclude that the composition Recall from (2.9) and (2.10) that we denote by h Recall also that we have the function g : B Y → A 1 defined in Proposition 3.6.We equip M ss X (r, m) with the orientation defined by the global critical chart description in Proposition 3.6 and Lemma A.10.We define the vanishing cycle complex ϕ M ss X (r,m) ∈ Perv(M ss X (r, m)) using this orientation.We set (3.11) ϕ M ss X (r,m) := H 1 (p X * ϕ M ss X (r,m) ) ∈ Perv(M ss X (r, m)) We need the following proposition: Proposition 3.10.We have isomorphisms ).We postpone the proof of this proposition until the next section.
Proof of Theorem 3.1.Let us take integers r, m, m ′ ∈ Z with r > 0. Let k(r) ∈ Z be integers as in Lemma 3.4.We take an exact sequence (3.1) such that L 2 is globally generated and deg(L 2 ) ≥ k(r) holds.Then by Proposition 3.6, the moduli stacks M ss X (r, m), M ss X (r, m ′ ) are written as the global critical loci inside the stacks M ss Y (r, m), M ss Y (r, m ′ ), respectively.Recall from Proposition 3.8 that the canonical bundles K M ss Y (r,m) and K M ss Y (r,m ′ ) are trivial, hence the natural orientation data in Lemma A.10 is a CY orientation data.
Let ϕ M ss X (r,m) , ϕ M ss X (r,m ′ ) be the associated perverse sheaves on M ss X (r, m), M ss X (r, m ′ ), defined as in (3.11).We have an isomorphism (3.12) ) by Proposition 3.7.By using the commutative diagram (3.9), we can rewrite the left hand side of (3.12) as . By Lemma 3.9, the map (3.10) is finite.Since the push-forward along a finite morphism preserves the perverse t-structures, we obtain )) for all i ∈ Z, and we have the same isomorphisms if we replace the integer m with m ′ .
Combining the isomorphisms (3.12) and (3.13), and taking the Euler characteristics, we conclude that )) as desired.

Cohomological integrality theorem for twisted Higgs bundles
In this section, we prove the cohomological integrality theorem in the sense of [DM20, §1.3] for twisted Higgs bundles.Since semistable twisted Higgs bundles form a category of homological dimension one, we can prove the cohomological integrality theorem using the techniques of [DM20, Mei], which treat the case of quivers.4.1.Mixed Hodge modules on stacks.Here we give a quick introduction to mixed Hodge modules, which is a sheaf theoretic version of mixed Hodge structures introduced by Morihiko Saito [Sai90].An advantage of working with mixed Hodge modules (rather than perverse sheaves) is the fact that the category of pure Hodge modules is semi-simple.In particular, an equality of Grothendieck group of the category of pure Hodge modules implies an isomorphism between them.This was used by Davison-Meinhardt [DM20] in their proof of the cohomological integrality theorem for quivers with potentials, and will be used in the proof of Theorem 4.6.
Let X be a separated scheme locally of finite type over complex number whose connected components are quasi-compact.For such X, we can define the category of mixed Hodge modules MHM(X) and its bounded derived category D b (MHM(X)) which admits a six-functor formalism (see [Sai89] for an overview).There exists an exact functor which restricts to a faithful functor MHM(X) → Perv(X).The functor rat is compatible with all six functors.A mixed Hodge module M is equipped with an increasing filtration called the weight filtration which we denote by The category of mixed Hodge modules over a point is equivalent to the category of graded polarizable mixed Hodge structures.Let a X : X → Spec C be the constant map to a point.Then we define objects , where Q denotes the mixed Hodge structure of weight zero and dimension one.
The category of mixed Hodge modules forms a stack in the smooth topology (see [Ach, Theorem 2.3]).This motivates the following definition of the category of mixed Hodge modules on an Artin stack X. Definition 4.1.Let X be a complex Artin stack.We let Sch sm,sep /X denote the category of separated schemes smooth and of finite type over X.A mixed Hodge module on X is a pair consisting of an assignment and a choice of an isomorphism satisfying the associativity relation.Mixed Hodge modules on X form a category MHM(X) in the natural way.We have a natural forgetful functor rat : MHM(X) → Perv(X).
Take a smooth surjective morphism from a separated finite type scheme t : T → X and we let pr i : T × X T → T denote the i-th projection.Then we can identify MHM(X) with the category of pairs (M, σ), where M is a mixed Hodge module on T and σ is an isomorphism At present we do not have a full six-functor formalism for mixed Hodge modules on Artin stacks.However we have some part of it which is sufficient for applications in this paper.Firstly, if we are given a smooth morphism q : X 1 → X 2 between Artin stacks, we can define a functor in the standard way.
Now assume that we are given a finite type morphism p : X → X from an Artin stack to a separated finite type scheme.We want to define the functor compatible with the functor rat.Here H n denotes the n-th cohomology with respect to the perverse t-structure on D(MHM(X)).We assume that the morphism p satisfies the following assumption: in the bounded derived category of sheaves on X with constructible cohomology and integer N , there exists a smooth morphism from a scheme q N : T N → X such that the natural map Here H n denotes the perverse t-structure on D b c (X).This assumption is automatically satisfied when X is of the form [Y /G] for some scheme Y and a linear algebraic group G (see [Dava,§2.3.2]).Let p : X → X be a morphism to a scheme.For a mixed Hodge module M ∈ MHM(X) and n ∈ Z, we define a mixed Hodge module where N is a sufficiently large integer.We can show that H n (p * M ) is independent of the choice of N and q N .If we take p as the constant map a X : X → Spec C, we can construct a mixed Hodge structure H n (X, M ) := H n (a X * M ).Similarly, we can extend the perverse sheaves H n (p * Q X ) and H n (p * D X ) to mixed Hodge modules, and the vector spaces H n (X) and H BM n (X) to mixed Hodge structures.
For a complex of mixed Hodge modules M ∈ D(MHM(X)), we define Lemma 4.2.Let X be a stack satisfying the condition ( * ), p : X → X be a morphism to a separated finite type complex scheme, and h : X → B be a proper morphism between separated finite type complex schemes.Take M ∈ D b (MHM(X)) and assume that H(p * M ) is pure.Then we have an isomorphism Proof.Take an integer n and an integer N such that N > n + dim h −1 (x) holds for each x ∈ B. Then [Dim04, Corollary 5.2.14] implies that we have an isomorphism Take a smooth morphism q : T → X such that we have isomorphisms Then what it is enough to prove the following isomorphism Saito's decomposition theorem implies an isomorphism Then using [Dim04, Corollary 5.2.14] again, we obtain the desired isomorphism.

Monodromic mixed Hodge modules.
Here we recall some basic properties of monodromic mixed Hodge modules.We do not give the precise definition here and refer the reader to [DM20, §2] and [BBBBJ15, §2.9] for the detailed discussion.Let X be a separated scheme locally of finite type over complex number whose connected components are quasi-compact.Then we can define an abelian category MMHM(X) of monodromic mixed Hodge modules on X. Roughly speaking, a monodromic mixed Hodge module consists of its underlying mixed Hodge module M and a monodromy operator acting on it.We have a natural functor forgetting the monodromy operator and a fully faithful functor

MHM(X) ֒→ MMHM(X)
which associates a mixed Hodge module M to a monodromic mixed Hodge module whose underlying mixed Hodge module is M and the monodromy operator is trivial.As similar to the usual mixed Hodge modules, monodromic mixed Hodge modules are also equipped with weight filtrations.The bounded derived category D b (MMHM(X)) admits a six-functor formalism, similarly to the usual mixed Hodge modules.The inclusion functor D b (MHM(X)) → D b (MMHM(X)) is compatible with these six operations.The forgetful functor D b (MMHM(X)) → D b (MHM(X)) is compatible with four operations f * , f !, f * , f ! for a morphism f between separated finite type complex schemes.However, the tensor product of monodromic mixed Hodge modules is not compatible with the tensor product of the underlying mixed Hodge modules.
For a regular function f : X → A 1 , we can define the monodromic vanishing cycle functor for (possibly unbounded) mixed Hodge module complexes (4.1) which enhances the usual vanishing cycle functor by incorporating the monodromy operator acting on it.The essential difference between monodromic and the usual mixed Hodge modules are the following: • Thom-Sebastiani isomorphism holds for the monodromic vanishing cycle functors (4.1).See [DM20, Proposition 2.13] for the precise statement and other basic properties.• There exists an object L 1/2 ∈ D b (MMHM(pt)) with an isomorphism where we put , which is concentrated in cohomological degree two, and is pure of weight two.
When X is an irreducible variety, we define the object IC X ∈ MMHM(X) as follows: where IC X denotes the intermediate extension of Q Xreg on the regular locus X reg ⊂ X.We will also use the following object: As in the previous subsection, we can define the notion of monodromic mixed Hodge modules for an Artin stack.In particular, we can define the object IC X ∈ MMHM(X) for a smooth Artin stack X.Moreover, we can define the functor for a morphism p : X → X from an Artin stack X to a scheme X satisfying the condition (*) in §4.1.
Let X be a separated scheme locally of finite type over C whose connected components are quasi-compact.We say that a (possibly unbounded) complex M ∈ D(MMHM(X)) is locally finite if for each connected component Z ⊂ X, the following conditions hold: We let D ≥,lf (MMHM(X)) ⊂ D ≥ (MMHM(X)) denote the full subcategory consisting of locally finite monodromic mixed Hodge complexes.We can see that the Grothendieck group K 0 (D ≥,lf (MMHM(X))) is isomorphic to the completion of K 0 (MMHM(X)) with respect to ideals {I i } i∈Z where I i is generated by objects whose weight is greater than i.
Let (X, m) be a monoid scheme where X is a separated and locally of finite type over complex number whose connected components are quasicompact and m : X × X → X is a finite morphism.For objects M, N ∈ D ≥,lf (MMHM(X)), we define The functor ⊠ m defines a symmetric monoidal structure on the category D ≥,lf (MMHM(X)).Therefore for each n ∈ Z >0 , we can define the symmetric product functor is the GIT semistable locus inside a certain quot scheme Quot with respect to a certain GL n -linearization.For a given integer f > n, we put We have a G n -action on Quot which passes through the GL n -action.We define a G n -action on A as follows: M ss Y (r, m), where we put The diagram (4.3) satisfies the following properties (cf.[MSa, Proposition 3.6]): • the horizontal morphisms are open immersions, • U f and M f are smooth schemes and the morphism π f is projective, . By the following proposition, we can compute the cohomology objects ) ) using the push-forward along the proper morphism π f : Proposition 4.3 (cf.[DM20, Lemma 4.1, Proposition 4.3]).The following statements hold: (1) For each n ∈ Z, there exists f ≫ 0 such that Proof.We just give an outline of the proof.See [DM20, Lemma 4.1, Proposition 4.3] for the details.Using the fact ) in the sense of (*) in §4.1.Hence the first assertion holds.
The second assertion now follows from the natural isomorphism between functors, which holds since the morphism π f : The following statement will be used in §5.3.Proposition 4.4.Let Z ⊂ M ss Y (r, m) be the critical locus of F and Z ⊂ M ss Y (r, m) be its good moduli space.Given a morphism q : Z → W between schemes, we have an isomorphism Fix an integer n.We let Z f denote the critical locus of the function F •π f .Take a sufficiently large integer f such that the following isomorphism holds: We have the following isomorphisms where the second isomorphism follows from Saito's decomposition theorem.If f is sufficiently large, we also have an isomorphism )) so we obtain the claim.4.4.Cohomological integrality theorem for L-Higgs bundles.Here we prove the cohomological integrality theorem for Y := Tot C (L), where L is a line bundle on a smooth projective curve C with deg(L) > 2g(C)−2.Since the category of µ-semistable one-dimensional sheaves on Y is homological dimension one, this can be proved in the same manner as [DM20, Theorem A] by applying the main result of [Mei].However we give a sketch of the proof for reader's convenience.
For a given rational number µ ∈ Q, we set For each positive integer n ∈ Z >0 , we have the following morphism: [DM, Examples 2.14 and 2.16]).We define functors as follows: The following statements hold: (1) The functor Sym ⊠ ⊕ is exact.
(2) The functor Sym ⊠ ⊕ sends pure objects to pure objects. ( Then the functors ϕ mmhm We use the following notations: Recall that we denote by p : M ss Y (r, m) → M ss Y (r, m) the canonical morphism to the good moduli space.Recall also that the definition of the object H * (BC * ) vir from (4.2).
Theorem 4.6.We have the following isomorphisms in D ≥,lf (MMHM(M ss Y (µ))): for a regular function Proof.We first construct the isomorphism (4.4).By the exactness of the functor Sym ⊠ ⊕ (see Proposition 4.5 (1)), the right hand side is isomorphic to its total cohomology.Hence it is enough to prove the isomorphism for each cohomology.By Proposition 4.3 (1), for each n ∈ Z and (r, m) ∈ Z >0 × Z, there exists f ≫ 0 such that we have an isomorphism Since the morphism π f is proper and the object IC M f is pure, it follows that the object in (4.6) is a pure mixed Hodge module.
On the other hand, since IC M ss Y (r,m) is pure, Proposition 4.5 (2) implies that the n-th cohomology of the right hand side of (4.4) is also pure.Now n-th cohomology of both sides of (4.4) are direct sums of simple pure mixed Hodge modules.Hence it is enough to prove the equality in the Grothendieck group K 0 (D ≥,lf (M ss Y (µ))), which holds by the main theorem of [Mei].
The second isomorphism (4.5) follows by applying the vanishing cycle functor ϕ mmhm Fµ to both sides of the isomorphism (4.4) and then using Proposition 4.3 (2) and Proposition 4.5 (3).
We end this section by proving Proposition 3.10 in the previous section: Proof of Proposition 3.10.Fix integers r > 0 and m ∈ Z, and put µ := m/r.Let g : B Y → A 1 be the function defined in (3.3).Recall from Proposition 3.6 that we have ⊂ M ss Y (r, m) for a line bundle L 2 with deg(L 2 ) ≫ 0. Hence the first isomorphism in Proposition 3.10 follows from Proposition 2.4.
For the second isomorphism, let us put . By taking the first cohomology of the isomorphism (4.5), we obtain , we get the second isomorphism in Proposition 3.10.
Remark 4.7.It is clear from the proof that Proposition 3.10 naturally extends to an isomorphism of monodromic mixed Hodge modules.

The case of Higgs bundles
In this section, we prove the cohomological integrality theorem and the cohomological χ-independence theorem for Higgs bundle moduli spaces on curves using the dimensional reduction theorem due to the first author [Kin].
We let ϕ T * [−1]Y denote the perverse sheaf on Y recalled in §2.2 with respect to this (−1)-shifted symplectic structure and orientation.The following theorem is called the dimensional reduction theorem.
Theorem 5.1 ([Kin, Theorem 4.14]).There exists a natural isomorphism in Here vdim Y := rank L Y denotes the virtual dimension of Y.
We now discuss the generalization of this theorem to the level of complexes in mixed Hodge modules.Firstly we discuss the case when Y is a derived scheme.To specify that Y is schematic, we write Y = Y, Y = Y and Y = Y.The following lemma is useful: Lemma 5.2.Let X be an algebraic variety and take complexes of mixed Hodge module M, N ∈ D b (MHM(X)) such that there exists an isomorphism η : rat(M ) ∼ = rat(N ) in D b (X).Assume that for each i < 0 the group Ext i (rat(M ), rat(N )) vanishes and we have an isomorphism of mixed Hodge structures H 0 (X, Hom(M, N )) ∼ = Q.Then η naturally extends to an isomorphism M ∼ = N in D b (MHM(X)).
Proof.Consider the natural map of mixed Hodge complexes The assumption implies an isomorphism Q ∼ = τ ≤0 RHom(M, N ) hence we obtain a map in D b (MMHM(X)) by adjunction.Then the following composition of morphisms in D b (MHM(X)) upgrades the isomorphism η up to scalar.
Proposition 5.3.Assume that the virtual dimension vdim Y is even.Then the dimensional reduction isomorphism (5.2) for a quasi-smooth derived scheme Y naturally upgrades to an isomorphism in D b (MHM(Y )): Proof.Using Lemma 5.2, we only need to prove that the mixed Hodge structure on is pure of weight zero.As this statement can be checked locally, using [BBJ19, Theorem 4.1], we may assume that there exists a smooth scheme U which admits a global étale coordinate, a trivial vector bundle E on U , and a section s ∈ Γ(U, E) such that Y is isomorphic to the derived zero locus Z(s).In this case the proof of [Kin,Theorem 3.1] shows that the dimensional reduction isomorphism (5.2) can be identified with Davison's local dimensional reduction theorem [Dav17, Theorem A.1].As the proof of this theorem works verbatim for complexes of mixed Hodge modules, we conclude that the claim holds.
Remark 5.4.We expect that the isomorphism (5.3) further upgrades to an isomorphism D b (MMHM(Y )) However, we could not prove this since we do not know whether the tensorhom adjunction holds for monodromic mixed Hodge modules.Instead, we can easily see that we have an isomorphism in D b (MMHM(Y )) since the monodromy operator acts trivially on both sides (see [Davc,Remark 3.9]).It is enough for our purposes.Now we discuss the stacky case of this proposition.Let Y be a quasismooth derived Artin stack such that its classical truncation Y = t 0 (Y) is of the form [Y /G] for some scheme Y and a linear algebraic group G.In this case, we can upgrade the dimensional reduction theorem to an isomorphism of mixed Hodge structures.
For a fixed i, take a smooth morphism q : T → Y of relative dimension d such that q !: H BM −i+vdim Y (Y) → H BM −i+vdim Y+2d (T ) and the map are isomorphisms.Here q : T → Y is the base change of q.Therefore we need to show that the following composition of isomorphism of vector spaces upgrades to an isomorphism of mixed Hodge structures.To prove this, we will show that the following morphism in D b (T ) upgrades an isomorphism in D b (MHM(T )).Here π T : T → T and π Y : Y → Y are natural projections and the second isomorphism is the dimensional reduction isomorphism.Using Lemma 5.2, we need to show that the mixed Hodge structure of (5.4) is weight zero.To prove this, take a smooth surjective morphism from a derived scheme h : U → Y such that vdim U is even.Write h = t 0 (h), t 0 (U ) = U , and X := t 0 (T * [−1]U ).Let T × Y U be the fibre product the natural projections and q U : T × Y U → U and qU : T × Y U → U be the base changes of q.Then we can construct a natural isomorphism in the same manner as η q .As we have seen in Proposition 5.3, the map η q U upgrades to an isomorphism in where the latter isomorphism follows from [Kin, Proposition 4.10].We also have a natural isomorphism Under these identifications, the proof of [Kin, Theorem 4.14] implies that h * T η q is equal to η q U [− dim h] up to a certain choice of the sign.This and the fact that η q U upgrades to an isomorphism in D b (MHM(T × Y U )) imply that the weight of the mixed Hodge structure (5.4) is zero.
The following statement can be proved in the same manner as the previous proposition: Proposition 5.6.We keep the notation from the previous proposition.Let p : Y → Z be a morphism to a separated finite type complex scheme.Then we have an isomorphism of mixed Hodge modules Remark 5.7.The argument as in Remark 5.4 implies that the isomorphism (5.5) upgrades to an isomorphism in D b (MMHM(Z))

BPS cohomology for Higgs bundles.
In [Dav16], Davison defined BPS sheaves and BPS cohomology for preprojective algebras.In this section, we introduce Higgs counterpart of these notions.
Let C be a smooth projective curve of genus g.We write S = Tot C (ω C ) and X = Tot C (O C ⊕ ω C ). Recall that M ss X (r, m) (resp.M ss S (r, m)) denotes the moduli stack of one-dimensional semistable sheaves of rank r and Euler characteristic m on X (resp.S), and M ss X (r, m) (resp.M ss S (r, m)) denotes the good moduli space of M ss X (r, m) (resp.M ss S (r, m)).We have the following commutative diagram: [Kin,Theorem 5.1] that there exists a natural equivalence of (−1)-shifted symplectic derived Artin stacks M ss X (r, m) ∼ = T * [−1]M ss S (r, m), where M ss X (r, m) (resp.M ss S (r, m)) denotes the derived enhancement of M ss X (r, m) (resp.M ss S (r, m)).Therefore (5.1) implies that there exists a canonical orientation X (r,m) .On the other hand, we have seen in Proposition 3.6 that there exist a line bundle L with deg L > 2g − 2 and a function f on the moduli stack M ss Tot C (L) (r, m) of semistable sheaves on Tot C (L) such that there exists an equivalence of (−1)-shifted symplectic derived Artin stacks M ss X (r, m) ∼ = Crit(f ).Therefore there exists an orientation where the first map is the projection to the first factor and the latter map is induced from the composition O Therefore we may assume that γ and γ ′ are contained in S × {t} for some t ∈ A 1 .Then the claim follows since the map S × {t} ⊂ X → Y defines an injection on the set of cycles.
The following corollary is an immediate consequence of the isomorphism (3.12) and the above lemma.
Corollary 5.14.Let us take integers r, m, m ′ such that r > 0. Then there exists an isomorphism in D b (MMHM(B X )): X (r,m ′ ) .Corollary 5.15.Let r, m, m ′ be as in the previous corollary.Then there exists an isomorphism in D b (MMHM(B S )): We now prove the cohomological integrality theorem for Higgs bundles.Recall that we have the following diagram: π / / M ss S (µ).For a rational number µ, we write Theorem 5.16.The monodromic mixed Hodge module BPS µ is contained in MHM(M ss S (µ)), i.e., it has a trivial monodromy operator.Further, we have an isomorphism in D ≥,lf (MHM(M ss S (µ))).Proof.Proposition 4.4 and Proposition 5.10 imply isomorphisms As we have seen in Remark 5.7, the left-hand side is monodromy-free, hence so is the BPS sheaf.The isomorphism (5.6) follows from the above isomorphism and an isomorphism which is a consequence of Proposition 5.6 and the equality vdim M ss S (r, m) = 2r 2 (g − 1).
Proof.The above theorem implies that there exists an embedding BPS r,m ֒→ H(p S * DQ M ss S (r,m) ) ⊗ L r 2 (g−1) .The purity of the right-hand side is proved in [Dava,Proposition 7.20], so we obtain the claim.
Example 5.18.Assume that (r, m) is coprime, in which case M ss S (r, m) is smooth and p S is a C * -gerbe.In this case, we have an isomorphism ) and D ≥,lf (MHM(N)) respectively.Combining the above corollary and the χ-independence theorem for BPS cohomology (= Corollary 5.14), we obtain the following χ-independence theorem for the Borel-Moore homology: Corollary 5.20.Let r, m, m ′ be integers such that r > 0 and gcd(r, m) = gcd(r, m ′ ).Then there exist isomorphisms in D ≥,lf (MHM(B S )) and D ≥,lf (MHM(pt)) respectively.
Remark 5.21.Based on P = W conjecture, it is conjectured in [FSY] that there exists an isomorphism of intersection cohomology groups IH(M ss S (r, m)) ∼ = IH(M ss S (r, m ′ )) preserving the perverse filtration for r, m, m ′ such that gcd(r, m) = gcd(r, m ′ ).At present we do not know how to prove this conjecture.However, once Davison's conjecture [Davb,Conjecture 7.7] on the structure of the BPS sheaf is established, it would be possible to deduce the χ-independence for intersection cohomology from the χ-independence for BPS cohomology (= Corollary 5.14).5.4.An example: g = 2, r = 2.Here we give an example where the intersection cohomology and the BPS cohomology are different.Let C be a smooth projective curve of genus 2, and put S := Tot C (ω C ).We consider the moduli space M S (2, 0).By taking the cohomology of the inclusion (5.7), we have an inclusion (5.8) IH(M S (2, 0)) ֒→ BPS 2,0 .
For the formula of Φ BPS , we used the isomorphisms (5.9).Note that the term t −10 appears by our shift convention so that the intersection and the BPS complexes are perverse sheaves, together with the fact dim M S (2, 0) = 10.From the formulas (5.10), it is obvious that the inclusion (5.8) is not an isomorphism.
It is shown in [BBJ19, Theorem 5.18] that any (−1)-shifted symplectic derived scheme is Zariski locally of this form.Now we discuss the canonical (−1)-shifted symplectic structure on the derived critical locus of a function on a general derived Artin stack.To do this, we need to recall the notion of Lagrangian structures.Definition A.5.Let (X, ω X ) be an n-shifted symplectic derived Artin stack and τ : L → X be a morphism of derived Artin stacks.An isotropic structure is a path from 0 to τ ⋆ ω X in A 2,cl (L, n).An isotropic structure η is called a Lagrangian structure if it induces an equivalence Let X be an n-shifted symplectic derived Artin stack and τ 1 : L 1 → X and τ 2 : L 2 → X be Lagrangians.These Lagrangian structures define a loop in A 2,cl (L 1 × X L 2 , n) hence a point in A 2,cl (L 1 × X L 2 , n − 1).It is shown in [PTVV13, Theorem 2.9] that this (n − 1)-shifted closed 2-form is shifted symplectic.
Proof.Take a smooth surjective morphism q : U → U and write X = Crit(f • q).Consider the following composition where we set s X = (q| X ) ⋆ s.We claim that the isomorphism o q descends to an orientation for (X, s).To do this, take an étale surjective morphism η : V → U × U U from a scheme V .We let pr i : U × U U be the i-th projection for i = 1, 2. Write Y = Crit(f • q • pr 1 •η) and define o q•pr i •η : ((q s in the same manner as o q .It is enough to prove the commutativity of the following diagram for each i = 1, 2: (pr i •η) red, * (q * K ⊗2 U | X red ) (pr i •η) red, * oq / / (pr i •η) red, * (q| X red ) * K X,s .
This follows from the commutativity of the diagram (2.1).
Proof of Proposition 2.4.We keep the notation as in the proof of the previous lemma.Let o X and o Y be natural orientations on (X, q ⋆ s) and (Y, η ⋆ pr ⋆ 1 q ⋆ s) coming from the descriptions as global critical loci.The construction of the orientation o in the previous lemma implies that we have the following natural commutative diagram of orientations Now define an isomorphism by the following composition for i = 1, 2.Here the third isomorphism follows from Lemma A.8 and Example 2.3.Similarly we can define an isomorphism θ q•pr i •η : (q • pr i •η) * ϕ X,s,o ∼ = (q The commutativity of the diagrams (A.4), (2.4), and (2.3) implies an equality (pr i •η) * θ q = θ q•pr i •η Corollary 1.5 (Corollary 5.20).Let r, m, m ′ be integers such that r > 0 and gcd(r, m) = gcd(r, m ′ ) hold.Then there exists an isomorphism H BM * (M Dol (r, m)) ∼ = H BM * (M Dol (r, m ′ )) preserving the Hodge structure and the perverse filtration introduced in [Dava, Proposition 7.24].
Theorem 3.1.Let X = Tot C (N ) be a local curve.For every positive integer r ∈ Z >0 and a class γ ∈ B X , Conjecture 2.10 holds.3.1.Global d-critical charts for moduli spaces on local curves.We first recall the main result of the companion paper [KM21]:Theorem 3.2.[KM21, Theorem 5.6, Proposition 5.7] Let C be a smooth projective curve and take a short exact sequence(3.1)0 → L 1 → N → L 2 → 0of locally free sheaves on C where L 1 and L 2 are rank one.Suppose that there exists an isomorphism det(N ) ∼ = ω C , and the inequality deg(L 2 ) > 2g(C) − 2 holds.Write X = Tot C (N ) and Y = Tot C (L 2 ).Let M X and M Y be the derived moduli stack of compactly supported coherent sheaves on X and Y respectively.

4. 3 .
Approximation by proper morphisms.Let L be a line bundle on a smooth projective curve C with deg(L) > 2g(C) − 2, and put Y := Tot C (L).For given integers r, m ∈ Z with r > 0, we denote by p : M ss Y (r, m) → M ss Y (r, m) the morphism from the moduli stack to its good moduli space.We fix a regular function F : M ss Y (r, m) → A 1 and denote by F := F • p : M ss Y (r, m) → A 1 the composition.Let us recall the construction of moduli spaces of framed objects following [DM20, MSa, Mei].We follow the notations in [MSa, §3.3].By construction, we have M ss Y (r, m) = [Quot ss / GL n ], where Quot ss ⊂ Quot Fµ and Sym ⊠ ⊕ commute.Proof.The same proofs as in [DM20, Propositions 3.5, 3.8, 3.11] work by using the finiteness of the morphism ⊕ : (M ss Y (µ)) ×n → M ss Y (µ) and Thom-Sebastiani isomorphism for the vanishing cycle functors ϕ mmhm (−) .

Example A. 3 .
Let Y be a derived Artin stack andT * [n]Y := Spec Y (Sym(L ∨ Y [−n])) be its n-shifted cotangent stack.Let λ ∈ A 1 (T * [n]Y, n) be the tautological 1-form.Then it is shown in [PTVV13, Proposition 1.21] and [Cal19, Theorem 2.2] that the n-shifted closed 2-form d cl L ∨ L ≃ L τ [n − 1].See [PTVV13, §2.2] for the detail.Example A.6.(1) Let Y be a derived Artin stack and λ ∈ A 1 (T * [n]Y, n) be the tautological 1-form.Then λ| Y is naturally equivalent to zero hence so isd cl dR λ| Y .Therefore the zero section map Y → T * [n]Y carries a natural isotropic structure.It is shown in [Cal19, Theorem 2.2]that this isotropic structure is a Lagrangian structure.(2)Let Y and λ be as above, and take a functionf ∈ Γ(Y, O Y [n]) of degree n.Let d dR f : Y → T * [n]Y be the map corresponding to the section d dR f ∈ Γ(Y, L Y [n]).Then the natural homotopy(d dR f ) ⋆ d cl dR λ ∼ d cl dR • d dR f ∼ 0 defines an isotropic structure on (d dR f ).It is shown in [Cal19,Theorem 2.15] that this isotropic structure is a Lagrangian structure.
r,m) .Therefore we have an isomorphism IC M ss S (r,m) ∼ = BPS r,m .In particular, for coprime pairs (r, m) and (r, m ′ ), Corollary 5.15 implies an isomorphism h S * IC M ss S (r,m) ∼ = h S * IC M ss S (r,m ′ ) .Now let (r, m) be a non-coprime pair.It follows from [Sim94, Theorem 11.1] and [Dava, Theorem 5.11] that M ss S (r, m) is normal.The connectedness of M ss S (r, m) is proved in [DP12, Claim 3.5 (iii)].Therefore the moduli space M ss S (r, m) is irreducible.Then using [Dava, Theorem 6.S and + : N × N → N denote the canonical monoid structures.The following statement is a direct consequence of Theorem 5.16 and Lemma 4.2.