Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces
aa r X i v : . [ m a t h . AG ] F e b LIE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES ANDMOTIVES OF MODULI SPACES
DAVID ALFAYA AND ANDR´E OLIVEIRA
Abstract.
Let L = ( L, [ · , · ] , δ ) be an algebraic Lie algebroid over a smooth projective curve ofgenus g ≥ L is a line bundle whose degree is less than 2 − g . Let r and d be coprimenumbers. We prove that the motivic class (in the Grothendieck ring of varieties) of the moduli spaceof L -connections of rank r and degree d over X does not depend on the Lie algebroid structure[ · , · ] and δ of L and neither on the line bundle L itself, but only the degree of L (and of course on r, d, g and X ). In particular it is equal to the motivic class of the moduli space of K X ( D )-twistedHiggs bundles of rank r and degree d , for D any divisor of positive degree. As a consequence,similar results (actually a little stronger) are obtained for the corresponding E -polynomials. Someapplications of these results are then deduced. Contents
1. Introduction 22. Moduli space of twisted Higgs bundles 53. L -connections, Λ-modules and moduli spaces 73.1. Lie algebroids and L -connections 73.2. Λ-modules and L -connections 93.3. Moduli spaces of L -connections and of Λ-modules 123.4. The L -Hodge moduli spaces 144. Grothendieck ring of varieties, motives and E -polynomials 164.1. Grothendieck ring of varieties, motives and E -polynomials 164.2. The plan 175. Invariance of the motive and E -polynomial with respect to the algebroid structure 175.1. Bialynicki-Birula stratification 185.2. Semiprojectivity of the moduli space of Higgs bundles 205.3. Semiprojectivity of the moduli space of Λ red L -modules 215.4. Invariance of the motive and E -polynomial with respect to the algebroid structure 296. Motives of moduli spaces of twisted Higgs bundles 306.1. Variations of Hodge structure and chains 306.2. Independence of the motives of Higgs moduli on the twisting line bundle 326.3. Independence of motives and E -polynomials from the Lie algebroid structure 357. Applications 357.1. Topological properties of moduli spaces of Lie algebroid connections 357.2. Chow motives and Voevodsky motives 387.3. Motives of moduli spaces of irregular or logarithmic connections 397.4. Explicit motives and E -polynomials for rank 2 and 3 398. Motives of moduli spaces of L -connections with fixed determinant 45 Mathematics Subject Classification.
Key words and phrases.
Lie algebroid connections, Higgs bundles, moduli space, motive, virtual class, Hodgestructure, E-polynomial. L -connections with fixed determinant 47References 511. Introduction
Consider the de Rham moduli space M dR ( r,
0) of rank r algebraic connections over a smoothprojective curve X and write K X for the canonical bundle of X . So M dR ( r,
0) parameterizes S -equivalence classes of pairs ( E, ∇ ) (all of them are semistable), with E a rank r and degree 0algebraic vector bundle on X and ∇ : E → E ⊗ K X an algebraic connection on E . In general, theintegrability condition ∇ = 0 must be imposed, but in this case this is automatic because K X isa line bundle. Consider also the closely-related Dolbeault moduli space M K X ( r, r anddegree 0 semistable Higgs bundles ( E, ϕ ), with E as before, and ϕ : E → E ⊗ K X a morphism of O X -modules, called the Higgs field. The link between these two moduli spaces is provided by thenon-abelian Hodge correspondence [Hit87, Sim92, Sim94, Sim95], which yields a homeomorphismbetween them and hence allows the study of the topology of the de Rham moduli, by studying theDolbeault one.Actually, more is true, as proved by Simpson in [Sim94]: both M K X ( r,
0) and M dR ( r,
0) are,respectively, the fiber over 0 and 1 of an isosingular algebraic family over the affine line, such thatany other fiber is isomorphic to M dR ( r, M λ ( r,
0) of λ -connections ( E, ∇ , λ ) of rank r and degree 0, with λ ∈ C . Here, the Leibniz rule fora λ -connection is obtained from the usual Leibniz rule by multiplying the summand with the deRham differential by λ . If we consider the moduli space of semistable λ -connections ( E, ∇ , λ ), with λ varying in C , then we obtain the so-called Hodge moduli space M Hod ( r, M Hod ( r, → C mapping to λ , produces the mentioned isosingular family.The existence of singularities on the moduli spaces M K X ( r,
0) and M dR ( r,
0) introduces, how-ever, serious difficulties on the study of their topology and geometry. This is ‘circumvented’ if weconsider instead d coprime with r and the corresponding moduli spaces M K X ( r, d ) of stable degree d and rank r Higgs bundles and M dR ( r, d ) of stable logarithmic connections, hence with poles atsome prescribed punctures on X , with fixed holonomy, depending on d , around the punctures. Thenon-abelian correspondence extends to this case, and M K X ( r, d ) and M dR ( r, d ) are also homeo-morphic. Moreover, a complete analogue of the picture of the preceding paragraph holds as well,by using the Hodge moduli space M Hod ( r, d ), defined in the same way. The substantial differenceis that all these moduli spaces (including all moduli M λ ( r, d ) of logarithmic λ -connections of rank r and degree d ) are now smooth.Using M Hod ( r, d ), and taking r and d coprime, Hausel and Thaddeus proved in [HT03] that M K X ( r, d ) and M dR ( r, d ) actually share deeper geometrical invariants, other than just topologicalones. Namely, their Hodge structure is pure and equal, so in particular both spaces share thesame E -polynomial. The basic feature of their proof is the use of the fact that, as proved in[Sim94, Sim97], M Hod ( r, d ) is a smooth semiprojective variety in the sense of Hausel—Rodriguez-Villegas [HRV15] (which implies that it comes endowed with a C ∗ -action), together with a surjective C ∗ -equivariant submersion onto C .More recently, it was verified that the smoothness and the semiprojective structure on the Hodgemoduli space M Hod ( r, d ) implies the existence of another fundamental correspondence between M dR ( r, d ) and M K X ( r, d ). Namely, in [HL19], Hoskins and Lehaleur established what they called This was actually done in [HT03] for the fixed determinant (and traceless) versions of these moduli spaces, butthe argument also works in the non-fixed determinant case.
IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 3 a “motivic non-abelian Hodge correspondence” by proving an equality of the Voevodsky motives of M dR ( r, d ) and M K X ( r, d ); their result indeed holds for any characteristic zero field, not just C . Infact, by considering d = 0 and the stacky version of these moduli spaces, a similar result was provedbefore in [FSS18], for motivic classes in the Grothendieck ring of stacks, but through a completelydifferent technique, namely point counting. This was recently generalized to the parabolic settingin [FSS20].In the present paper, we prove the equality of the motivic class in the Grothendieck ring ofvarieties (and other invariants) between two generalizations of M K X ( r, d ) and M dR ( r, d ), as weexplain in the next paragraphs.The Hodge moduli space M Hod ( r, d ) is just a particular case of a much more general construction,by Simpson [Sim94], arising from the moduli space M Λ ( r, d ) of Λ-modules (see Definition 3.8),where Λ is a sheaf of rings of differential operators on X . Now, as proved by Tortella in [Tor11,Tor12], there is an equivalence of categories between a certain subclass of such sheaves (consisting ofthe split and almost polynomial ones) and algebraic Lie algebroids on X . In turn, such equivalenceinduces an equivalence of categories between integrable L -connections, where L is a Lie algebroidon X and Λ L -modules, with Λ L the split almost polynomial sheaf of rings of differential operatorscorresponding to L . This correspondence preserves semistability, hence one can think of M Λ L ( r, d )as the moduli space of L -connections of rank r and degree d .We now briefly recall what are these objects. Let T X = K − X be the tangent bundle of X .An algebraic Lie algebroid L = ( V, [ · , · ] , δ ) on X consists of an algebraic vector bundle V → X ,equipped with a Lie bracket [ · , · ] : V ⊗ C V → V and an anchor algebraic map δ : V → T X , suchthat [ · , · ] and δ are related by a Leibniz rule relation. Then an integrable (or flat) L -connection ofrank r and degree d on X is a pair ( E, ∇ L ), where E → X is an algebraic vector bundle of rank r and degree d , together with a C -linear algebraic map ∇ L : E → E ⊗ V ∗ , satisfying a generalizationof the Leibniz rule to the Lie algebroid setting, i.e. where the usual differential is replaced by an L -differential d L , and so that the integrability condition ∇ L = 0 holds. In this language, an algebraicconnection on X is just a T X -connection, where T X is the canonical Lie algebroid structure withunderlying bundle T X . Note also that if we consider the trivial algebroid L = ( V, , L -connection on X is simply a V ∗ -twisted Higgs bundle on X . i.e. so that the Higgsfield ϕ is twisted by V ∗ rather than K X and, furthermore, ϕ ∧ ϕ = 0.More generally, we can consider integrable ( λ, L )-connections ( E, ∇ L , λ ) for each λ ∈ C , thesebeing the analogues of the above mentioned λ -connections (which are hence ( λ, T X )-connections).As before, we see that for L = ( V, [ · , · ] , δ ), an integrable (0 , L )-connection is just a V ∗ -twistedHiggs bundle on X and an integrable (1 , L )-connection is an integrable L -connection. Then, thegeneralized version of the above Hodge moduli space is the L -Hodge moduli space M Λ red L ( r, d ) whichparameterizes S -equivalence classes of semistable integrable ( λ, L )-connections on X . As before itcomes with the natural map M Λ red L ( r, d ) → C , ( E, ∇ L , λ ) λ , whose fiber over zero is hence themoduli space of rank r and degree d V ∗ -twisted Higgs bundles and every other fiber is isomorphicto the fiber over 1, namely the moduli space of integrable L -connections of rank r and degree d .The first main result of this paper is the following (see Theorem 5.15). Theorem 1.1.
For any Lie algebroid L = ( L, [ · , · ] , δ ) such that L is a line bundle with deg( L ) < − g , where g ≥ is the genus of X , and ( r, d ) = 1 , the L -Hodge moduli space M Λ red L ( r, d ) is asmooth semiprojective variety. Moreover the projection M Λ red L ( r, d ) → C is a surjective submersion. This generalizes the above mentioned result by Simpson for the Hodge moduli space for anyLie algebroid on the given conditions. The two equivalent interpretations of the “same object” — L -connections and Λ L -modules — are actually required in this proof. For instance, in order toprove smoothness, we make use of the deformation theory for integrable L -connections, developed D. ALFAYA AND A. OLIVEIRA in [Tor12], since deformation theory for Λ-modules is not yet well-understood in the requiredgenerality. On the other hand, the proof of the existence of limits of a natural C ∗ -action on the L -Hodge moduli (which is a condition for being semiprojective) is carried out by explicitly usingΛ L -modules rather than L -connections.Let K ( V ar C ) be the Grothendieck ring of varieties and let ˆ K ( V ar C ) be its completion with respectto the Lefschetz motive L = [ A ]. Let L = ( L, [ · , · ] , δ ) be a Lie algebroid of rank 1. Then, using theprevious semiprojectivity result, and making use of the Bialynicki-Birula decompositions of boththe L -Hodge moduli space and of the L − -twisted Higgs bundle moduli space M L − ( r, d ), we provethe following result, concerning the motives of the moduli spaces M Λ L ( r, d ) of L -connections on X (note that integrability is automatic because L is a line bundle) and also concerning their Hodgestructures and E -polynomials (cf. Theorem 6.7 and Theorem 7.4). Theorem 1.2.
Suppose the genus of X is g ≥ . Let L = ( L, [ · , · ] , δ ) and L ′ = ( L ′ , [ · , · ] ′ , δ ′ ) beany two Lie algebroids on X , such that L and L ′ are line bundles with deg( L ) = deg( L ′ ) < − g .Suppose ( r, d ) = 1 . Then I ( M Λ L ( r, d )) = I ( M Λ L′ ( r, d )) . where I ( X ) denotes one of the following:(1) The virtual motive [ X ] ∈ ˆ K ( V ar C ) ;(2) The Voevodsky motive M ( X ) ∈ DM eff ( C , R ) for any ring R . In this case, moreover, themotives are pure;(3) The Chow motive h ( X ) ∈ Chow eff ( C , R ) for any ring R ;(4) The Chow ring CH • ( X, R ) for any ring R .Moreover, the mixed Hodge structures of the moduli spaces are pure and if d ′ is any integer coprimewith r , then E ( M Λ L ( r, d )) = E ( M Λ L′ ( r, d ′ )) . Finally, if L = L ′ = K ( D ) for some effective divisor D , then there is an actual isomorphism ofpure mixed Hodge structures H • ( M Λ L ( r, d )) ∼ = H • ( M Λ L′ ( r, d ′ )) . This is proved as follows. We prove it for ˆ K ( V ar C ) in Theorem 6.7. The results for the othermotives (Theorem 7.4) follow by the same techniques using technical results of [HL19]. For Theorem6.7, firstly we use the semiprojectivity of the L -Hodge and L ′ -Hodge moduli spaces described beforeto show that the classes of M Λ L ( r, d ) and M Λ L′ ( r, d ) equals, respectively, that of the twisted Higgsbundles moduli spaces M L − ( r, d ) and M L ′− ( r, d ), which correspond to the particular case oftrivial algebroids; see Theorem 5.17. Notice that this is a generalization (for k = C ) of the “motivicnon-abelian Hodge correspondence” of [HL19] to any Lie algebroid L on the given conditions.Secondly, by studying the Bialynicki-Birula decomposition of M L − ( r, d ), we can prove that itsmotive only depends on the degree of the twisting line bundle L − , but not on the twisting itself;see Theorem 6.6.Notice also that our setting includes, for example, the moduli spaces of logarithmic connections(without fixed residues on the poles), corresponding to the Lie subalgebroid T X ( − D ) ⊂ T X , andhence to the case L − = K X ( D ) for an effective and reduced divisor on X . It also includes themoduli spaces of wild connections (again without fixing the corresponding Stokes data), by taking T X ( − D ′ ) ⊂ T X , thus L − = K X ( D ′ ), where D ′ is an effective and non-reduced divisor on X .Hence, we have the following direct application of the above theorem (see Corollary 7.6): Corollary 1.3.
The motive and E -polynomial of any moduli space of irregular connections ona smooth projective curve X of genus at least equals that of any moduli space of logarithmic connections X , with singular divisor of the same degree. IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 5
Moreover, using the above theorem, we provide the formulas for the motivic classes and for the E -polynomials of M Λ L ( r, d ), for any rank 1 Lie algebroid L of degree less than 2 − g , when r = 2 , d coprime with r ; see Corollaries 7.7 and 7.9.In addition, for a Lie algebroid L under the same conditions, we also compute the homotopygroups of the moduli spaces M Λ L ( r, d ), up to some bound depending on the rank of the L -connectionand on the genus of the curve X . See Theorem 7.2.Our main results hold for Lie algebroids L whose underlying bundle is a line bundle, but weexpect that at least some of them hold for higher rank Lie algebroids (see Remark 5.16). Inaddition, we prove some results in the higher rank setting, mostly concerning non-emptiness of themoduli spaces of L -connections; see section 3.3.Here is a brief description of the contents of the paper. In section 2 we give a quick introductionto V -twisted Higgs bundles. In Section 3 we introduce Lie algebroids, sheaves of rings of differentialoperators, L -connections and Λ-modules and, following [Tor11, Tor12], give a short overview of theequivalence of categories between integrable L -connections and Λ L -modules. This is done overany base variety. Then we introduce the moduli spaces of Λ L -modules / integrable L -connectionsover a curve X , prove some non-emptiness results about them, and also introduce the L -Hodgemoduli spaces. The purpose of Section 4 is to introduce the Grothendieck ring of varieties andmotives and to give an overview of our strategy to prove the main results. Sections 5 and 6 arethe technical core of the paper. In Section 5 we prove that the motivic class of the moduli spacesof Lie algebroid connections equals that of the corresponding twisted Higgs bundle moduli (usingthe semiprojectivity of the L -Hodge moduli, which is also proved) and in Section 6 we show thatthe motivic class of the twisted Higgs bundles moduli is independent of the twisting. In Section 7we deduce some applications of the results proved in the previous sections, including the versionsof the main theorem for Voevodsky motives, Chow motives and Chow rings. Finally, in Section 8we show how to achieve similar results for the same moduli spaces of L -connections ( E, ∇ L ) butwhere the determinant of E and the trace of ∇ L are fixed. Acknowledgments.
This research was funded by MICINN (grants MTM2016-79400-P, PID2019-108936GB-C21 and “Severo Ochoa Programme for Centres of Excellence in R&D” SEV-2015-0554)and by CMUP – Centro de Matem´atica da Universidade do Porto – financed by national fundsthrough FCT – Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P., under the project with referenceUIDB/00144/2020. The first author was also supported by a postdoctoral grant from ICMATSevero Ochoa project. He would also like to thank the hospitality of CMUP during the researchvisits which took place in the course of the development of this work. Finally, we thank VicenteMu˜noz, Jos´e ´Angel Gonz´alez and Jaime Silva for helpful discussions.2.
Moduli space of twisted Higgs bundles
Throughout the paper we will only be considering algebraic objects (vector bundles, Lie alge-broids, connections, etc.) on smooth projective varieties over C , this being implicitly assumedwhenever the corresponding adjective is missing. We will also always take the usual identificationbetween algebraic vector bundles and locally free sheaves.Let Y be a smooth complex projective variety and let V be an algebraic vector bundle on Y ,thus a locally free O Y -module. Definition 2.1. A V -twisted Higgs bundle on Y is a pair ( E, ϕ ) consisting by an algebraic vectorbundle E on Y and a map of O Y -modules ϕ : E −→ E ⊗ V, called the Higgs field , such that ϕ ∧ ϕ = 0 ∈ H (End( E ) ⊗ Λ V ) . D. ALFAYA AND A. OLIVEIRA If ϕ ∈ H (End( E ) ⊗ V ), by ϕ ∧ ϕ ∈ H (End( E ) ⊗ Λ V ) in the previous definition we mean thefollowing. Let p : V ⊗ V → Λ V be the quotient map. Then ϕ ∧ ϕ = (Id E ⊗ p ) ◦ ( ϕ ⊗ Id V ) ◦ ϕ. A local version is given as follows. Take an open set U ⊂ Y where both E and V are trivialized,and let w , . . . , w k be a local trivializing basis of V ; then we can write ϕ | U = P i G i ⊗ w i , with G i a local section of End( E ), so an O X ( U )-valued matrix, and( ϕ ∧ ϕ ) | U = X i Suppose that r ≥ and d are coprime and that L is a line bundle with deg( L ) > g − , where g ≥ is the genus of X . Then the moduli space M L ( r, d ) is a smooth connected, soirreducible, quasi-projective variety of dimension r deg( L ) . Remark 2.3. There are choices of the twist V which do not satisfy the assumptions of the Lemmaand for which M V ( r, d ) is nonetheless a smooth variety. For instance, the classical case V = K X from [Hit87] and which was also considered in [BGL11, Proposition 3.3] . IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 7 However, a condition similar to the one presented in the previous lemma is to be expected ingeneral since there exist many examples of low degree twists for which the moduli space is notsmooth and even not irreducible.For example, if L = O X ( x ) for a curve X with genus at least 4 and x ∈ X , then M O X ( x ) ( r, d ) is,in general, a singular variety. For generic stable vector bundles E , we have H (End( E )( x )) ∼ = C by [BGM13, Lemma 2.2] so M O X ( x ) ( r, d ) contains an open subvariety U ⊂ M O X ( x ) ( r, d ) which is aline bundle over the open locus of such bundles in the moduli space M ( r, d ) of stable vector bundles.Nevertheless, suppose that we pick the curve X so that there exists a stable vector bundle F on X with a nonzero section ϕ ∈ H (End ( F )( x )) , ϕ = 0 . Then ( F, ϕ ) ∈ M O X ( x ) ( r, d ) will not belongto the closure of U , so the moduli space will not be irreducible, and the point ( F, ∈ M O X ( x ) ( r, d ) will belong to two different irreducible components, making the moduli space a singular variety. Consider a V -twisted Higgs bundle ( E, ϕ ) on X . Since ϕ ∧ ϕ = 0, the Higgs field ϕ : E → E ⊗ V induces maps ∧ i ϕ : Λ i E −→ Λ i E ⊗ S i V, for i = 1 , . . . , r . If we define s i = tr( ∧ i ϕ ) ∈ H ( S i V ) , then this yields the Hitchin map (2.1) H : M V ( r, d ) −→ r M i =1 H ( S i V ) , H ( E, ϕ ) = ( s , . . . , s r ) . We call W := L ri =1 H ( S i V ) the Hitchin base . Lemma 2.4. The Hitchin map H : M V ( r, d ) −→ W is proper.Proof. When V is a line bundle, this was proven by Nitsure [Nit91, Theorem 6.1]. In the case V = K X , besides being proven by Hitchin [Hit87], it was also proven by Faltings [Fal93, TheoremI.3] following an argument by Langton, and the same argument works word by word for an arbitrarytwisting V . (cid:3) L -connections, Λ -modules and moduli spaces Lie algebroids and L -connections. Let Y be a smooth complex projective variety withtangent bundle T Y . We will be mostly interested in the case Y is the smooth projective curve X ,but we will actually need to consider a higher dimensional variety in sections 3.4 and 5.3. Definition 3.1. An algebraic Lie algebroid on Y is a triple L = ( V, [ · , · ] , δ ) consisting on • an algebraic vector bundle V on Y , • a C -bilinear and skew-symmetric map [ · , · ] : V ⊗ C V → V , called the Lie bracket , • a vector bundle map δ : V → T Y , called the anchor ,satisfying the following properties:(1) [ u, [ v, w ]] + [ v, [ w, u ]] + [ w, [ u, v ]] = 0 (Jacobi rule) ,(2) [ u, f v ] = f [ u, v ] + δ ( u )( f ) v (Leibniz rule) ,for any local sections u, v, w of V and any local function f in O Y .The rank of a Lie algebroid L , denoted by rk( L ) , is the rank of the underlying vector bundle V . Example 3.2. (1) The canonical example of Lie algebroid is the tangent bundle T Y , together with the Liebracket of vector fields and the identity anchor. Denote it as T Y = ( T Y , [ · , · ] Lie , Id) . D. ALFAYA AND A. OLIVEIRA (2) More generally, an algebraic foliation F Y ⊂ T Y gives also rise to a Lie algebroid F Y simplyby restricting the Lie bracket [ · , · ] Lie and by taking the inclusion F Y ֒ → T Y as the anchormap.(3) Any algebraic vector bundle can be seen as a Lie algebroid with the zero bracket and thezero map as the anchor; this is called a trivial algebroid . The Lie bracket of a Lie algebroid L = ( V, [ · , · ] , δ ) endows V with a structure of a sheaf of C -Liealgebras, which is not a sheaf of O Y -Lie algebras unless the anchor δ is zero.A Lie algebroid map f : L → L ′ between L = ( V, [ · , · ] V , δ V ) and L ′ = ( V ′ , [ · , · ] V ′ , δ V ′ ) is aalgebraic C -Lie algebra bundle map f : V → V ′ such δ V ′ ◦ f = δ V . For example, the anchor δ : V → T Y is a Lie algebroid map δ : L → T Y . A Lie algebroid isomorphism is a Lie algebroidmap which is an isomorphism of the underlying bundles; in that case, the Lie algebroids are saidto be isomorphic .Let L = ( V, [ · , · ] , δ ) be a Lie algebroid. We now define a differential on the complex of exteriorpowers Ω •L = Λ • V ∗ , d L : Ω k L −→ Ω k +1 L , generalizing the classical de Rham complex d : Ω kY −→ Ω k +1 Y on Ω • Y = Λ • T ∗ Y . In degree 0, define d L : O Y → V ∗ as the composition of the canonical differential d : O Y → T ∗ Y with the dual of theanchor, δ t : T ∗ Y → V ∗ . Thus, given v ∈ V and f a local algebraic function on Y ,(3.1) d L ( f )( v ) = df ( δ ( v )) = δ ( v )( f ) . The map d L : O Y → V ∗ is clearly a V ∗ -valued derivation. Now we extend it to higher orderexterior powers through the usual recursive equation, but using the anchor map δ . For ω ∈ Ω n L and v , . . . , v n +1 local sections of V , take d L ( ω )( v , . . . , v n +1 ) = n +1 X i =1 ( − i +1 δ ( v i )( ω ( v , . . . , ˆ v i , . . . , v n +1 ))+ X ≤ i Let L = ( V, [ · , · ] , δ ) be a Lie algebroid on Y . An L -connection on Y is a pair ( E, ∇ L ) , where E is an algebraic vector bundle and where ∇ L is a C -linear algebraic vector bundlemap ∇ L : E −→ E ⊗ Ω L = E ⊗ V ∗ , such that ∇ L ( f s ) = f ∇ L ( s ) + s ⊗ d L ( f ) , for s a local section of E and f a local algebraic function on Y . The rank of an L -connection isthe rank of the underlying algebraic vector bundle. We will also refer to ∇ L : E → E ⊗ Ω L as an L -connection on the algebraic vector bundle E .Note that, for a trivial algebroid L , the map ∇ L : E → E ⊗ Ω L is actually O Y -linear rather thanjust C -linear.Any L -connection ( E, ∇ L ) can be extended to a map ∇ L : E ⊗ Ω •L −→ E ⊗ Ω • +1 L , as ∇ L ( s ⊗ ω ) = ∇ L ( s ) ∧ ω + s ⊗ d L ( ω ) . IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 9 Definition 3.4. Let ( E, ∇ L ) be an L -connection on Y . The composition ∇ L : E → E ⊗ Ω L iscalled the curvature of ( E, ∇ L ) . An L -connection is integrable or flat if its curvature vanishes. Example 3.5. (1) Recall the canonical Lie algebroid T Y = ( T Y , [ · , · ] Lie , Id) given by the tangent bundle of Y .Then, a flat T Y -connection is just a flat algebraic connection in the usual sense.(2) If L is a trivial algebroid, i.e. L = ( V, , , then a flat L -connection is simply a pair ( E, ∇ L ) formed by a vector bundle E and an O Y -linear map ∇ L : E → E ⊗ V ∗ such that ∇ L ∧ ∇ L = 0 . This is precisely a V ∗ -twisted Higgs bundle from Definition 2.1. Alternatively, we can think of L -connections, for any Lie algebroid L = ( V, [ · , · ] , δ ), from a slightlydifferent point of view. An L -connection ∇ L : E → E ⊗ Ω L induces a O Y -linear map(3.3) ¯ ∇ L : V −→ End C ( E ) , v 7→ ∇ L ,v , which, by (3.1), satisfies(3.4) ∇ L ,v ( f s ) = f ∇ L ,v ( s ) + δ ( v )( f ) s. In particular, identifying the local function f ∈ O Y with the endomorphism E → E , s f s , wehave that ∇ L ,v ◦ f − f ◦ ∇ L ,v is O Y -linear, ∇ L ,v ◦ f − f ◦ ∇ L ,v ∈ End O Y ( E ) . Hence, ∇ L ,v is a section of the locally free sheaf Diff ( E ) of differentials of order at most 1. Notethat, since End C ( E ) is a sheaf of associative algebras, the vector bundle Diff ( E ) inherits canonicallya Lie algebroid structure D ( E ) = (Diff ( E ) , [ · , · ] D ( E ) , δ D ( E ) ) through the commutator, by taking[ A, B ] D ( E ) = AB − BA ∈ Diff ( E ) , for each A, B ∈ Diff ( E ), and, by making use of the O Y -module structure of Diff ( E ), δ D ( E ) ( A )( f ) = Af − f A ∈ O Y ⊆ End O Y ( E ) , for every f ∈ O Y . Now, note that the L -connection ∇ L is integrable if and only if[ ∇ L ,u , ∇ L ,v ] D ( E ) = ∇ L , [ u,v ] , for all u, v ∈ V . Furthermore, it follows by (3.4) that, for every v , δ D ( E ) ( ∇ L ,v ) = δ ( v ) . We conclude that the map (3.3) can be canonically upgraded to a Lie algebroid map ¯ ∇ L : L → D ( E ),sometimes called a representation of L in D ( E ). In addition, the correspondence ∇ L ¯ ∇ L yieldsa bijective correspondence(3.5) (cid:8) integrable L -connections on E (cid:9) ←→ (cid:8) representations of L in D ( E ) (cid:9) . -modules and L -connections. In [Tor11, § 3] and [Tor12, § L -connections and Simpson’s notion of Λ-modules [Sim94],to be introduced below. This will be important for us to deduce properties of the correspondingmoduli spaces.Let S be a smooth variety and let X → S be a smooth variety over S . We start with the notionof sheaf of rings of differential operators as defined in [Sim94]. Definition 3.6. A sheaf of rings of differential operators on X over S is a sheaf of O X -algebras Λ over X , with a filtration by subalgebras Λ ⊆ Λ ⊆ · · · ⊆ Λ , verifying the following properties:(1) Λ = S ∞ i =0 Λ i and Λ i · Λ j ⊆ Λ i + j , for every i, j ;(2) Λ = O X ;(3) the image of p − ( O S ) in O X lies in the center of Λ ; (4) the left and right O X -module structures on Gr i (Λ) := Λ i / Λ i − are equal;(5) the sheaves of O X -modules Gr i (Λ) are coherent;(6) the morphism of sheaves Gr (Λ) ⊗ · · · ⊗ Gr (Λ) → Gr i (Λ) induced by the product is surjective.Moreover, Λ is said to be polynomial if Λ ∼ = Sym • (Gr (Λ)) and almost polynomial if Gr • (Λ) ∼ =Sym • (Gr (Λ)) . Remark 3.7. Simpson’s definition is a slightly more general in point (2) , allowing Λ to be aquotient of O X and, therefore, allowing Λ to be supported on a subscheme of X . However, we willonly be interested on sheafs of rings supported on the whole scheme X . Definition 3.8. Let Λ be a sheaf of rings of differential operators over X , flat over S . A Λ-module is a pair ( E, ∇ Λ ) consisting of an algebraic vector bundle E on X , flat on S , with an action ∇ Λ : Λ ⊗ O X E → E satisfying the usual module conditions. Moreover, E is required to be locallyfree as a Λ -module and its inherent O X -module structure coincides with the O X -module structureinduced by the inclusion O X ⊆ Λ . The rank of a Λ -module is the rank of the underlying bundle. The notions of maps and isomorphisms between Λ-modules over X are the obvious ones, as inthe Lie algebroids and Higgs bundles cases.Let us now briefly recall the relation between integrable L -connections and Λ-modules. Fordetails, see [Tor11, § 3] and [Tor12, § L = ( V, [ · , · ] , δ ) on Y . Consider the associated Lie algebroid b L = ( O Y ⊕ V, [ · , · ] , δ )given by [( f, u ) , ( g, v )] = ( δ ( u )( g ) − δ ( v )( f ) , [ u, v ]) , δ ( f, u ) = δ ( u ) , with u, v ∈ V and f, g ∈ O Y . Note that this means that, if O Y also denotes the trivial algebroid( O Y , , split short exact sequence(3.6) 0 −→ O Y −→ b L −→ L −→ b L ∼ = O Y ⊕ L as Lie algebroids or, equivalently that thereis a splitting ζ : L → b L of the sequence (3.6), which in this case is simply given by ζ ( v ) = (0 , v ), v ∈ V .From the Lie algebroid b L , hence from L , there is an associated an almost polynomial sheaf ofrings of differential operators Λ L whose weight 1 piece Λ L , ⊂ Λ L in corresponding filtration isisomorphic to O Y ⊕ V . This is roughly constructed as follows. The universal enveloping algebra ofthe Lie algebra ( O Y ⊕ V, [ · , · ] ) is the sheaf of O Y -algebras, defined by U = T • ( O Y ⊕ V ) / h x ⊗ y − y ⊗ x − [ x, y ] | x, y ∈ O Y ⊕ V i , with T • ( O Y ⊕ V ) being the tensor algebra on O Y ⊕ V . If i : O Y ⊕ V ֒ → U is the canonical inclusion,then let U † ⊂ U be the subalgebra generated by i ( O X ⊕ V ) and takeΛ L = U † / ( i ( f, · i ( g, v ) − i ( f g, f v ) | f, g ∈ O Y , v ∈ V ) . This is the universal enveloping algebra of the Lie algebroid L . Then the graded structure from thetensor algebra T • ( O Y ⊕ V ) induces a filtered algebra structure on Λ L which satisfies the axioms ofan almost polynomial sheaf of rings of differential operators. We thus get the correspondence(3.7) L 7→ Λ L . IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 11 Since this construction started by taking the trivial extension of Lie algebroids in (3.6), the sheafΛ L is called a split almost polynomial sheaf of differential operators.Conversely, take an almost polynomial sheaf of rings of differential operators Λ. Then the O Y -module Λ ⊂ Λ inherits a Lie algebroid structure in the following way [Tor11, Proposition 28][Tor12, § u, v of Λ and f of O Y , define(3.8) [ u, v ] Λ = uv − uv and δ Λ ( u )( f ) = uf − f u, using the product in Λ. Let us prove that [ u, v ] Λ ∈ Λ and that δ Λ ( u )( f ) ∈ O Y . As Λ is almostpolynomial, then Gr • (Λ) is abelian. Let sb : Λ → Gr • (Λ) be the symbol map sending an elementto the highest graded class where it is nonzero. One can check that sb is multiplicative. We havesb( uv ) − sb( vu ) = sb( u ) sb( v ) − sb( v ) sb( u ) = 0 ∈ Gr • (Λ)As uv, vu ∈ Λ , we conclude that uv ≡ vu mod Λ , that is, uv − vu ∈ Λ , proving the firstclaim. For the second claim, we use the fact that the left and right O Y -module structures onGr (Λ) = Λ / Λ agree. Hence uf − f u ∈ Λ = O Y and the second claim follows. Now, usingthe associativity of Λ and the fact that elements of O Y commute with all elements of Λ, one seesthat [ · , · ] Λ is O Y -bilinear, skew-symmetric and verifies the Jacobi identity and that δ Λ is so thatLeibniz rule also holds. So (3.8) gives the mentioned Lie algebroid structure on Λ .Now, it is clear that the induced Lie algebroid structure on O Y = Λ ⊂ Λ is trivial, thus (3.8)descends to give a Lie algebroid structure on Gr (Λ), which we denote by L Λ : L Λ = (Gr (Λ) , [ · , · ] Gr (Λ ) , δ Gr (Λ ) ) . Moreover, the following short exact sequence of Lie algebroids0 −→ ( O X , , −→ (Λ , [ · , · ] Λ , δ Λ ) −→ L Λ −→ , hence the sheaf Λ is split. Thus, we have the correspondence(3.9) Λ 7→ L Λ . By [Tor12, Theorem 4.4], Λ is split almost polynomial if and only if it is isomorphic to the uni-versal enveloping algebra of Gr (Λ). Hence, the upshot of this discussion is the following theorem. Theorem 3.9. The correspondences (3.7) and (3.9) induce inverse equivalences of categories: (cid:26) isomorphism classes ofLie algebroids on Y (cid:27) ←→ isomorphism classes of splitalmost polynomial sheaves of ringsof differential operators on Y . L 7−→ Λ L (3.10)Moreover, (3.10) induces a correspondence between integrable L -connections and Λ L -modules.This goes roughly as follows (see again the above mentioned references for details).Let L = ( V, [ · , · ] , δ ) be a Lie algebroid with corresponding sheaf Λ L . So we have a short exactsequence 0 → O Y → Λ L , → L → L , = O Y ⊕ L ⊂ Λ L . Given a Λ L -module ∇ Λ L : Λ L ⊗ E → E , define the L -connection ( E, ∇ L ) by taking(3.11) ∇ L : E → E ⊗ V ∗ , ∇ L ( s )( v ) = ∇ Λ L ((0 , v ) ⊗ s ) , for s and v local sections of E and V respectively. This is a flat L -connection.Conversely, from a flat L -connection ( E, ∇ L ), define ∇ Λ L , : Λ L , ⊗ E → E, ∇ Λ L , (( f, v ) ⊗ s ) = f s + ∇ L ( s )( v ) , where f ∈ O Y , v ∈ V and s ∈ E . By successive compositions, this defines a (Λ L , ) ⊗• -modulestructure on E , which descends to a Λ L -module structure(3.12) ∇ Λ L : Λ L ⊗ E → E because ∇ L is integrable and satisfies Leibniz.Now, from [Tor12, Proposition 5.3], we have the following. Theorem 3.10. The correspondences (3.11) and (3.12) induce inverse equivalences of categories: (3.13) (cid:26) isomorphism classes ofintegrable L -connections on Y (cid:27) ←→ (cid:26) isomorphism classes of Λ L -modules on Y (cid:27) . Remark 3.11. The results of [Tor11] and [Tor12] are actually more general, in the sense that thereis no split condition on the almost polynomial sheaves of differential operators. The correspondencethere includes, in both sides, a certain cohomology class Q , which is the obstruction to the existenceof a Lie algebroid splitting of the above short exact sequence. The case stated above corresponds to Q = 0 , which is actually the one of interest to us, since that is the only split case for which thecorresponding moduli spaces (to be introduced in the next section) are non-empty. Moduli spaces of L -connections and of Λ -modules. Let us return now to the case where Y is our fixed smooth complex projective curve X and where X → S is an S -family of smoothcomplex projective curves over a scheme S . An L -connection ( E, ∇ L ) is (semi)stable if for everysubbundle 0 = F ( E preserved by ∇ L , i.e. such that ∇ L ( F ) ⊆ F ⊗ Ω L , we have µ ( F ) < µ ( E ) (resp. ≤ )and that a Λ-module ( E, ∇ Λ ) on a fiber X s of X → S is (semi)stable if for every 0 = F ( E preserved by ∇ Λ , i.e. such that ∇ Λ (Λ ⊗ F ) ⊆ F we have µ ( F ) < µ ( E ) (resp. ≤ ) . Note that if the vector bundle E is semistable, then any L -connection and any Λ-module are alsosemistable. Clearly, an L -connection is (semi)stable if and only if the corresponding Λ L -module is(semi)stable. Define the degree of an L -connection or of a Λ-module over X to be the degree of theunderlying vector bundle.For any sheaf of rings of differential operators Λ on X over S , and any rank r ≥ d ∈ Z ,Simpson [Sim94] proved that there exists a moduli space M Λ ( X , r, d ) of semistable Λ-modules ofrank r and degree d and that it is a complex quasi-projective variety over S . If the family X isclear from the context, we denote the moduli space simply by M Λ ( r, d ). If s ∈ S is a point, let Λ s be the pullback of Λ to the fiber X s of X → S . A closed point of M Λ ( X , r, d ) over a point s ∈ S represents a semistable Λ s -module ( E, ∇ Λ s ) over X s .Then, Theorem 3.10 also identifies M Λ L ( r, d ) with the moduli space of integrable L -connectionsof rank r and degree d [Tor12]. Remark 3.12. Alternatively, a moduli space of (possibly non-integrable) Lie algebroid connectionswas also constructed by Krizka [Kri09] using analytic tools. When L has rank , integrability isautomatic and, therefore, this construction gives an alternative description for the moduli space ofintegtrable Lie algebroid connections. On the other hand, in an unpublished work [Kri10] , the sameauthor works with flat connections and proposes a general analytical construction of such modulispace. We saw in Example 3.5 that when L has the trivial algebroid structure L = ( V, , L -connection is precisely the same thing as a V ∗ -twisted Higgs bundle. Moreover, it is obvious that IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 13 (semi)stability is preserved under this identification. Hence, the moduli space of integrable L -connections coincides with the moduli space of V ∗ -twisted Higgs bundles. Therefore, for a trivialLie algebroid L = ( V, , M Λ ( V, , L ( r, d ) = M V ∗ ( r, d )between the moduli spaces of integrable L -connections, Λ L -modules and V ∗ -twisted Higgs bundlesover X , all of the same rank r and degree d .Note that, depending on the choice of the rank, degree and Λ, the moduli space M Λ ( X , r, d ) maybe singular or even empty. For instance, if Λ is the sheaf of differential operators D X on a curve X , then M D X ( r, d ) coincides with the moduli space of semistable ( T X -)connections on X whoseunderlying vector bundle has rank r and degree d . But the flatness condition (which is automaticsince T X is a line bundle) implies that d = 0, so the moduli space is empty for d = 0. Moreover,for d = 0 and r ≥ 2, the moduli space of flat connections is singular due to the existence of strictlysemistable objects. In the remaining part of this section we will to prove sufficient conditions forthese moduli spaces to be nonempty and smooth varieties.The existence of algebraic connections (i.e. T X -connections in the language of Lie algebroids) onan algebraic vector bundle E over X has been studied by Atiyah in [Ati57]. There it was proved theexistence of a class in Ext ( T X , End( E )) — now known as the Atiyah class of E — whose vanishingis equivalent to the existence of such a algebraic connection. A generalization of this picture to L -connections was carried out in [Tor11, Section 2.4.4] [Tor12, § E and let L = ( V, [ · , · ] , δ ) be a Lie algebroid. Let a ( E ) ∈ Ext ( T X , End( E )) be the Atiyah class of E .Define the L -Atiyah class of E as a L ( E ) = δ ∗ ( a ( E )) ∈ Ext ( V, End( E )) . Proposition 3.13. [Tor11, Proposition 17] An algebraic vector bundle E admits an algebraic L -connection if and only if a L ( E ) = 0 . From this we can exhibit some concrete examples of L -connections. Corollary 3.14. Let E be a semistable vector bundle on the curve X . Let L = ( V, [ · , · ] , δ ) be aLie algebroid such that the vector bundle V is semistable and − µ ( V ) > g − . Then E admits an L -connection. Moreover, if rk( L ) = 1 , then E admits an integrable L -connection.Proof. The L -Atiyah class a ( E ) is an element ofExt ( V, End( E )) ∼ = H (End( E ) ⊗ V ∗ ) ∼ = H (End( E ) ⊗ K X ⊗ V ) ∗ , by Serre duality. Since both E and V are semistable then End( E ) ⊗ K X ⊗ V is also semistable.Moreover µ (End( E ) ⊗ K X ⊗ V ) = 2 g − µ ( V ) < , so H (End( E ) ⊗ K X ⊗ V ) = 0. Thus a ( E ) = 0, and the result follows from Proposition 3.13.If in addition rk( L ) = 1, then Ω L = Λ V ∗ = 0 thus any L -connection on E is automaticallyflat. (cid:3) A Lie algebroid L is called transitive if the anchor δ is surjective and intransitive otherwise. Corollary 3.15. Let L be any intransitive Lie algebroid on X and let E be a semistable algebraicvector bundle. Then E admits an integrable L -connection.Proof. As L = ( V, [ · , · ] , δ ) is intransitive, the anchor map δ : V → T X is not surjective. Then,as T X is a line bundle, there exists a point x ∈ X such that δ | x = 0. Thus, δ factors through¯ δ : V → T X ( − x ) ⊂ T X . Actually, the line bundle T X ( − x ) inherits a natural Lie algebroid structure,denoted by T ( − x ), from the one of T = ( T X , [ · , · ] Lie , Id), since the Lie bracket of two local vector fields which annihilate at x also annihilates at x . Thus, the anchor in T X ( − x ) is just the inclusion T X ( − x ) ֒ → T X . Since we also know the anchor maps are also Lie algebroid maps, we have acommutative diagram of Lie algebroid maps L TT ( − x ) . δ ¯ δ Now, T X ( − x ) is a line bundle such that − deg( T X ( − x )) = 2 g − > g − 2, so the previouscorollary shows that E admits an integrable T ( − x )-connection. Moreover, by (3.5) such connectionis determined by a representation T ( − x ) → D ( E ). Pre-composing it with ¯ δ yields a representation L → D ( E ) and, hence, again by (3.5), gives rise to an integrable L -connection on E . (cid:3) The previous results can be now immediately used to prove, under certain conditions, non-emptiness of the moduli spaces of L -connections of any rank and degree. Proposition 3.16. For any rank r and degree d and any Lie algebroid L such that either(1) rk( L ) = 1 and deg( L ) < − g , or(2) L is intransitive.Then the moduli space M Λ L ( r, d ) is nonempty.Proof. Let L be any Lie algebroid verifying either of the two given conditions. Choose anysemistable vector bundle E over X . By Corollary 3.14, if rk( L ) = 1 and deg( L ) < − g , orby Corollary 3.15, if L is intransitive, we conclude that E admits an integrable L -connection ∇ L . Moreover, since E is semistable, then so is ( E, ∇ L ), which therefore represents a point in M Λ L ( r, d ). (cid:3) The L -Hodge moduli spaces. Let us now introduce the deformation of a split almostpolynomial sheaf of rings of differential operators (cf. Definition 3.6) to its associated graded ring,as described in [Sim94, p. 86].Let L = ( V, [ · , · ] , δ ) be a Lie algebroid over X and let Λ L be its associated split almost polynomialsheaf of rings of differential operators. By construction, we know thatGr • (Λ L ) ∼ = Sym • ( V ) . We can associate to Λ L a sheaf of rings of differential operators Λ red L on X × C over C whose fiberover 1 is Λ L and whose fiber over 0 is isomorphic to its graded algebra Sym • ( V ). Let λ be thecoordinate of C and let p X : X × C → X be the projection. We define Λ red L as the subsheaf of p ∗ X (Λ L ) generated by sections of the form P i ≥ λ i u i , where u i is a local section of Λ L ,i ⊆ Λ L . Thissubsheaf is a sheaf of filtered algebras on X × C over C , which coincides with the Rees algebraconstruction of the sheaf of filtered algebras Λ and one can verify that it satisfies all properties ofDefinition 3.6, making it a sheaf of rings of differential operators on X × C over C .On the other hand, given the Lie algebroid L = ( V, [ · , · ] , δ ) and λ ∈ C , we can define another Liealgebroid L λ as(3.14) L λ = ( V, λ [ · , · ] , λδ ) , Then, the following can be checked through direct computation(3.15) Λ red L | X ×{ λ } ∼ = Λ L λ . Observe that if λ = 0 then multiplicacion by λ defines an isomorphism of Lie algebroids L λ ∼ −→ L , v λv IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 15 and for λ = 0, we have that L = ( V, , 0) is the trivial algebroid over V . This implies the followingproperties of the fibers of Λ red L over λ ∈ C which are also well known consequences of the Reesconstruction.(1) For every λ = 0 we have Λ red L | X ×{ λ } ∼ = Λ red L | X ×{ } ∼ = Λ L .(2) If λ = 0 then Λ red L | X ×{ } ∼ = Gr • (Λ L ) ∼ = Sym • ( V ).Now, consider the moduli space M Λ red L ( r, d ) of Λ red L -modules of rank r and degree d . By Simpson’sconstruction, it is a quasiprojective variety with a map(3.16) π : M Λ red L ( r, d ) −→ C and, by (3.15), it parametrizes S -equivalence classes of triples ( E, ∇ L λ , λ ) where λ = π ( E, ∇ L λ , λ ) ∈ C and ( E, ∇ L λ ) is a semistable integrable L λ -connection of rank r and degree d on X . By properties(1) and (2) above, it becomes clear that(3.17) π − ( λ ) = M Λ L λ ( r, d ) ∼ = M Λ L ( r, d ) = π − (1) , for every λ ∈ C ∗ and π − (0) ∼ = M V ∗ ( r, d ) . so M Λ red L ( r, d ) is a variety which “interpolates” between the moduli space lf L -connections and themoduli space of V ∗ -twisted Higgs bundles.Let us explore more precisely what is an L λ -connection, for L λ as in (3.14). By (3.1) and(3.2) we see that if (Ω •L , d L ) is the Chevalley–Eilenberg–de Rham complex of L , then Ω •L λ = Ω •L and (Ω •L λ , λd L ) is the Chevalley–Eilenberg–de Rham complex of L λ . Hence, in an L λ -connection( E, ∇ L λ ), the map ∇ L λ : E → E ⊗ V ∗ verifies(3.18) ∇ L λ ( f s ) = f ∇ L λ ( s ) + λs ⊗ d L ( f ) . Motivated by the classical example of a λ -connection on an algebraic vector bundle E (seethe example below), we also call a L λ -connection to be a ( λ, L )-connection, and consider it as atriple ( E, ∇ L , λ ), where ∇ L : E → E ⊗ V ∗ verifies (3.18). Then we identify M Λ red L ( r, d ) with themoduli space of ( λ, L )-connections and, in analogy with the terminology for the moduli space of λ -connections, we call it the L -Hodge moduli space . Example 3.17. (1) A ( λ, T X ) -connection is a λ -connection on an algebraic vector bundle E .(2) An L -connection is a the same thing as a (1 , L ) -connection.(3) For L = ( V, [ · , · ] , δ ) , a (0 , L ) -connection is a V ∗ -twisted Higgs bundle. The moduli space M Λ red L ( r, d ) is endowed with a C ∗ -action scaling the ( λ, L )-connection,(3.19) t · ( E, ∇ L , λ ) = ( E, t ∇ L , tλ ) , t ∈ C . Indeed, since ( E, ∇ L ) is flat and semistable, then so is ( E, t ∇ L ) for every t . Furthermore, ( E, t ∇ L λ )is a ( tλ, L )-connection. Thus the map π is C ∗ -equivariant with respect to this C ∗ -action and thestandard one on C .Note that for λ = 0, the action (3.19) restricts to the usual C ∗ -action on the Higgs bundle modulispace by scaling the Higgs field (cf. (5.8)), and which will play an important role in section 6.Finally, observe that since it is clear that ( E, t ∇ L , t ) is semistable if and only if ( E, ∇ L , 1) is, sothe C ∗ -action (3.19) induces an isomorphism π − ( C ∗ ) ∼ = π − (1) × C ∗ = M Λ L ( r, d ) × C ∗ Grothendieck ring of varieties, motives and E -polynomials Grothendieck ring of varieties, motives and E -polynomials. The main goal of thispaper is to compare the class of the moduli spaces M Λ L ( r, d ) in the Grothendieck ring of varietiesby varying the Lie algebroid L . In this brief section we recall such ring and its basic properties.Denote by V ar C the category of quasi-projective varieties over C . For each Y ∈ V ar C , let [ Y ]denote the corresponding isomorphism class. Consider the group obtained by the free abelian groupon isomorphism classes [ Y ], modulo the relation[ Y ] = [ Y ′ ] + [ Y \ Y ′ ] , where Y ′ ⊂ Y is a Zariski-closed subset. In particular, in such group,[ Y ] + [ Z ] = [ Y ⊔ Z ] , where ⊔ denotes disjoint union. If we define the product[ Y ] · [ Z ] = [ Y × Z ] , in this quotient, then we obtain a commutative ring, known as the Grothendieck ring of varieties and denoted by K ( V ar C ). Then 0 = [ ∅ ] and 1 = [Spec( C )] are the additive and multiplicative unitsof this ring.The following is an extremely useful property of K ( V ar C ), which follows directly from the defi-nitions, and which we will repeatedly use without further notice. Proposition 4.1. If π : Y → B is an algebraic fiber bundle (thus Zariski locally trivial), with fiber F , then [ Y ] = [ F ] · [ B ] . The class of the affine line, sometimes called the Lefschetz object , is denoted by L := [ A ] = [ C ] . Of course, L n = [ A n ] = [ C n ]. We will consider the localization K ( V ar C )[ L − ], and then thedimensional completionˆ K ( V ar C ) = X r ≥ [ Y r ] L − r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ Y r ] ∈ K ( V ar C ) with dim Y r − r −→ −∞ . Notice that we have a map K ( V ar C ) −→ ˆ K ( V ar C ). Observe also that L n − K ( V ar C ), for every n , with inverse equal to − P ∞ k =0 L − kn . This is the reason why we had tointroduce the completion ˆ K ( V ar C ): there will be computations in which we will need to invertelements of the form L n or L n − Definition 4.2. Let Y be a quasi-projective variety. The class [ Y ] in K ( V ar C ) or in ˆ K ( V ar C ) iscalled the motive , or motivic class , of Y . There are other notions of motive in different, but related, categories, such as Chow motive orVoevodsky motive. These will not appear anywhere in this paper, except in section 7.2.The motive [ Y ] is an important invariant of Y , from which it is possible to read of geometricinformation, such as the E -polynomial. If Y ∈ V ar C is d -dimensional variety, with pure Hodgestructure, then its E -polynomial is defined as E ( Y ) = E ( Y )( u, v ) = d X i =0 h p,qc ( Y ) u p v q , IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 17 where h p,qc ( Y ) stands for the dimension of the compactly supported cohomology groups H p,qc ( Y ).For instance, E ( C ) = uv . Actually, the E -polynomial can be seen as a ring map(4.1) E : ˆ K ( V ar C ) −→ Z [ u, v ] (cid:20)(cid:20) uv (cid:21)(cid:21) with values in the Laurent series in uv , which takes values in Z [ u, v ] when restricted to K ( V ar C ).Hence two varieties with the same motive have the same E -polynomial. In particular, Y ′ ⊂ Y is aclosed subvariety, then E ( Y ) = E ( Y ′ ) + E ( Y \ Y ′ ) and for an algebraic fiber bundle Y → B withfiber F , we have E ( Y ) = E ( F ) E ( B ).4.2. The plan. The goal of this paper is to prove that given any two rank 1 Lie algebroids L and L ′ , over the genus g curve X , such that deg( L ) = deg( L ′ ) < − g , we have an equality of motivicclasses [ M Λ L ( r, d )] = [ M Λ L′ ( r, d )] ∈ ˆ K ( V ar C ) , where d is coprime with r . We will prove this in two steps.Take any rank 1 Lie algebroid L = ( L, [ · , · ] , δ ) with deg( L ) < − g on X . Take r ≥ d ∈ Z such that ( r, d ) = 1. We first prove that, under the given assumptions, the motive of the modulispace of L -connections is invariant with respect to the Lie algebroid structure on L , i.e., that themotive of the moduli space of rank r and degree d flat L -connections is the same as the motive ofthe moduli space of flat L = ( L, , , L )-connections) of the same rankand degree, which just means the moduli space of L − -twisted Higgs bundles or rank r and degree d , [ M Λ L ( r, d )] = [ M L − ( r, d )] . Then, we prove that [ M L − ( r, d )] is also invariant with respect to the choice of the twisting linebundle L − as long as we fix its degree. Thus, we prove that for any pair of line bundles L, L ′ withdeg( L ) = deg( L ′ ) < − g and any d coprime with r we have[ M L − ( r, d )] = [ M L ′− ( r, d )] . Combining the two results we conclude that[ M Λ L ( r, d )] = [ M Λ L′ ( r, d )] , for every L and L ′ of degree less than 2 − g and every d and r coprime.In particular, such equalities imply by (4.1) that the E -polynomials of these moduli spaces arealso equal. Then, we can go further by using results by Maulik–Shen [MS20a] or of Groechening,Wyss and Ziegler [GWZ20] which imply that for every line bundle N → X such that deg( N ) > g − 2, and every d and d ′ coprime with r , we have E ( M N ( r, d )) = E ( M N ( r, d ′ )) . Hence, this implies, together with the above equality of motivic classes, that E ( M Λ L ( r, d )) = E ( M Λ L′ ( r, d ′ )) , for every L and L ′ of degree less than 2 − g and every d and d ′ coprime to r .5. Invariance of the motive and E -polynomial with respect to the algebroidstructure As outlined in the previous section, our first objective is to prove that the motive (and, thus,the E -polynomial) of the moduli space of L -connections is invariant with respect to the algebroidstructure by proving that given a rank 1 algebroid L = ( L, [ · , · ] , δ ) such that deg( L ) < − g wehave [ M Λ L ( r, d )] = [ M L − ( r, d )] In order to prove it, we will generalize the strategy used by Hausel and Rodriguez-Villegas toshow that the moduli spaces of Higgs bundles and certain logarithmic connections share the same E -polynomials [HRV15] by using the semiprojectivity of the Hodge moduli space (moduli space of λ -connections). In our case, we will show that for every algebroid L satisfying the given hypothesisthe moduli space M Λ red L ( r, d ) is a smooth semiprojective variety over C which interpolates between M Λ L ( r, d ) and M L − ( r, d )]. The C ∗ -action on this moduli space induces Bialynicki-Birula strat-ifications on M L − ( r, d ) and M Λ red L ( r, d ) which allow us to decompose the corresponding motivesand prove the desired motivic equality.In order to follow this approach on motivic classes (Hausel and Rodriguez-Villegas results wereexplicit equalities of Hodge structures), we will first explore some results on motivic decompositionsof semiprojective varieties. Then we will prove the regularity properties of each of the involvedmoduli spaces needed to guarantee that the interpolating scheme is a semiprojective variety inducingthe desired motivic equality.5.1. Bialynicki-Birula stratification. Let Y be a smooth quasi-projective variety endowed withan algebraic C ∗ -action, denoted as Y → Y , x t · x , x ∈ Y , t ∈ C ∗ . Definition 5.1. [HRV15, Definition 1.1] The variety Y is semiprojective if the following conditionsare satisfied:(1) for each x ∈ Y the limit lim t → t · x exists in Y ;(2) the fixed-point locus of the C ∗ action Y C ∗ is proper. Every semiprojective variety Y admits a canonical stratification as follows. Let Y C ∗ = [ µ ∈ I F µ be the decomposition of the C ∗ -fixed-point loci into connected components. Then, for each F µ , wecan consider the subsets U + µ = { x ∈ M | lim t → t · x ∈ F µ } and U − µ = { x ∈ M | lim t →∞ t · x ∈ F µ } . By (1) of Definition 5.1, every point in Y belongs to exactly one of the subsets U + µ , hence there isa decomposition Y = [ µ ∈ I U + µ , called the Bialynicki-Birula decomposition of Y .The arguments in [BB73, SS4], [Kir84] and [HRV15, SS1] prove the following lemma (see [HL19,Apendix A] for a compact complete proof). Lemma 5.2. Using the above notations, the following properties hold.(1) For every µ ∈ I , the map U + µ → F µ defined by x lim t → t · x and the map U − µ → F µ givenby x lim t →∞ t · x are Zariski locally trivial fibrations in affine spaces.(2) For every µ ∈ I , U + µ is a locally closed subset of Y .(3) There exists an order of the components { µ i } ni =1 such that ⊂ U + µ ⊂ . . . ⊂ [ i ≤ j U + µ i ⊂ . . . ⊂ n [ i =1 U + µ i = Y is a stratification of Y . IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 19 For each p ∈ F µ the tangent space T p Y splits as follows T p Y = T p ( U + µ | p ) ⊕ T p F µ ⊕ T p ( U − µ | p ) . Define(5.1) N + µ = dim T p ( U + µ | p ) , N µ = dim T p F µ and N − µ = dim T p ( U − µ | p ) . Clearly N + µ = dim( U + µ ) − dim( F µ ) is the rank of the affine bundle U + µ → F µ and, as we assumedthat Y is smooth,(5.2) N + µ + N µ + N − µ = dim Y. Lemma 5.3. Let Y be a smooth complex semiprojective variety and consider the above notations.Then the motivic class of Y decomposes as [ Y ] = X µ ∈ I L N + µ [ F µ ] . Proof. As Y is semiprojective, we have a Bialynicki-Birula decomposition which, in virtue of prop-erties (2) and (3) of Lemma 5.2, forms a stratification0 ⊂ U + µ ⊂ . . . ⊂ [ i ≤ j U + µ i ⊂ . . . ⊂ n [ i =1 U + µ i = Y. As the Grothendieck class is additive on closed subvarieties, we have(5.3) [ Y ] = X µ ∈ I [ U + µ ] . By property (1) of Lemma 5.2, each µ ∈ I is a Zariski locally trivial affine fibration over F µ whosefiber has dimension N + µ , so we have(5.4) [ U + µ ] = [ C N + µ ][ F µ ] = L N + µ [ F µ ] . Now, (5.3) and (5.4) prove the lemma. (cid:3) On the other hand, we have the following proposition that is a generalization for motives of[HRV15, Corollary 1.3.3]. Proposition 5.4. Let Y be a smooth complex semiprojective variety together with a surjective C ∗ -equivariant submersion π : Y → C covering the standard scaling action on C . Then in ˆ K ( V ar C ) we have [ π − (0)] = [ π − (1)] and [ Y ] = L [ π − (0)] . Proof. Clearly, the fixed-point locus of Y is concentrated in π − (0). As π − (0) is a smooth closedsubspace of Y , then π − (0) is also a smooth semiprojective variety. Moreover, since π is C ∗ -equivariant, then the fixed points of the C ∗ -action on Y are precisely those of π − (0). Let Y C ∗ = π − (0) C ∗ = [ µ ∈ I F µ be the decomposition of the fixed-point locus into connected components. As we have discussedabove, both Y and π − (0) admit Bialynicki-Birula stratifications of the form Y = [ µ ∈ I ˜ U + µ and π − (0) = [ µ ∈ I U + µ , where ˜ U + µ = n p ∈ Y (cid:12)(cid:12)(cid:12) lim t → t · p ∈ F µ o and U + µ = n p ∈ π − (0) (cid:12)(cid:12)(cid:12) lim t → t · p ∈ F µ o . Moreover, ˜ U + µ and U + µ are affine bundles over F µ of rank ˜ N + µ and N + µ respectively. On the otherhand, let ˜ U − µ = n p ∈ Y (cid:12)(cid:12)(cid:12) lim t →∞ t · p ∈ F µ o and U − µ = n p ∈ π − (0) (cid:12)(cid:12)(cid:12) lim t →∞ t · p ∈ F µ o . Then ˜ U − µ and U − µ are also affine bundles over F µ , of rank ˜ N − µ and N − µ respectively, and we havethe following decomposition of the tangent spaces T p Y and T p ( π − (0)) at each p ∈ F µ , T p Y = T p ( ˜ U + µ | p ) ⊕ T p ( ˜ U − µ | p ) ⊕ T p F µ and T p ( π − (0)) = T p ( U + µ | p ) ⊕ T p ( U − µ | p ) ⊕ T p F µ . Using the smoothness assumption, this yields(5.5) dim Y = ˜ N + µ + ˜ N − µ + dim( F µ ) and dim π − (0) = N + µ + N − µ + dim( F µ ) . Since C ∗ -action contracts the points of Y to the 0 fibre of π − (0), then, for each µ ∈ I , all thepoints points p of Y such that lim t →∞ t · p ∈ F µ must lie in π − (0). Thus ˜ U − µ = U − µ and we have˜ N − µ = N − µ . On the other hand, as π : Y → C is a submersion of smooth varieties, we have thatdim Y = dim π − (0) + 1, so from (5.5) we conclude that for each µ we have(5.6) ˜ N + µ = N + µ + 1 . Thus, using the Bialynici-Birula decompositions of Y and π − (0), we can apply Lemma 5.3 todecompose the corresponding motives as [ π − (0)] = P µ ∈ I L N + µ [ F µ ], and(5.7) [ Y ] = X µ ∈ I L ˜ N + µ [ F µ ] = L [ π − (0)] . On the other hand, the C ∗ -action yields an isomorphism π − ( C ∗ ) ∼ = π − (1) × C ∗ , so we can write[ Y ] = [ π − (0)] + [ π − ( C ∗ )] = [ π − (0)] + ( L − π − (1)] , and, by (5.7), this shows that [ π − (1)] = [ π − (0)] in ˆ K ( V ar C ). (cid:3) Semiprojectivity of the moduli space of Higgs bundles. Let L be a line bundle overthe genus g curve X . In this section we show the well-known fact that the moduli space M L ( r, d )of L -twisted Higgs bundles over X is a smooth semiprojective variety, under the usual conditionson the degree of L and on r and d . In the next section, we will prove the analogous result for the L -Hodge moduli space and that will a more substantial amount of work.The moduli M L ( r, d ) admits a natural C ∗ -action by scaling the Higgs field(5.8) t · ( E, ϕ ) = ( E, tϕ ) . Note that this is a particular case of (3.19).Recall now the Hitchin map from (2.1). Then the Hitchin base W = L ri =1 H ( L i ) also admits anatural C ∗ -action given by t · ( s , . . . , s r ) = ( ts , t s , . . . , t r s r ) , which makes the Hitchin map H : M L ( r, d ) → W a C ∗ -equivariant map.Let us first prove that the C ∗ -action on M L ( r, d ) verifies the first condition on Definition 5.1. Lemma 5.5. Let ( E, ϕ ) be a semistable Higgs bundle on X . Then the limit lim t → ( E, tϕ ) existsin M L ( r, d ) .Proof. As H : M L ( r, d ) → W is C ∗ -equivariant, we havelim t → H ( E, tϕ ) = lim t → t · H ( E, ϕ ) = 0 , IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 21 thus the map C ∗ → W given by t H ( E, tϕ ) extends to C → W . By Lemma 2.4, H is proper,so by the valuative criterion of properness the map C ∗ → M L ( r, d ) given by t ( E, tϕ ) must alsoextend to a map C → M L ( r, d ), thus providing the desired limit. (cid:3) Now we consider the second condition on Definition 5.1. Lemma 5.6. The fixed-point set of the C ∗ -action on M V ( r, d ) is a proper scheme contained in H − (0) .Proof. Since the Hitchin map H is C ∗ -equivariant, the fixed-point set M L ( r, d ) C ∗ must be a closedsubset of H − ( W C ∗ ) = H − (0). By Lemma 2.4, H is proper, so H − (0) is proper and hence so is M L ( r, d ) C ∗ . (cid:3) Proposition 5.7. Suppose that r and d are coprime. Suppose the line bundle L is such that deg( L ) > g − . Then the moduli space M L ( r, d ) is a smooth complex semiprojective variety.Proof. By Lemma 2.2 the moduli space M L ( r, d ) is a smooth complex variety. Then Lemmas 5.5and 5.6 prove that the action t · ( E, ϕ ) = ( E, tϕ ) satisfies the semiprojectivity conditions. (cid:3) Semiprojectivity of the moduli space of Λ red L -modules. Our aim in this section is toprove that the L -Hodge moduli space M Λ red L ( r, d ) is also a smooth semiprojective variety, wheneverrk( L ) = 1, deg( L ) < − g and r and d are coprime. This is going to take considerably more effortthan the case of Higgs bundles from the previous section. In particular, we will need to explicitlyuse both interpretations, provided by Theorem 3.10, of the points parameterized by M Λ red L ( r, d ),namely semistable Λ red L -modules and semistable ( λ, L )-connections (which are automatically flatsince rk( L ) = 1). For instance, the proof that condition (1) of Definition 5.1 is goint to be provedby closely following an argument by Simpson, via Λ-modules, but all the arguments required toprove smoothness of the moduli will be carried out by taking the L -connections point of view,because the deformation theory of such objects has been developed, contrary to the deformationtheory of Λ-modules.Recall the C ∗ -action (3.19) on the L -Hodge moduli space M Λ red L ( r, d ) by sending t · ( E, ∇ L , λ ) ( E, t ∇ L , tλ ) . We will show that with this C ∗ -action M Λ red L ( r, d ) becomes a smooth semiprojective variety. Recallalso the surjective map π : M Λ red L ( r, d ) → C defined in (3.16). Lemma 5.8. Let L be any Lie algebroid on X . Let ( E, ∇ L , λ ) ∈ M Λ red L ( r, d ) be any ( λ, L ) -connection. Then the limit lim t → ( E, t ∇ L , tλ ) exists in π − (0) ⊂ M Λ red L ( r, d ) .Proof. The proof is analogous to [Sim97, Corollary 10.2]. We will use (3.13) to consider the Λ L λ -module ( E, ∇ Λ L , λ ), with ∇ Λ L : Λ L λ ⊗ E → E , instead of the ( λ, L )-connection (i.e. L λ -connection)( E, ∇ L , λ ).Consider the C ∗ -flat family of relative Λ red L | X × C ∗ -modules over π C : X × C → C , where π C ( x, t ) = tλ , given by (cid:16) E , ∇ Λ red L (cid:17) = ( π ∗ X E, tπ ∗ X ∇ Λ L ) , where π X : X × C ∗ → X is the projection. For t = 0, the generic fibre of the family is semistable, asfor any t = 0 we clearly have that the corresponding ( tλ, L )-connection ( E, t ∇ L , tλ ) is semistableif and only if ( E, ∇ L , λ ) is semistable. By [Sim97, Theorem 10.1], there exists a family ( E , ∇ Λ red L )of Λ red L -modules over π C : X × C → C , flat over C , such that ( E , ∇ Λ red L ) | X × C ∗ ∼ = ( π ∗ X E, t ∇ Λ L ) andsuch that ( E , ∇ Λ red L ) | X ×{ } is semistable. Thus, ( E , ∇ Λ red L ) | X ×{ } ∈ π − (0) is the limit at t = 0 ofthe C ∗ -orbit of ( E, ∇ L , λ ) in M Λ red L ( r, d ). (cid:3) Now we will focus on the regularity of the L -Hodge moduli space. Now we will use L -connectionsin our study. We start with a simple lemma. Lemma 5.9. Let L be any Lie algebroid on the curve X , and let ( E, ∇ L ) and ( E ′ , ∇ ′L ) be semistable L -connections.(1) If µ ( E ) > µ ( E ′ ) then Hom(( E, ∇ L ) , ( E ′ , ∇ ′L )) = 0 .(2) Suppose ( E, ∇ L ) and ( E ′ , ∇ ′L ) are stable and µ ( E ) = µ ( E ′ ) . Let ψ ∈ Hom(( E, ∇ L ) , ( E ′ , ∇ ′L )) be a non-zero map. Then it is an isomorphism.(3) If ( E, ∇ L ) is stable, then its the only endomorphisms are the scalars, i.e. End( E, ∇ L ) ∼ = C .Proof. The proof is classical. (1) and (2) are completely analogous to [BGL11, Lemma 3.2]. Toprove (3), let α : ( E, ∇ L ) → ( E, ∇ L ) be any endomorphism. Choose any point x ∈ X . Then α induces an endomorphism of the fiber E x . Let λ ∈ C be an eigenvalue of such morphism. As ∇ L is C -linear, then α − λ Id ∈ End( E, ∇ L ). By (2), we know that this map is either zero or anisomorphism. Nevertheless, we know that λ is an eigenvalue of α x , so α − λ Id has a nontrivialkernel at the fiber over x and, therefore, it cannot be an isomorphism. Thus, α − λ Id = 0, so α = λ Id. (cid:3) Given any Lie algebroid L , the deformation theory of flat L -connections was studied in Chapter 5of [Tor11]. In particular, it follows from Theorem 47 of loc. cit. that that the Zariski tangent spaceto the moduli space at an integrable L -connection ( E, ∇ L ) is isomorphic to H ( X, C • ( E, ∇ L )),where C • ( E, ∇ L ) is the complex(5.9) C • ( E, ∇ L ) : End( E ) [ − , ∇ L ] −→ End( E ) ⊗ Ω L [ − , ∇ L ] −→ . . . [ − , ∇ L ] −→ End( E ) ⊗ Ω rk( L ) L and that the obstruction for the deformation theory lies in H ( X, C • ( E, ∇ L )).In the next lemma we only consider rank 1 Lie algebroids. Lemma 5.10. Let L be Lie algebroid of rank on X and let ( E, ∇ L ) be a stable L -connection ofrank r and degree d . Then the dimension of the Zariski tangent space to M Λ L ( r, d ) at ( E, ∇ L ) isgiven by dim T ( E, ∇ L ) M Λ L ( r, d ) = 1 − r deg( L ) + dim (cid:0) H ( C • ( E, ∇ L )) (cid:1) . Proof. Since rk( L ) = 1, then the deformation complex (5.9) only has two terms, C • ( E, ∇ L ) : End( E ) [ − , ∇ L ] −→ End( E ) ⊗ Ω L , thus the hypercohomology of the complex C • ( E, ∇ L ) fits in the following exact sequence0 −→ H ( C • ( E, ∇ L )) −→ H (End( E )) −→ H (End( E ) ⊗ Ω L ) −→ H ( C • ( E, ∇ L )) −→ H (End( E )) −→ H (End( E ) ⊗ Ω L ) −→ H ( C • ( E, ∇ L )) −→ . Therefore,dim( H ( C • ( E, ∇ L ))) = dim( H ( C • ( E, ∇ L )))+dim( H ( C • ( E, ∇ L )))+ χ (End( E ) ⊗ Ω L ) − χ (End( E )) . We can compute each term in the previous expression working in an analogous way to [BGL11,Proposition 3.3]. By construction, H ( C • ( E, ∇ L )) corresponds to sections of End( E ) belongingto the kernel of the commutator [ − , ∇ L ], so H ( C • ( E, ∇ L )) ∼ = H (End( E, ∇ L )). By stability of( E, ∇ L ), point (3) of Lemma 5.9 shows that dim( H ( C • ( E, ∇ L ))) = dim( H (End( E, ∇ L ))) = 1.On the other hand, χ (End( E )) = r (1 − g ) and χ (End( E ) ⊗ Ω L ) = − r deg( L ) + r (1 − g ). Hencedim T ( E, ∇ L ) M Λ L ( r, d ) = dim( H ( C • ( E, ∇ L ))) = 1 − r deg( L ) + dim (cid:0) H ( C • ( E, ∇ L )) (cid:1) , as claimed. (cid:3) IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 23 Let L = ( V, [ · , · ] , δ ) be any Lie-algebroid on X , so no constrains on the algebraic vector bundle V .Recall the associated Lie algebroid L λ given by (3.14). Then L = ( V, , 0) is the trivial algebroidwith underlying bundle V . Now we aim to study the first order deformations of a semistableintegrable L -connection ( E, ∇ L ) of rank r and degree d (i.e. a semistable V ∗ -twisted Higgsbundle) inside the not just of π − (0) = M Λ L ( r, d ) = M V ∗ ( r, d ) but, rather insider the L -Hodgemoduli space M Λ red L ( r, d ). So we allow deformations of ( E, ∇ L ) not only along π − (0) but also to π − ( λ ) for some λ = 0. Recall that here π : M Λ red L ( r, d ) → C is the projection (3.16). Lemma 5.11. Let L = ( V, [ · , · ] , δ ) be any Lie algebroid. Then the Zariski tangent space to the L -Hodge moduli space M Λ red L ( r, d ) at a point ( E, ∇ L , lying over the fiber is T ( E, ∇ L , M Λ red L ( r, d ) ∼ = ( c, C, λ ε ) ∈ (cid:18) C ( U , End( E )) × C ( U , End( E ) ⊗ Ω L ) × C (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂c = 0 ∂C = ˜ ∇ L c + λ ε ω ˜ ∇ L C = − λ ε d L ( ∇ L ) n ( ∂η, ˜ ∇ L η, (cid:12)(cid:12)(cid:12) η ∈ C ( U , End( E )) o where U = { U α } is an open cover of X such that E is trivial over each open subset U α , where ω ∈ C ( U , End( E ) ⊗ Ω L ) is some -cocycle. Moreover, if π : M Λ red L ( r, d ) −→ C is the map sending ( E, ∇ L , λ ) to λ then its differential dπ : T ( E, ∇ L , M Λ red L ( r, d ) −→ C is just [( c, C, λ ε )] λ ε .Proof. We will proceed analogously to [Tor11, § E, ∇ L , ∈ π − λ (0). Then, by definition,( E, ∇ L ) is a V ∗ -twisted semistable Higgs bundle. Fix an open cover U = { U α } of X such that E istrivial over each open subset U α . We will use the usual notation U αβ := U α ∩ U β , etc. to denote theintersections of the open subsets. For each α and β , let g αβ : U αβ −→ GL( r, C ) be the transitionfunctions of E , and let G α be the matrix valued function representing the Higgs field ∇ L in thelocal coordinates over U α .The first order deformations of ( E, ∇ L ) are given by families of Λ red L -module over each Spec( C [ ε ] /ε ) −→ Spec( C [ λ ]). The possible maps Spec( C [ ε ] /ε ) −→ Spec( C [ λ ]) are given by the choice of the imageof ε , which must be of the form λ ε λ for some λ ε ∈ C . Fix the value λ ε . Then a family overSpec( C [ ε ] /ε ) −→ Spec( C [ λ ]) for that parameter λ ε is a triple ( E ′ , ∇ ′L , λ ε ε ) such that E ′ is a vectorbundle over X × Spec( C [ ε ] /ε ) and ∇ ′L is a ( λ ε ε, L )-connection over E ′ . Relative to the open cover { U α × Spec( C [ ε ] /ε ) } , we can write the transition functions of E ′ as g ′ αβ = g αβ + εg αβ , where g αβ ∈ O X ( U αβ ) × gl r . Similarly, we can write locally ∇ ′L over U α as ∇ ′L ,α = λ ε εd L + G α + εG α , where G α ∈ Ω L ( U α ) ⊗ gl r .Moreover, define the 1-cocycle c ∈ C ( U , End( E )) in the following way. Given an isomorphismFor each U αβ , let(5.10) ( c αβ ) ( α ) = g αβ g βα , where we use the notation ( − ) ( α ) to denote the matrix with respect to the basis given by thetrivialization over U α .Since E and E ′ are vector bundles, the following equations must be satisfied: g αβ g βα = 1 , g ′ αβ g ′ βα = 1 , g αβ g βγ g γα = 1 and g ′ αβ g ′ βγ g ′ γα = 1 . A direct computation with the first two equations yields g βα = − g βα g αβ g βα , thus(5.11) c ( α ) βα = g αβ c ( β ) βα g βα = g αβ g βα = − g αβ g βα = − c ( α ) αβ . On the other hand, the last couple of equations imply g αβ g βγ g γα + g αβ g βγ g γα + g αβ g βγ g γα = 0. Wecan rewrite each summand of the last equation in terms of the cocycle c as follows g αβ g βγ g γα = g αβ g βα = c ( α ) αβ , g αβ g βγ g γα = g αβ g βγ g γβ g βα = c ( α ) βγ and g αβ g βγ g γα = g αγ g γα = c ( α ) γα . Thus, we obtain(5.12) c ( α ) αβ + c ( α ) βγ + c ( α ) γα = 0 , so c ∈ C ( U , End( E )) in (5.10) is a closed 1-cocycle.On the other hand, as ( E, ∇ L ) is a V ∗ -twisted Higgs bundle and ( E ′ , ∇ ′L , λ ε ε ) is a ( λ ε ε, L )-connection, then, on U αβ , we must have ∇ L ,β = g βα ∇ L ,α g αβ and ∇ ′L ,β = g βα ∇ ′L ,α g αβ . Expandingeach side of the last expression and taking into account that ε = 0 we obtain λ ε εd L + G β + εG β = g ′ βα ( λ ε εd L + G α + εG α ) g ′ αβ = λ ε εd L + λ ε εg βα d L g αβ + g βα G α g αβ + ε (cid:0) g βα G α g αβ + g βα G α g αβ + g βα G α g αβ (cid:1) , hence we conclude that(5.13) G β = g βα G α g αβ + g βα G α g αβ + g βα G α g αβ + λ ε g βα d L g αβ . Define the 0-cocycle C ∈ C ( U , End( E ) ⊗ Ω L ) by taking C ( α ) α = G α , for each α . Then, the equality (5.13) written in terms of the cocycles c and C , reads as C ( α ) β = g αβ G β g βα = g αβ g βα G α + G α + G α g αβ g βα + λ ε ( d L g αβ ) g βα = − c ( α ) αβ G α + C ( α ) α + G α c ( α ) αβ + λ ε ( d L g αβ ) g βα = C ( α ) α + h G α , c ( α ) αβ i + λ ε ( d L g αβ ) g βα . Let us finally consider the 1-cocycle ω ∈ C ( U , End( E ) ⊗ Ω L ) defined as(5.14) ω ( α ) αβ = ( d L g αβ ) g βα , for each α, β . Observe that( d L g αβ ) g βα + g αβ ( d L g βα ) = d L ( g α βg β α ) = d L (1) = 0so d L g βα = − g βα ( d L g αβ ) g βα , and we get ω ( α ) βα = g αβ ( d L g βα ) g αβ g βα = g αβ ( d L g βα ) = − ( d L g αβ ) g βα = − ω ( α ) αβ , thus(5.15) C ( α ) β − C ( α ) α = [ G α , c ( α ) αβ ] + λ ε ω ( α ) αβ . On the other hand 0 = d L (1)= d L ( g αβ g βγ g γα )= ( d L g αβ ) g βγ g γα + g αβ ( d L g βγ ) g γα + g αβ g βγ ( d L g γα )= ( d L g αβ ) g βα + g αβ ( d L g βγ ) g γβ g βα + g αγ ( d L g γα ) g αγ g γα = ω ( α ) αβ + ω ( α ) βγ + ω ( α ) γα .ω is hence a closed 1-cocycle. IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 25 Finally, flatness of ∇ L and ∇ ′L implies that for each α we have 0 = ∇ L ,α = G α ∧ G α and0 = ( ∇ ′L ,α ) = ( λ ε εd L + G α + εG α ) = λ ε εd L ( G α ) + G α ∧ G α + εG α ∧ G α + εG α ∧ G α . Denote by d L ( ∇ L ) ∈ C ( U , End( E ) ⊗ Ω L ) the 0-cocycle defined locally as d L ( ∇ L ) ( α ) = d L ( G α ).Then, we can write the previous equation in terms of C and d L ( ∇ L ) as follows. The flatnessequations yield G α ∧ G α + G α ∧ G α = − λ ε d L ( G α ), so(5.16) ˜ ∇ L C ( α ) α = − λ ε d L ( ∇ L ) ( α ) . where ˜ ∇ L = [ − , ∇ L ] : End( E ) → End( E ) ⊗ Ω L is the induced map on End( E ) by ∇ L . Wecan express equations (5.11), (5.12), (5.15) and (5.16) globally as follows. Each deformation of( E, ∇ L , 0) is given by a triple ( c, C, λ ε ) , with c ∈ C ( U , End( E )), C ∈ C ( U , End( E ) ⊗ Ω L ) and λ ε ∈ C , such that(5.17) ∂c = 0 ∂C = ˜ ∇ L c + λ ε ω ˜ ∇ L C = − λ ε d L ( ∇ L ) . On the other hand, two such triples ( c, C, λ ε ) and (¯ c, ¯ C, ¯ λ ε ) give rise to equivalent deformations( E ′ , ∇ ′L , λ ε ) and ( E ′ , ∇ ′L , ¯ λ ε ) of ( E, ∇ L , 0) if and only if λ ε = ¯ λ ε and there exists a 0-cocycle oflocal automorphisms ξ α : U α × Spec( C [ ε ] /ε ) → GL r of the form ξ α = Id + εη α with η α : U α → gl r such that(5.18) g ′ αβ ( g ′ αβ ) − = ξ β ξ − α and(5.19) ∇ ′L ,α = ξ − α ∇ ′L ,α ξ α . Here, following the previous notation, we write E ′ and ∇ ′L locally in the corresponding trivializationover U as g ′ αβ = g αβ + εg αβ = g αβ + εg αβ and ∇ ′L ,α = ¯ λ ε εd L + G α + εG α = λ ε εd L + G α + εG α . We have that ξ β ξ − α = (Id + εη β )(Id − εη α ) = Id + ε ( η β − η α ) and g ′ αβ ( g ′ αβ ) − = ( g αβ + εg αβ )( g βα + εg βα ) = Id + ε ( g αβ g βα + g αβ g βα ) = Id + ε ( c αβ − c αβ ) . From (5.18), we obtain c αβ − c αβ = η β − η α , so c − c = ∂η . On the other hand, ξ − α ∇ ′L ,α ξ α = λ ε εd L + λ ε ε (Id − εη α ) d L (Id + εη α ) + (Id − εη α ) G α (Id + εη α ) + ε (Id − εη α ) G α (Id + εη α )= λ ε εd L + G α + ε (cid:0) G α η α − η α G α + G α (cid:1) . Hence, (5.19) yields G α = [ G α , η α ] + G α or, equivalently, C α − C α = [ G α , η α ] = ˜ ∇ L η α , and thus, C − C = ˜ ∇ L η . We finally conclude that the deformation space of M Λ red L ( r, d ) at ( E, ∇ L , 0) is T ( E, ∇ L , M Λ red L ( r, d ) ∼ = ( c, C, λ ε ) ∈ (cid:18) C ( U , End( E )) × C ( U , End( E ) ⊗ Ω L ) × C (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂c = 0 ∂C = ˜ ∇ L c + λ ε ω ˜ ∇ L C = − λ ε d L ( ∇ L ) n ( ∂η, ˜ ∇ L η, (cid:12)(cid:12)(cid:12) η ∈ C ( U , End( E )) o and the map dπ : T ( E, ∇ L , M Λ red L ( r, d ) −→ C is just [( c, C, λ ε )] λ ε . (cid:3) Remark 5.12. While the deformation theory of M Λ L ( r, d ) is governed by a nice deformationcomplex (as expectable for this type of deformation problems), we have not been able to provide, ingeneral, a natural cohomological interpretation for the deformation theory of M Λ red L ( r, d ) .This can, however, be achieved in certain cases. For instance, suppose rk( L ) = 1 and suppose E is a stable vector bundle over X . By Corollary 3.14, E admits an integrable L -connection ∇ L , : E → E ⊗ Ω L . Let us consider the family ( π ∗ X E, λπ ∗ X ∇ L , , λ ) over X × C , where π X : X × C → X is the projection, and consider the infinitesimal family over Spec( C [ ε ] /ε ) around given by ( π ∗ X E, επ ∗ X ∇ L , , ε ) . We can now express the family locally in a similar way to the previousLemma. Given an open cover { U α × Spec( C [ ε ] /ε ) , let g αβ : U αβ → GL( r, C ) be the transitionfunctions of E . As ε ∇ L , is an ( ε, L ) -connection, we can express it locally over U α as ε ∇ L , ,α = εd L + εG α for some G α ∈ Ω L ( U α ) ⊗ gl r . As G α comes from an actual L -connection, we must have ε ∇ L , ,β = g βα ε ∇ L , ,α g αβ on the overlaps U αβ . Plugging in the local representation of ∇ L yields εd L + εG β = εd L + εg βα d L g αβ + εg βα G α g αβ . Thus, g αβ G β g βα = d L g αβ g βα + G α = ω ( α ) αβ + G α . Let Ω ∈ C ( U , End( E ) ⊗ Ω L ) be the -cocycle defined as Ω ( α ) α = G α . Then ω ( α ) αβ = Ω ( α ) β − Ω ( α ) α , hence ω = ∂ Ω . Now, since L has rank , the integrability condition is automatic and so T ( E, ∇ L , M Λ red L ( r, d ) ∼ = (cid:26) ( c, C, λ ε ) ∈ (cid:18) C ( U , End( E )) × C ( U , End( E ) ⊗ Ω L ) × C (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∂c = 0 ∂C = ˜ ∇ L c + λ ε ∂ Ω (cid:27)n ( ∂η, ˜ ∇ L η, (cid:12)(cid:12)(cid:12) η ∈ C ( U , End( E )) o . Then, the map ( c, C, λ ε ) ( c, C − λ ε Ω , λ ε ) induces an isomorphism T ( E, ∇ L , M Λ red L ( r, d ) ∼ = (cid:26) ( c, D, λ ε ) ∈ (cid:18) C ( U , End( E )) × C ( U , End( E ) ⊗ Ω L ) × C (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∂c = 0 ∂D = ˜ ∇ L c (cid:27)n ( ∂η, ˜ ∇ L η, (cid:12)(cid:12)(cid:12) η ∈ C ( U , End( E )) o ∼ = H ( C • ( E, ∇ L )) × C , yielding the desired cohomological interpretation of the deformation space. It is clear in this casethat H ( C • ( E, ∇ L )) parameterizes the deformations along the fiber p − λ (0) of the projection p λ : M Λ red L ( r, d ) → C from (3.16) , and C parameterizes the deformations of λ i.e. along the target of p λ .However, a vector bundle E need not admit an integrable L -connection and, therefore, the as-sociated cocycle ω in (5.14) may not be exact. Moreover, for higher rank L , the presence of theintegrability condition breaks the previous trivialization of the deformation theory and the corre-sponding cohomological description.We believe that the somehow unnatural presentation of the deformation theory of Lemma 5.11 isa reflection of the fact that the moduli space of Higgs bundles admits a broader range of deformationsthan the ones considered in this section, as suggested in [Tor11, Section 7.3] . More precisely, foreach family of Lie algebroid structures over V , L −→ X × T on T , we obtain a moduli space M Λ L ( r, d ) −→ T over T . Each family going through the trivial Lie algebroid ( V, , gives rise IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 27 to a deformation of the moduli space M V ∗ ( r, d ) . More generally, the infinitesimal deformationsof the trivial algebroid structure give rise to deformations of M V ∗ ( r, d ) . We expect that if weconsidered the whole space of such infinitesimal deformations, then that space would get indeed anatural cohomological interpretation.In Lemma 5.11 we are only considering “radial sections” of such deformation space, correspond-ing to families over C of the form (3.14) , therefore obtaining “sections” or “cuts” of the wholedeformation space, and these deformations do not seem to exhibit a cohomological description any-more. The preceding Lemmas 5.10 and 5.11 now allow us to show that the L -Hodge moduli space underthe following conditions. Lemma 5.13. Let r ≥ and d be coprime and L = ( L, [ · , · ] , δ ) be a Lie algebroid such that rk( L ) = 1 and deg( L ) < − g . Then:(1) M Λ L ( r, d ) is a smooth variety, whose connected components have all dimension − r deg( L ) ;(2) M Λ red L ( r, d ) is a smooth variety, whose connected components have all dimension − r deg( L ) .Moreover, the map π : M Λ red L ( r, d ) → C from (3.16) is a smooth submersion.Proof. Let us start by proving that M Λ L ( r, d ) is a smooth variety of dimension 1 − r deg( L ). Let( E, ∇ L ) ∈ M Λ L ( r, d ). Consider the map C ∗ → M Λ red L ( r, d ) given by t ( E, t ∇ L , t ). By Lemma5.8, the limit of the C ∗ -action at zero exists, so this map extends to a curve γ : C → M Λ red L ( r, d ),which is a section of the map π : M Λ red L ( r, d ) → C . Let ( E , ∇ L , , 0) := γ (0).Consider the map ρ : C → Z given by ρ ( λ ) = dim (cid:16) γ ∗ T M Λ red L ( r, d ) (cid:17)(cid:12)(cid:12)(cid:12) λ . We know from (3.17) that the C ∗ -action produces an isomorphism between any nonzero fiber of π and π − (1), yielding an isomorphism π − ( C ∗ ) ∼ = M Λ L ( r, d ) × C ∗ . Therefore, for every λ = 0 we have ρ ( λ ) = dim (cid:16) γ ∗ T M Λ red L ( r, d ) (cid:17)(cid:12)(cid:12)(cid:12) λ = dim T ( E, ∇ L ) M Λ L ( r, d ) + 1The map ρ is upper semicontinuous, so applying Lemma 5.10, we get(5.20) ρ (0) ≥ dim T ( E, ∇ L ) M Λ L ( r, d ) + 1 = 2 − r deg( L ) + dim (cid:0) H ( C • ( E, ∇ L )) (cid:1) ≥ − r deg( L ) . Note that this is where we used the fact that r and d are coprime, because in such a case everysemistable L -connection is actually stable, so Lemma 5.10 applies at every point of M Λ L ( r, d ).On the other hand, we have thatdim T ( E , ∇ L , , M Λ red L ( r, d ) = dim ker dπ + dim Im dπ ≤ dim ker dπ + 1 . The kernel of dπ | ( E , ∇ L , , can be computed explicitly through our formula for the Zariski tangentspace given in Lemma 5.11ker dπ | ( E , ∇ L , , ∼ = ( c, C, ∈ (cid:18) C ( U , End( E )) × C ( U , End( E ) ⊗ Ω L ) × C (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂c = 0 ∂C = ˜ ∇ L , c ˜ ∇ L C = 0 n ( ∂η, ˜ ∇ L , η, (cid:12)(cid:12)(cid:12) η ∈ C ( U , End( E )) o ∼ = H (cid:18) End( E ) [ − , ∇ L , ] −→ End( E ) ⊗ Ω L (cid:19) ∼ = T ( E , ∇ L , ) M L − ( r, d ) . Using Lemma 2.2, we know that M L − ( r, d ) is smooth of dimension 1 − r deg( L ), sodim ker dπ | ( E , ∇ L , , = 1 − r deg( L )and, therefore,(5.21) ρ (0) = dim T ( E , ∇ L , , M Λ red L ( r, d ) ≤ dim ker dπ | ( E , ∇ L , , + 1 ≤ − r deg( L ) . From (5.20) and (5.21), we conclude that ρ (0) = 2 − r deg( L ) , that is, H ( C • ( E, ∇ L )) = 0 for all ( E, ∇ L ) ∈ M Λ L ( r, d ).The upshot is that the deformation theory at ( E, ∇ L ) is unobstructed and the dimension of theZariski tangent space T ( E, ∇ L ) M Λ L ( r, d ) is 1 − r deg( L ) for each ( E, ∇ L ). As a consequence, by[FM98] the moduli space M Λ L ( r, d ) is a smooth variety of dimension 1 − r deg( L ), completing theproof of (1).Let us now consider point (2), i.e. the regularity of the L -Hodge moduli space and the map π .First note that we have the isomorphism π − ( C ∗ ) ∼ = M Λ L ( r, d ) × C ∗ , and by (1), M Λ L ( r, d ) is smooth, so π − ( C ∗ ) is also smooth and the map π | π − ( C ∗ ) : π − ( C ∗ ) → C ∗ is clearly a smooth submersion. Therefore, it is enough to study the deformation of the elements inthe zero fiber of π and then check that the dimension of the corresponding Zariski tangent spacescoincides with the expected one and that the differential of the map π is surjective at those points.Let us consider the subvariety π − ( C ∗ ) ⊂ M Λ red L ( r, d )given by the closure of π − ( C ∗ ) in the L -Hodge moduli. The C ∗ -flow through any point of M Λ red L ( r, d ) has a limit at 0 in π − (0), due to Lemma 5.8, so π − (0) ∩ π − ( C ∗ ) = ∅ . By (1) and (3.17), we have dim π − ( λ ) = 1 − r deg( L ) for every λ = 0. Hence, by semicontinuity,each component of π − (0) ∩ π − ( C ∗ ) has dimension at least 1 − r deg( L ). By Lemma 2.2 thevariety π − (0) = M L − ( r, d ) is smooth and connected of dimension 1 − r deg( L ), so we concludethat π − (0) ∩ π − ( C ∗ ) = π − (0)and thus π − ( C ∗ ) = M Λ red L ( r, d ).As π − ( C ∗ ) ∼ = M Λ L ( r, d ) × C ∗ , then we know that for any ( E ′ , ∇ ′L , λ ′ ) ∈ π − ( C ∗ ) we havedim T ( E ′ , ∇ ′L ,λ ′ ) M Λ red L ( r, d ) = dim T ( E ′ , ∇ ′L /λ ) M Λ L ( r, d ) + 1 = 2 − r deg( L ) IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 29 so, by semicontinuity, for each ( E, ∇ L , ∈ π − (0) ∩ π − ( C ∗ ) = π − (0) we have2 − r deg( L ) ≤ dim T ( E, ∇ L , M Λ red L ( r, d ) = 1 − r deg( L ) + dim Im dπ ≤ − r deg( L ) . Hence, we havedim T ( E, ∇ L , M Λ red L ( r, d ) = 2 − r deg( L ) and dim Im dπ | ( E, ∇ L , = 1 , for each ( E, ∇ L , ∈ π − (0). So the map π is a smooth submersion with equidimensional fibersand the dimension of the Zariski tangent space of the moduli space M Λ red L ( r, d ) at any point isconstant and coincides with the dimension of the scheme which is therefore smooth. (cid:3) Remark 5.14. We will see in Theorem 7.2 below that, under the stated conditions, M Λ L ( r, d ) isactually connected, hence so is M Λ red L ( r, d ) . Combining the previous result yields the desired semiprojectivity and regularity of the L -Hodgemoduli space. Theorem 5.15. Let X be a smooth projective curve of genus g ≥ . Let L = ( L, [ · , · ] , δ ) be a rankLie algebroid on X such that rk( L ) = 1 and deg( L ) < − g . Then the moduli space M Λ red L ( r, d ) ,with the C ∗ -action t · ( E, ∇ L , λ ) = ( E, t ∇ L , tλ ) , is a semiprojective variety.If, moreover, r and d are coprime and r ≥ , then it is a smooth semiprojective variety andthe map π : M Λ red L ( r, d ) → C from (3.16) is a surjective C ∗ -equivariant submersion covering thestandard action on C .Proof. By the GIT construction of [Sim94], M Λ red L ( r, d ) is a complex quasi-projective variety.Lemma 5.8 ensures that for every ( E, ∇ L , λ ) ∈ M Λ red L ( r, d ) the limit lim t → ( E, t ∇ L , tλ ) exists.Moreover, the fixed-point set corresponds to the fixed-point set of the C ∗ -action in π − (0), whichcoincides with the moduli space of L − -twisted Higgs bundles. Then Lemma 5.6 implies that M Λ red L ( r, d ) C ∗ is proper. So the M Λ red L ( r, d ) is a semiprojective variety. In the coprime case, thesmoothness claim follows from Lemma 5.13. (cid:3) Remark 5.16. We expect that the above results still hold true for higher rank Lie algebroids L =( V, [ · , · ] , δ ) with V polystable such that µ ( V ) < − g . Indeed most of the above arguments gothrough immediately in this situation, except in two related steps. First, Lemma 5.10 really requiresrank Lie algebroids, because it is only in that setting that the deformation complex (5.9) has onlytwo terms. If rk( L ) = n , then (5.9) has n terms, and it is not clear how to proceed to computethe dimension of M Λ L ( r, d ) . Similarly, Lemma 2.2 also requires the twisting to be a line bundle,and the corresponding result for higher rank twistings is not yet known, by similar reasons (noticethat the infinitesimal study carried out in [BR94] is done for any twisting, but it does not take intoaccount the integrability condition on the Higgs field). Invariance of the motive and E -polynomial with respect to the algebroid structure. We continue with our fixed base curve X , of genus g ≥ 2. Now that we have established the requiredregularity conditions and the semiprojectivity of the L -Hodge moduli space, for L = ( L, [ · , · ] , δ ) ofrank 1 and degree less than 2 − g , we can address the invariance of the motive with respect to thealgebroid structure of L by keeping L fixed, and when we vary λ in C . Hence the variation on theLie algebroid structure we are considering is the one given by (3.14). By (3.17), this is clearly trueif one changes the Lie algebroid structure by varying from λ ∈ C ∗ to λ ′ ∈ C ∗ , so the main point isthat the motivic class remains unchanged when we go to the trivial algebroid structure, thus λ = 0.This is one of the contents of the next theorem. Theorem 5.17. Let L = ( L, [ · , · ] , δ ) be Lie algebroid on X such that L is a line bundle with deg( L ) < − g . If r and d are coprime, then the following equalities hold in ˆ K ( V ar C )[ M Λ L ( r, d )] = [ M L − ( r, d )] , [ M Λ red L ( r, d )] = L [ M L − ( r, d )] and we have an isomorphism of Hodge structures H • ( M Λ L ( r, d )) ∼ = H • ( M L − ( r, d )) In particular, E ( M Λ L ( r, d )) = E ( M L − ( r, d )) , E ( M Λ red L ( r, d )) = uvE ( M L − ( r, d )) . Moreover, both M Λ L ( r, d ) and M Λ red L ( r, d ) have pure mixed Hodge structures.Proof. By Theorem 5.15, the moduli space M Λ red L ( r, d ) is a smooth semiprojective variety for the C ∗ -action (3.19). Moreover, the map π from (3.16) is a surjective C ∗ -equivariant submersion coveringthe standard C ∗ -action on C . Then Proposition 5.4 gives the desired motivic equalities,[ M L − ( r, d )] = [ π − (0)] = [ π − (1)] = [ M Λ L ( r, d )] and [ M Λ red L ( r, d )] = L [ π − (0)] = L [ M L − ( r, d )] , which yield the corresponding equalities of E -polynomials, E ( M Λ L ( r, d )) = E ( M L − ( r, d )) and E ( M Λ red L ( r, d )) = uvE ( M L − ( r, d )) . Moreover, by [HRV15, Corollary 1.3.3], the fibers M L − ( r, d ) = π − (0) and M Λ L ( r, d ) = π − (1)have isomorphic cohomology supporting pure mixed Hodge structures. Finally, as M Λ red L ( r, d ) isalso smooth and semiprojective, its cohomology is also pure by [HRV15, Corollary 1.3.2]. (cid:3) Motives of moduli spaces of twisted Higgs bundles Continuing with the plan outlined in Section 4, after proving that we can reduce the computationof the motivic classes of the moduli spaces of L -connections to the computation of the motivic classesof moduli spaces of twisted Higgs bundles, the next step is to analyze the structure of the motiveof the latter moduli space.In this section we will prove that, under certain assumptions, such motive is independent onthe twisting line bundle, up to its degree, and we will provide tools to decompose the motive ofthe moduli space that will be useful later on, in section 7, to compute explicitly the motives and E -polynomials of the moduli spaces in low ranks. In order to do this, it will be useful to introducethe notion of variation of Hodge structure.6.1. Variations of Hodge structure and chains. Let L be a line bundle over the curve X . An L -twisted variation of Hodge structure of type r = ( r , . . . , r k ) and multidegree d = ( d , . . . , d k ) isan L -twisted Higgs bundle ( E, ϕ ) of the form(6.1) ( E • , ϕ • ) = k M i =1 E i , ··· ϕ ··· ϕ ··· ... ... ... ... ... ··· ϕ k − , where E i are vector bundles on X , with rk( E i ) = r i and deg( E i ) = d i , and ϕ i : E i → E i +1 ⊗ L foreach i = 1 , . . . , k , with ϕ k = 0.Let r = P ki =1 r i and d = P ki =1 d i and denote byVHS L ( r, d ) ⊂ M L ( r, d )the subscheme of the moduli space of L -twisted Higgs bundles corresponding to semistable varia-tions of Hodge structure of type r and multi-degree d . IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 31 On the other hand, recall that an algebraic chain on X is a quiver bundle of type A, hence asequence of algebraic vector bundles ( E , . . . , E k ), together with maps ϕ i : E i → E i +1 . We denotea chain by the symbol ( ˜ E • , ˜ ϕ • ).Given real numbers α = ( α , . . . , α k ), we define the α -degree of the chain ( ˜ E i , ˜ ϕ i ) as(6.2) deg α ( ˜ E • , ˜ ϕ • ) = k X i =1 (deg( E i ) + rk( E i ) α i )and the α -slope as µ α ( ˜ E • , ˜ ϕ • ) = deg α ( ˜ E • , ˜ ϕ • ) P ki =1 rk( E i ) . We say that ( ˜ E • , ˜ ϕ • ) is of type r = ( r , . . . , r k ) if rk( E i ) = r i for each i = 1 , . . . , k and we call d = (deg( E ) , . . . , deg( E k )) its multidegree .A subchain ( ˜ F • , ˜ ϕ • ) of ( ˜ E • , ˜ ϕ • ) is a collection ( F , . . . , F k ) of subbundles of ( E , . . . , E k ), i.e. F i ⊂ E i , such that ϕ i ( F i ) ⊂ F i +1 , so that ( ˜ F • , ˜ ϕ • | F • ) is itself a chain. An algebraic chain ( ˜ E • , ˜ ϕ • )is (semi)stable if for any subchain ( ˜ F • , ˜ ϕ • | F • ) ⊂ ( ˜ E • , ˜ ϕ • ), we have µ α ( ˜ F • , ˜ ϕ • | F • ) < µ α ( ˜ E • , ˜ ϕ • ) (resp. ≤ ) . Denote by HC α ( r, d )the moduli space of α -semistable algebraic chains on X of type r and multidegree d .Now, given a variation of Hodge structure ( E • , ϕ • ) of type r = ( r , . . . , r k ) and multidegree d = ( d , . . . , d k ), an algebraic chain ( ˜ E • , ˜ ϕ • ) can be constructed as follows. Take˜ E i = E i ⊗ L i − k . Then ϕ i induces a map ˜ ϕ i = ϕ i ⊗ Id L i − k : ˜ E i −→ ˜ E i +1 , thus ( ˜ E • , ˜ ϕ • ) is a chain of type r and multidegree d L = ( d + r (1 − k ) deg( L ) , . . . , d k ). Thisconstruction is reversible, giving a variation of Hodge structure from an algebraic chain, and hencegiving a bijection between these two kinds of objects.It turns out that their (semi)stability conditions also match, if one chooses a particular set ofreal numbers α . Indeed, if α L = (( k − 1) deg( L ) , . . . , deg( L ) , α L -degree (6.2) of anychain ( ˜ E • , ˜ ϕ • ), isdeg α L ( ˜ E • , ˜ ϕ • ) = k X i =1 (cid:16) deg( E i ⊗ L i − k ) + rk( E i )( k − i ) deg( L ) (cid:17) = k X i =1 deg( E i )so µ α ( ˜ E • , ˜ ϕ • ) = µ ( E • , ϕ • ) , where ( E • , ϕ • ) is the corresponding variation of Hodge structure.The proof of the next lemma follows by the exact same argument as in [ ´ACGP01, Proposition3.5], by replacing the canonical line bundle K X by L . Proposition 6.1. A variation of Hodge structure ( E • , ϕ • ) is (semi)stable (as an L -twisted Higgsbundle) if and only if for every choice of subbundles F i ⊂ E i with ϕ i ( F i ) ⊂ F i +1 we have µ ( F • , ϕ • | F • ) < µ ( E • , ϕ • ) ( resp. ≤ ) . Hence ( E • , ϕ • ) is (semi)stable if and only if the corresponding chain ( ˜ E • , ˜ ϕ • ) is α L -(semi)stable. So the following corollary is immediate. Corollary 6.2. Fix an algebraic line bundle L over X . Let r = ( r , . . . , r k ) , d = ( d , . . . , d k ) and d L = ( d + r (1 − k ) deg( L ) , d + r (2 − k ) deg( L ) , . . . , d k ) . The previously described correspondencebetween chains and L -twisted variations of Hodge structure induces an isomorphism, VHS L ( r, d ) ∼ = HC α L ( r, d L ) , for α L = (( k − 1) deg( L ) , . . . , deg( L ) , . Independence of the motives of Higgs moduli on the twisting line bundle. We willuse Corollary 6.2 to show that the motivic class of the moduli spaces of L -twisted Higgs bundlesfor coprime rank and degree only depend on the degree of the twisting line bundle L , whenever itis big enough.Recall from (5.8) that the moduli space M L ( r, d ) has a natural C ∗ -action given by scalling theHiggs field. Suppose that d is coprime with r ≥ L is a line bundle with deg( L ) > g − 2. Then by Proposition 5.7, the moduli space M L ( r, d ) is a smooth semiprojective variety.Accordingly, it admits a Bialynicki-Birula stratification M L ( r, d ) = [ µ ∈ I U + µ , which, by Lemma 5.3, induces the decomposition(6.3) [ M L ( r, d )] = X µ ∈ I L N + µ [ F µ ]of its motivic class, where N + µ = dim( U + µ ) − dim( F µ ) is the rank of the affine bundle U + µ → F µ ,corresponding to those Higgs bundles ( E, ϕ ) ∈ M L ( r, d ) such that lim t → ( E, tϕ ) lies in the C ∗ -fixedpoint set F µ . The characterization of the fixed points under the C ∗ -action carried out by Simpsonin [Sim92, § 4] also applies to the L -twisted case, obtaining the following lemma. Lemma 6.3. Let ( E, ϕ ) be any L -twisted Higgs bundle such that ( E, ϕ ) ∼ = ( E, tϕ ) for some t ∈ C ∗ which is not a root of unity. Then E has the structure of an L -twisted variation of Hodge structure (6.1) . Reciprocally, any L -twisted variation of Hodge structure is a fixed point of the C ∗ -action. Given any multirank r = ( r , . . . , r k ) and multidegree d = ( d , . . . , d k ), define(6.4) | r | = k X j =1 r j , | d | = k X j =1 d j and ∆ L = { ( r, d ) | VHS L ( r, d ) = ∅} . The previous lemma says that the semistable C ∗ -fixed points are precisely those in VHS L ( r, d ) ⊂M L ( r, d ) for each suitable choice of r and d . Thus, we rewrite (6.3) as(6.5) [ M L ( r, d )] = X ( r,d ) ∈ ∆ L | r | = r, | d | = d L N + L,r,d [VHS L ( r, d )] , where N + L,r,d is the notation for N + µ in this case. We will also use the notations N − L,r,d for N − µ and N L,r,d for N µ ; see (5.1).Next, we will focus on the computation and invariance with respect to L of the ranks N ± L,r,d in(6.5). Suppose that r and d are coprime, so that the moduli space M L ( r, d ) is smooth. Following[Hit87, Kir84] and working as in [BGL11], we will proceed by analyzing the Bialynicki-Birula strat-ification from a Morse-theoretic point of view. The moduli space M L ( r, d ) has a K¨ahler structure IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 33 which is preserved by the action of S ⊂ C ∗ . Therefore, the C ∗ -action induces a Hamiltonian actionof S , with moment map µ : M L ( r, d ) −→ R , µ ( E, ϕ ) = 12 k ϕ k , where the L -norm is given with respect to the (harmonic) metric solving the Hitchin equationscorresponding to the stable Higgs bundle ( E, ϕ ) under the Hitchin-Kobayashi correspondence; cf.[Hit87].By [Fra59] the map µ becomes a perfect Morse-Bott function in M L ( r, d ) and we have thefollowing lemma. Lemma 6.4. Suppose that r and d are coprime and that deg( L ) > g − . Let r = ( r , . . . , r k ) and d = ( d , . . . , d k ) be such that r = r + · · · + r k , d = d + · · · + d k and VHS L ( r, d ) is non-empty.Then VHS L ( r, d ) is a component of the critical point set of µ and if M L,r,d is its Morse index, then M L,r,d = 2 N − L,r,d . In particular, N + L,r,d + N L,r,d + M L,r,d / M L ( r, d )) = 1 + r deg( L ) . Proof. M L ( r, d ) is smooth by Lemma 2.2. Then, by [Kir84, Theorem 6.18, Example 9.4 andCorollary 13.2], we conclude that, for each ( r, d ) in the given conditions, the component VHS L ( r, d )of the fixed-point locus M L ( r, d ) C is a component the critical point set of µ and that the affinebundle U − r,d −→ VHS L ( r, d ) coincides with the downwards Morse flow of µ . Then, for each point p ∈ VHS L ( r, d ), we have N − L,r,d = dim (cid:16) T p (cid:16) U − r,d | p (cid:17)(cid:17) = 12 dim R (cid:16) U − r,d | p (cid:17) = 12 M L,r,d . The last statement follows from (5.2) and again Lemma 2.2. (cid:3) Lemma 6.5. Let L and L ′ be line bundles on X such that deg( L ) = deg( L ′ ) > g − . Supposethat r and d are coprime. Then the Morse index M L,r,d of VHS L ( r, d ) ⊂ M L ( r, d ) is the same asthe Morse index M L ′ ,r,d of VHS L ′ ( r, d ) ⊂ M L ′ ( r, d ) .Proof. Either by [BR94, Theorem 2.3] or by [Tor11, Theorem 47], the tangent space to the modulispace M L ( r, d ) of L -Higgs bundles at a point ( E, ϕ ) is isomorphic to H ( X, C • ( E, ϕ )), where C • ( E, ϕ ) is the following complex C • ( E, ϕ ) : End( E ) [ − ,ϕ ] −→ End( E ) ⊗ L. At a variation of Hodge structure ( E • , ϕ • ) ∈ VHS L ( r, d ), this deformation complex decomposes as(6.6) C • ( E, ϕ ) = k − M l = − k +1 C • l ( E, ϕ )where C • l ( E, ϕ ) : M j − i = l Hom( E i , E j ) [ − ,ϕ ] −→ M j − i = l +1 Hom( E i , E j ) ⊗ L and, thus, the tangent space decomposes as H ( C • ( E, ϕ )) = k − M l = − k +1 H ( C • l ( E • , ϕ • )) . Then, the computations in [BGL11, § 5] show that if r and d are coprime and deg( L ) > g − L ( r, d ) ⊂ M L ( r, d ) is(6.7) M L,r,d = 2 k − X l =1 dim( H ( C • l ( E • , ϕ • ))) = − k − X l =1 χ ( C • l ( E • , ϕ • )) , where, for each l = 1 , . . . , k − − χ ( C • l ( E • , ϕ • )) = k − l − X i =1 χ (Hom( E i , E i + l +1 ) ⊗ L ) − k − l X i =1 χ (Hom( E i , E i + l ))= k − l − X i =1 ( − r i + l +1 d i + r i d i + l +1 + r i r i + l deg( L ) + r i r i + l +1 (1 − g )) − k − l X i =1 ( − r i + l d i + r i d i + l + r i r i + l (1 − g )) . (6.8)Thus χ ( C • l ( E • , ϕ • )) depends on the degree of L , but not on L itself, and hence the same is true forthe Morse index. (cid:3) Theorem 6.6. Let X be a smooth complex projective curve of genus g ≥ . Let L and L ′ be linebundles over X such that deg( L ) = deg( L ′ ) > g − . Assume that the rank r and degree d arecoprime. Then the virtual motives of the corresponding moduli spaces [ M L ( r, d )] and [ M L ′ ( r, d )] areequal in K ( V ar C ) . Moreover, if d ′ is any integer coprime with r , then E ( M L ( r, d )) = E ( M L ′ ( r, d ′ )) .Finally, if L = L ′ = K ( D ) for some effective divisor D , then there is an actual isomorphism ofpure mixed Hodge structures H • ( M L ( r, d )) ∼ = H • ( M L ′ ( r, d ′ )) .Proof. By Corollary 6.2, for each k = 1 , . . . , r and each r = ( r , . . . , r k ) and d = ( d , . . . , d k ) with | r | = r and | d | = d (recall (6.4)), we haveVHS L ( r, d ) ∼ = HC α L ( r, d L ) , where α L = (( k − 1) deg( L ) , . . . , deg( L ) , 0) and d L = ( d L,i ) with d L,i = d i + r i ( i − k ) deg( L ). Asdeg( L ) = deg( L ′ ) we have α L = α L ′ and d L = d ′ L , so we obtain an isomorphism(6.9) VHS L ( r, d ) ∼ = HC α L ( r, d L ) = HC α L ′ ( r, d L ′ ) ∼ = VHS L ′ ( r, d ) . In particular, we have ∆ L = ∆ L ′ ; cf. (6.4).On the other hand, as M L ( r, d ) is smooth of dimension 1 + r deg( L ), then for each ( r, d ) ∈ ∆ L = ∆ L ′ , we have, using Lemma 6.4,(6.10) N + L,r,d = 1 + r deg( L ) − dim(VHS L ( r, d )) − M L,r,d / . By Lemma 6.5 and (6.9) we have N + L,r,d = N + L ′ ,r,d .Therefore, using (6.5), we conclude that[ M L ( r, d )] = r X k =1 X ( r,d ) ∈ ∆ L | r | = r, | d | = d L N + L,r,d [VHS L ( r, d )] = r X k =1 X ( r,d ) ∈ ∆ L ′ | r | = r, | d | = d L N + L ′ ,r,d [VHS L ′ ( r, d )] = [ M L ′ ( r, d )] . Then(6.11) E ( M L ( r, d )) = E ( M L ′ ( r, d ))is direct from (4.1). IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 35 Take now any d ′ also coprime with r . To prove that M L ( r, d ) and M L ′ ( r, d ′ ) have the same E -polynomial, we proceed as follows. By [MS20a, Theorem 0.1] (see also [GWZ20, Theorem 7.15]),(6.12) E ( M K X ( D ) ( r, d )) = E ( M K X ( D ) ( r, d ′ )) , where K X denotes the canonical bundle of X and D is an effective divisor. To prove it for anytwisting line bundles L, L ′ of the same degree (greater than 2 g − x ∈ X and let m = deg( L ) + 2 − g > 0. Then deg( K X ( mx )) = deg( L ) = deg( L ′ ) so, applying (6.11) and (6.12),yields E ( M L ( r, d )) = E ( M K X ( mx ) ( r, d )) = E ( M K X ( mx ) ( r, d ′ )) = E ( M L ′ ( r, d ′ )) , as claimed.Finally, if L = L ′ = K ( D ) with D effective, the isomorphism H • ( M L ( r, d )) ∼ = H • ( M L ′ ( r, d ′ )) ofHodge structures follows immediately from [MS20a, Theorem 0.1]. (cid:3) Independence of motives and E -polynomials from the Lie algebroid structure. Wecan combine all the previous invariance results to prove our main theorem. Given a Lie algebroid, L on the curve X , recall that M Λ L ( r, d ) denotes the moduli space of semistable integrable L -connections of rank r and degree d or, equivalently, of semistable Λ L -modules, where Λ L is the splitalmost polynomial sheaf of rings of differential operators associated to L , under the equivalenceprovided by Theorem 3.9. If rk( L ) = 1, then every L -connection is automatically integrable, so inthat case M Λ L ( r, d ) is the moduli space of all semistable L -connections of rank r and degree d . Theorem 6.7. Let X be a smooth projective curve of genus g ≥ and let L and L ′ be any twoLie algebroids on X such that rk( L ) = rk( L ′ ) = 1 and deg( L ) = deg( L ′ ) < − g . Supposethat r and d are coprime. Then [ M Λ L ( r, d )] = [ M Λ L′ ( r, d )] in ˆ K ( V ar C ) . Moreover, if d ′ is anyinteger coprime with r , then E ( M Λ L ( r, d )) = E ( M Λ L′ ( r, d ′ )) . Finally, if L = L ′ = K ( D ) forsome effective divisor D , then there is an actual isomorphism of pure mixed Hodge structures H • ( M Λ L ( r, d )) ∼ = H • ( M Λ L′ ( r, d ′ )) .Proof. This follows directly from Theorems 5.17 and 6.6, and by (4.1). (cid:3) Hence to compute the motivic class or the E -polynomial of M Λ L ( r, d ), it is enough to do it forthe moduli space M K X ( D ) ( r, 1) of K X ( D )-twisted Higgs bundles of rank r and degree 1, for somedivisor D of the appropriate (positive) degree.7. Applications Now, let us analyze some consequences of the preceding results. In the next section we deducesome topological properties of the moduli spaces of L -connections. These properties are not ob-tained by using the motivic results proved before, but rather the Bialynicki-Birula stratification ofthe L -Hodge moduli space. In the subsequent sections, we will give a direct application of Theorem6.7 related to the moduli spaces of logarithmic and irregular connections on X , and we will alsoprovide explicit formulas for the motivic classs and E -polynomials, of Theorem 6.7, for r = 2 , Topological properties of moduli spaces of Lie algebroid connections. Similarly tohow we used the smoothness of the moduli space of twisted Higgs bundles to prove the smoothnessof the moduli space of L -connections back in section 5.3, we can also use the regularity propertiesand the Bialynicki-Birula stratification of M Λ red L ( r, d ) to transfer other known properties of themoduli spaces of twisted Higgs bundles to moduli spaces of L -connections. As an example, in thissection we prove that, under certain conditions, the moduli space of L -connections is irreducibleand compute some its homotopy groups, by showing that they are isomorphic to the ones of themoduli space of vector bundles. Let M ( r, d ) denote the moduli space of semistable vector bundles of rank r and degree d on thecurve X . Lemma 7.1. Let X be a smooth projective curve of genus g ≥ and let L be a rank Lie algebroidon X such that deg( L ) < − g . Suppose that r ≥ and d are coprime. Then the loci in M Λ L ( r, d ) corresponding to semistable L -connections whose underlying vector bundle is not stablehas codimension at least ( g − r − . In particular, with the given bounds on g and r it hascodimension at least .Proof. Let L = ( L, [ · , · ] .δ ). By Proposition 5.7 and Theorem 5.15 we know that the moduli spaces M Λ red L ( r, d ) and π − (0) = M L − ( r, d ) ⊂ M Λ red L ( r, d ) are smooth semiprojective varieties for the C ∗ -action (3.19) and its restriction (5.8) to M L − ( r, d ) respectively. Recall that here π is the map(3.16).The fixed-point locus of this action is concentrated in M L − ( r, d ) and, by Lemma 6.3, it corre-sponds to the subset of variations of Hodge structure (recall (6.4)), M L − ( r, d ) C ∗ = M Λ red L ( r, d ) C ∗ = [ ( r,d ) ∈ ∆ L − | r | = r, | d | = d VHS L − ( r, d ) . In this decomposition there is a distinguished component, namely the one for which r = { r } and d = { d } . It parameterizes points of the form ( E, , 0) with E stable (because ( r, d ) = 1), and it istherefore isomorphic to the moduli space M ( r, d ). Let M L − ( r, d ) = [ ( r,d ) ∈ ∆ L − | r | = r, | d | = d U + L − ,r,d and M Λ red L ( r, d ) = [ ( r,d ) ∈ ∆ L − | r | = r, | d | = d ˜ U + L − ,r,d be the corresponding Bialynicki-Birula decompositions, hence where U + L − ,r,d = n ( E, ∇ L , ∈ M L − ( r, d ) (cid:12)(cid:12)(cid:12) lim t → ( E, t ∇ L ) ∈ VHS L − ( r, d ) o , ˜ U + L − ,r,d = n ( E, ∇ L , λ ) ∈ M Λ red L ( r, d ) (cid:12)(cid:12)(cid:12) lim t → ( E, t ∇ L , tλ ) ∈ VHS L − ( r, d ) o are affine bundles over VHS L − ( r, d ) of rank N + L − ,r,d and ˜ N + L − ,r,d respectively.Let us write U + = U + L − , { r } , { d } and ˜ U + = ˜ U + L − , { r } , { d } the affine bundles lying over M ( r, d ). Let S and ˜ S denote the subsets of M L − ( r, d ) and M Λ red L ( r, d ) respectively corresponding to triples( E, ∇ L , λ ) with E not stable. If E is a stable vector bundle then for every ( E, ∇ L , λ ) ∈ M Λ red L ( r, d )we have lim t → ( E, t ∇ L , tλ ) = ( E, , ∈ M ( r, d ) ⊂ M Λ red L ( r, d ) C ∗ , so S ⊂ M L − ( r, d ) \ U and ˜ S ⊂ M Λ red L ( r, d ) \ ˜ U . Actually, by [BGL11, Proposition 5.1], S = M L − ( r, d ) \ U (the proof is given for the moduli space with fixed determinant, but the proof alsoworks for fixed degree) and in [BGL11, Proposition 5.4] it is proven that codim( S ) ≥ ( g − r − U + L − ,r,d = U + , thencodim( U + L − ,r,d ) ≥ ( g − r − . On the other hand, we know ˜ N L − ,r,d = N L − ,r,d +1 (see (5.6)), hence dim ˜ U + L − ,r,d = dim U + L − ,r,d +1.By Lemma 5.13, dim M Λ red L ( r, d ) = dim( M L − ( r, d )) + 1 so we conclude thatcodim( ˜ U + L − ,r,d ) = codim( U + L − ,r,d ) ≥ ( g − r − . IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 37 Finally, define S ′ = ˜ S ∩ π − (1). As C ∗ preserves the stability of the underlying bundle, then ˜ S is C ∗ -invariant and the restriction of this action to ˜ S gives an isomorphism˜ S ∩ π − ( C ∗ ) ∼ = S ′ × C ∗ . Therefore, dim( S ′ ) = dim( ˜ S ∩ π − ( C ∗ )) − ≤ dim( ˜ S ) − . Finally, as dim M Λ L ( r, d ) = dim M Λ red L ( r, d ) − 1, we conclude thatcodim( S ′ ) ≥ codim( ˜ S ) ≥ ( g − r − , completing the proof. (cid:3) Consider an L -connection ( E, ∇ L ) of rank r and degree d on X . Let E be the C ∞ vector bundleon X underlying the algebraic vector bundle E . Note that E is independent of the choice of the L -connection, as long as the rank and degree are still r and d . Let G ( E ) be the unitary gauge groupfor a fixed Hermitian metric on E . In other words, G ( E ) = Ω ( u ( E )), where Ω ( u ( E )) stands forthe space of C ∞ -sections of the C ∞ -bundle of unitary endomorphisms of E . Theorem 7.2. Let X be a smooth complex curve of genus g ≥ and let L be a algebroid on X such that rk( L ) = 1 and deg( L ) < − g . Suppose that r and d are coprime and that r ≥ . Then, M Λ L ( r, d ) is connected, hence irreducible. If, moreover, ( r, g ) = (2 , , then its higher homotopygroups are given as follows: • π ( M Λ L ( r, d )) ∼ = H ( X, Z ) ∼ = Z g ; • π ( M Λ L ( r, d )) ∼ = Z ; • π k ( M Λ L ( r, d )) ∼ = π k − ( G ( E )) , for every k = 3 , . . . , g − r − − .Proof. Let L = ( L, [ · , · ] .δ ). By Lemma 5.13 we know that the moduli space M Λ L ( r, d ) is a smoothvariety whose components are all of the same dimension 1 − r deg( L ). Let S ′ ⊂ M Λ L ( r, d ) be thesubspace of L -connections ( E, ∇ L ) with E not stable. Define U ′ = M Λ L ( r, d ) \ S ′ .It turns out that the C ∗ -flow provides a deformation retraction from U ′ to M ( r, d ). To be precise,consider the forgetful map π M : U ′ −→ M ( r, d ), given by π M ( E, ∇ L ) = E . By Corollary 3.14 themap is surjective and we have the following explicit description of each fiber π − M ( E ) = {∇ L : E → E ⊗ L ∗ | ∇ L ( f s ) = f ∇ L ( s ) + s ⊗ d L ( f ) , ∀ s ∈ E, ∀ f ∈ O X } . Observe that if ∇ L , ∇ ′L ∈ π − E ( E ), then ∇ L − ∇ ′L ∈ H (End( E ) ⊗ L ∗ ), so π − M ( E ) is an affine spaceon H (End( E ) ⊗ L ∗ ). Moreover, H (End( E ) ⊗ L ∗ ) ∼ = H (End( E ) ⊗ K ⊗ L ) ∗ = 0because, since E is stable, End( E ) is semistable and so End( E ) ⊗ K X ⊗ L is a semistable vectorbundle with deg(End( E ) ⊗ K X ⊗ L ) = r (deg( K X ) + deg( L )) < 0. Thus, by Riemann-Roch, foreach E ∈ M ( r, d ), dim( π − M ( E )) = dim H (End( E ) ⊗ L ∗ ) = 1 − r deg( L ) − g is constant and thus the map π M is equidimensional. Let E → X × M ( r, d ) be the universal bundleover M ( r, d ) (i.e., the bundle whose fiber over X × { E } is isomorphic to E ); it exists since ( r, d ) = 1.Let π X : X × M ( r, d ) → X be the projection. Then we conclude that U ′ is a torsor for the vectorbundle R ( π X ) ∗ (End( E ) ⊗ π ∗ X L ∗ ) −→ M ( r, d ) . It follows that, the homotopy groups of U ′ verify π k ( U ′ ) ∼ = π k ( M ( r, d )), for every k ≥ S ′ ) ≥ ( g − r − M Λ L ( r, d ) is smooth, this implies that π k ( M Λ L ( r, d )) ∼ = π k ( U ′ ) ∼ = π k ( M ( r, d )) , for every k = 0 , . . . , g − r − − 2. The moduli M ( r, d ) is connected, hence so is M Λ L ( r, d ),and thus irreducible because it is smooth. As for the higher homotopy groups, the results followfrom [DU95, Theorem 3.1]. (cid:3) Remark 7.3. By taking the trivial Lie algebroid L = ( L, , in the above theorem, one getsthe results for the moduli space of M L − ( r, d ) of L − -twisted Higgs bundles. The irreducibilityconclusion was proved in [BGL11] by the same arguments, and actually the higher homotopy groupsof M L − ( r, d ) would also follow directly from [BGL11] by the same argument as above.On the other hand, improvements on the bound of k for which the isomorphism π k ( M Λ L ( r, d )) ∼ = π k − ( G ( E )) holds have been achieved, for twisted Higgs bundles, in certain particular situations (cf. [Hau98] and [ZnR18] ), hence we might expect that such isomorphism also holds, in this generality,for higher values of k . Chow motives and Voevodsky motives. Theorem 6.7 shows that, under the stated con-ditions, there is an equality of motives[ M Λ L ( r, d )] = [ M Λ L′ ( r, d )] ∈ ˆ K ( V ar C )in the (completed) Grothendieck ring of varieties.Nevertheless, the techniques that we use to prove this equality (namely, semiprojectivity of the L -Hodge moduli space and the exposed relations between the Bialynicki-Birula decompositions ofthe corresponding moduli spaces of twisted Higgs bundles) also allow us to obtain isomorphisms forother types of invariants. Given a complex scheme X and a ring R , let us consider the following. • Let M ( X ) ∈ DM eff ( C , R ) denote the Voevodsky motive of X , where DM eff ( C , R ) is thecategory of effective geometric motives as defined by Voevodsky in [Voe00]. • Let h ( X ) ∈ Chow eff ( C , R ) be the Chow motive of X , where Chow eff ( C , R ) is the categoryof effective Chow motives; see for example [Man68, Sch94, dBn01]. • Let CH • ( X, R ) denote the chow ring of X with coefficients in R .Moreover, recall that we say that X has a pure Voevodsky motive if M ( X ) belongs to the heartof DM eff ( C , R ) which is equivalent to Chow eff ( C , R ) through Voevodsky’s embedding (c.f. [HL19,Section 6.3]). Theorem 7.4. Let X be a smooth projective curve of genus g ≥ and let L and L ′ be any twoLie algebroids on X such that rk( L ) = rk( L ′ ) = 1 and deg( L ) = deg( L ′ ) < − g . Suppose that r and d are coprime. Then, for every ring R , the Voevodsky motive of the moduli space M Λ L ( r, d ) ispure and we have M ( M Λ L ( r, d )) ∼ = M ( M Λ L′ ( r, d )) ∈ DM eff ( C , R ) ,h ( M Λ L ( r, d )) ∼ = h ( M Λ L′ ( r, d )) ∈ Chow eff ( C , R ) , CH • ( M Λ L ( r, d )) ∼ = CH • ( M Λ L′ ( r, d )) . Proof. The proof is analogous to the one in Theorem 5.17 and Theorem 6.7, but we now use thetechnical theorems from Appendices A and B of [HL19] to perform the necessary computations inDM eff ( C , R ) instead of ˆ K ( V ar C ). By Theorem 5.15, for every rank one algebroid L = ( L, [ · , · ] , δ )satisfying the hypothesis of the theorem the moduli space M Λ L ( r, d ) is a smooth quasiprojectivesemiprojective variety with a C ∗ -equivariant submersion π : M Λ L ( r, d ) −→ C such that π − (0) = M L ∗ ( r, d ) and π − (1) = M Λ L ( r, d ). Then [HL19, Theorem B.1] and [HL19, Corollary B.2] yieldisomorphisms M ( M L − ( r, d )) = M ( π − (0)) ∼ = M ( M Λ red L ( r, d )) ∼ = M ( π − (1)) = M ( M Λ L ( r, d ))CH • ( M L − ( r, d )) = CH • ( π − (0)) ∼ = CH • ( M Λ red L ( r, d )) ∼ = CH • ( π − (1)) = CH • ( M Λ L ( r, d )) IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 39 As M L − ( r, d ) is also smooth and semiprojective by Proposition 5.7, its motive is pure by [HL19,Corollary A.5], so the motive of M Λ L ( r, d ) is also pure and, therefore, the above isomorphism ofVoevodsky motives induces an isomorphism of the Chow motives. Thus, we can assume withoutloss of generality that the algebroid structures for L and L ′ are trivial (i.e., that we have modulispaces of Higgs bundles). Furthermore, purity of the Voevodsky motives and representability ofthe Chow groups as certain spaces of morphisms in DM eff ( C , R ) (c.f. [HL19, Corollary B.2]) implythat it is enough to prove that the Voevodsky motives are isomorphic to conclude the desiredisomorphisms between Chow motives or Chow rings. The Bialynicki-Birula decomposition of themoduli spaces of Higgs bundles yield the following motivic decompositions [HL19, Theorem A.4],in which we follow the notation from Section 6. M ( M L − ( r, d )) ∼ = r M k =1 M ( r,d ) ∈ ∆ L − | r | = r, | d | = d M (VHS L − ( r, d )) { N − L − ,r,d } M ( M ( L ′ ) − ( r, d )) ∼ = r M k =1 M ( r,d ) ∈ ∆ ( L ′ ) − | r | = r, | d | = d M (VHS ( L ′ ) − ( r, d )) { N − ( L ′ ) − ,r,d } By Corollary 6.2 we have ∆ L − = ∆ ( L ′ ) − . Calling this set ∆, then Corollary 6.2 and Lemma 6.5imply that for each ( r, d ) ∈ ∆, we have VHS L − ( r, d ) ∼ = VHS ( L ′ ) − ( r, d ) and N − L − ,r,d = N − ( L ′ ) − ,r,d ,so we obtain a term-by-term isomorphism of the previous Voevodsky motives. (cid:3) This result can be considered as an extension to moduli spaces of Lie algebroid connections (over C ) of [HL19, Theorem 4.2], in which it is proved that there exists an isomorphism between theVoevodsky motives and Chow rings of the de Rham and K -twisted Higgs moduli spaces.7.3. Motives of moduli spaces of irregular or logarithmic connections. Recall the canonicalLie algebroid T X = ( T X , [ · , · ] Lie , Id) on our smooth projective curve X . Consider an effective divisor D = P ni =1 k i x i on X , with k i ≥ 1. Let M conn ( D, r, d ) be the moduli space of rank r and degree d semistable singular ( T X -)connections, with poles of order at most k i over each x i ∈ D . Theseconnections are irregular if k i > i .Take the Lie subalgebroid T X ( − D ) ⊂ T X , thus with underlying bundle T X ( − D ) ⊂ T X , theinduced Lie bracket of vector fields, and the inclusion anchor map. Then M Λ T X ( − D ) ( r, d ) = M conn ( D, r, d ), hence we have the following direct corollary of Theorem 6.7. Corollary 7.5. If D and D ′ are any two effective divisors on X with deg( D ) = deg( D ′ ) and r and d are coprime, then [ M conn ( D, r, d )] = [ M conn ( D ′ , r, d )] ∈ ˆ K ( V ar C ) and E ( M conn ( D, r, d )) = E ( M conn ( D ′ , r, d )) . In particular, by taking D ′ to be a simple divisor, we conclude the following. Corollary 7.6. The motivic class and E -polynomial of any moduli space of irregular connectionson a smooth projective curve X of genus at least equals that of any moduli space of logarithmic connections X , with singular divisor of the same degree. Explicit motives and E -polynomials for rank and . Fix a rank 1 Lie algebroid L =( L, [ · , · ] , δ ) on the curve X , such that deg( L ) < − g . In this section, we provide explicit formulaefor the motivic classes and E -polynomials of the moduli spaces of L -connections of rank 2 and 3 andcoprime degree. Theorem 6.7 allows us to perform all computations by just considering the trivialLie algebroid ( L, , L − -twisted Higgs bundles of corresponding rankand degree. Recollection of properties of motives. We first need to introduce some notation and recall,without proof, some facts on the theory of motivic classes in ˆ K ( V ar C ). For details, see for example[Hei07, Kap00].The symmetric product of a variety gives rise to the λ -operator defined, for each n ≥ 0, as(7.1) λ n : ˆ K ( V ar C ) → ˆ K ( V ar C ) , λ n ([ Y ]) = [Sym n ( Y )] . For example λ n ( L k ) = L nk . With these operators, ˆ K ( V ar C ) acquires the structure of a λ -ring . Inparticular, the relation(7.2) λ n ([ Y ] + [ Z ]) = X i + j = n λ i ([ Y ]) λ j ([ Z ]) , holds.The motive of our fixed genus g ≥ X splits as [ X ] = 1 + h ( X ) + L , where h ( X ) ∈ ˆ K ( V ar C ) is such that the motive of the Jacobian of X is given by(7.3) [Jac( X )] = g X i =0 λ i ( h ( X )) . Define the ˆ K ( V ar C )-valued polynomial(7.4) P X ( x ) = g X i =0 λ i ( h ( X )) x i ∈ ˆ K ( V ar C )[[ x ]]and note that P X (1) = [Jac( X )]. Consider also the zeta function of X , defined as Z ( X, x ) = X k ≥ λ k ([ X ]) x k ∈ ˆ K ( V ar C )[[ x ]] . Using that λ n ( h ( X )) = L n − g λ g − n ( h ( X )), if n = 0 , . . . , g , and that λ n ( h ( X )) = 0 if n > g ,it follows that λ n ([ X ]) = coeff x Z ( X, x ) x n = coeff x P X ( x )(1 − x )(1 − L x ) x n . Motives of M Λ L ( r, d ) for r = 2 , . Now we move on to the motives of moduli spaces. Wewant to compute the motive of M L − ( r, d ), for r = 2 , d coprime with r . This will be doneby using the formula (6.5), and so we will need to consider the moduli space of rank r and degree d vector bundles (which we think of consisting of Higgs bundles which are variations of Hodgestructure of type ( r )) and then variations of Hodge structure of type (1 , 1) for r = 2 and type (1 , , 1) and (1 , , 1) for r = 3. We will not fill the full details of the computations, and leave them tothe reader.In this section, we use the notation d L for the degree of the line bundle L , so that d L < − g .Recall that M ( r, d ) denotes the moduli space of stable vector bundles of rank r and degree d over the curve X .Let us start with rank 2 case. Let d be odd. By Example 3.4 of [GPHS14] or equation (3.9),page 41 of [S´an14], the motivic class of M (2 , d ) is given by(7.5) [ M (2 , d )] = [Jac( X )] P X ( L ) − L g [Jac( X )] ( L − L − , where P X is the polynomial given in (7.4). Notice that this formula is obtained by the one in[GPHS14] by multiplying by L − C ∗ -gerbe over M (2 , d ). IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 41 We move on the motivic class of subvarieties of M L − (2 , d ) corresponding to variations of Hodgestructure of type (1 , L − -twisted-Higgs bundle ( E, ϕ ) lies in VHS L − ((1 , , ( d , d − d )) ifit is stable and(7.6) E = E ⊕ E , ϕ = (cid:0) ϕ (cid:1) , with E , E line bundles of degree d and d − d respectively and ϕ : E → E ⊗ L − nonzero. Thefact that ϕ = 0 and stability ( E is ϕ -invariant) impose conditions on the degree d and, indeed,VHS L − ((1 , , ( d , d − d )) = ∅ ⇐⇒ d/ < d ≤ ( d − d L ) / . In such a case, the map (cid:0) E ⊕ E , (cid:0) ϕ (cid:1)(cid:1) (div( ϕ ) , E ), where div( ϕ ) denotes the divisor ofthe section ϕ ∈ H ( E − E L − ), yields the isomorphism(7.7) VHS L − ((1 , , ( d , d − d )) ∼ = Sym d − d − d L ( X ) × Jac d − d ( X ) . Hence, since the ‘Jacobian’ of degree d − d line bundles on X is isomorphic to the Jacobian Jac( X )of (degree 0 line bundles on) X ,(7.8) [VHS L − ((1 , , ( d , d − d ))] = λ d − d − d L ([ X ])[Jac( X )] , where we are using the λ -operations defined in (7.1).Now we consider the rank 3 case. Fix d coprime with 3, so that every semistable L − -twistedHiggs bundle is stable. We have[ M (3 , d )] = [Jac( X )]( L − L − ( L − (cid:16) L g − (1 + L + L )[Jac( X )] − L g − (1 + L ) [Jac( X )] P X ( L ) + P X ( L ) P X ( L ) (cid:17) . (7.9)by [GPHS14, Remark 3.5] or [S´an14, Theorem 4.7]. As in the r = 2 case, this is obtained by thestated formula in [GPHS14] by multiplying by L − , 2) in M L − (3 , d ). A stable L − -twistedHiggs bundle ( E, ϕ ) lies in VHS L − ((1 , , ( d , d − d )) if it is of the form (7.6), with the onlydifference that now E has rank 2. Let I ⊂ E be the line bundle such that the saturation of theimage of ϕ equals IL − . Then both E and E ⊕ I are ϕ -invariant subbundles of E . Checking thestability for them imposes conditions of d , and actually we have that(7.10) VHS L − ((1 , , ( d , d − d )) = ∅ ⇐⇒ d/ < d < d/ − d L / . Moreover, by Corollary 6.2, there is an isomorphism with the moduli space of α L − = ( − d L , L − ((1 , , ( d , d − d )) ∼ = HC α L − ((1 , , ( d + d L , d − d )) . Since d is coprime with 3, then α L − is not a critical value (i.e. a value of the stability parameterwhere semistability changes), so the motive of HC α L − ((1 , , ( d + d L , d − d )) can be read off from[GPHS14, Example 6.4], by adapting the computation to the L − -twisting setting, or, perhaps moredirectly, from Theorem 3.2 of [S´an14], where the author considers the moduli space of − d L -stabletriples of type ((2 , d − d − d L , d )) (cf. [BGPG04]), which is isomorphic to the moduli space of α L − -stable chains. From this, we conclude that[VHS L − ((1 , , ( d , d − d ))] = [Jac( X )] L − (cid:18) L ⌊ d/ ⌋− d + d + g +1 λ d −⌊ d/ ⌋− d − d L − ([ X ] + L ) − λ d −⌊ d/ ⌋− d − d L − ([ X ] L + 1) (cid:19) = [Jac( X )] L − (cid:18) L ⌊ d/ ⌋− d + d + g +1 × d −⌊ d/ ⌋− d − d L − X i =0 λ i ([ X ])( L d − ⌊ d/ ⌋− d − d L − − i − L i ) (cid:19) . (7.12)The motives of the subvarieties of M L − (3 , d ) corresponding to variations of Hodge structure oftype (2 , 1) are directly obtained from the ones of type (1 , 2) by making use of the isomorphism(7.13) VHS L − ((2 , , ( d , d − d )) ∼ = VHS L − ((1 , , ( d − d, − d ))arising from duality.Finally, we deal with variations of Hodge structure of type (1 , , 1) in M L − (3 , d ). Similarly tothe previous cases, it follows thatVHS L − ((1 , , , ( d , d , d − d − d )) = ∅ ⇐⇒ ( d , d ) ∈ ∆ − d L ( d ) , where(7.14) ∆ − d L ( d ) = (cid:8) ( a, b ) ∈ Z | a − b ≤ − d L , a + 2 b − d ≤ − d L , a > d/ , a + b > d/ (cid:9) , and in that case, we have the following isomorphism(7.15)VHS L − ((1 , , , ( d , d , d − d − d )) ∼ = Sym − d + d − d L ( X ) × Sym d − d − d − d L ( X ) × Jac d − d − d ( X ) . Thus,(7.16) [VHS L − ((1 , , , ( d , d , d − d − d ))] = λ − d + d − d L ([ X ]) λ d − d − d − d L ([ X ])[Jac( X )] . Now we have the promised corollary of Theorem 6.7. Corollary 7.7. Let X be a smooth projective curve of genus g ≥ and let L be a rank Liealgebroid on X . Write d L = deg( L ) and suppose that d L < − g . Then,(1) if (2 , d ) = 1 , [ M Λ L (2 , d )] = L − d L +4 − g (cid:16) [Jac( X )] P X ( L ) − L g [Jac( X )] (cid:17) ( L − L − L − d L +2 − g [Jac( X )] ⌊ d − dL ⌋ X d = ⌊ d/ ⌋ +1 λ d − d − d L ([ X ]) . IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 43 (2) if (3 , d ) = 1 , [ M Λ L (3 , d )] = L − d L +9 − g [Jac( X )]( L − L − ( L − (cid:16) L g − (1 + L + L )[Jac( X )] − L g − (1 + L ) [Jac( X )] P X ( L ) + P X ( L ) P X ( L ) (cid:17) + L − d L +5 − g [Jac( X )] L − ⌊ d − dL ⌋ X d = ⌊ d/ ⌋ +1 (cid:18) L ⌊ d/ ⌋− d + d + g +1 λ d −⌊ d/ ⌋− d − d L − ([ X ] + L ) − λ d −⌊ d/ ⌋− d − d L − ([ X ] L + 1) (cid:19) + L − d L +5 − g [Jac( X )] L − ⌊ d − dL ⌋ X d = ⌊ d/ ⌋ +1 (cid:18) L ⌊− d/ ⌋ + d + g +1 λ d −⌊− d/ ⌋− d − d L − ([ X ] + L ) − λ d −⌊− d/ ⌋− d − d L − ([ X ] L + 1) (cid:19) + L − d L +3 − g [Jac( X )] X ( d ,d ) ∈ ∆ − dL ( d ) λ − d + d − d L ([ X ]) λ d − d − d − d L ([ X ]) , with ∆ − d L ( d ) defined in (7.14) . Remark 7.8. It is easy to see that the motivic class [ M Λ L (3 , d )] is indeed the same by replacing d for − d (for the last sum, one should use the bijection between ∆ − d L ( d ) and ∆ − d L ( − d ) givenby ( a, b ) ( − d + a + b, − b ) ). Of course, this had to occur since duality yields an isomorphismbetween the moduli spaces M Λ L (3 , d ) and M Λ L (3 , − d ) (and, of course, the mentioned bijection ∆ − d L ( d ) ≃ ∆ − d L ( − d ) is provided by duality).Proof. As we are considering cases in which the rank and the degree are coprime, we can applyTheorem refthm:equalMotive, and assume without loss of generality that L has the trivial Liealgebroid structure ( L, , M L ( r, d ) corresponds to the moduli space of L − -twistedHiggs bundles of rank r and degree d .Then, everything follows from the decomposition (6.5), using the formula (6.10) (with L replacedby L − ), N + L − ,r,d = 1 − r deg( L ) − dim(VHS L − ( r, d )) − M L − ,r,d / L in each summand. All of them are straightforward, using (6.7) and (6.8).We leave the details of the computations for the reader. In rank 2, use (7.7) followed by (7.5) and(7.8).In rank 3, use (7.11) and (7.15), knowing that, in the (1 , L − ( X, (1 , , ( d , d − d ))) = dim(HC α L − − ss ((1 , , ( d + d L , d − d )))= 3 g − − d + d − d L by Theorem A (2) of [BGPG04] and that the dimension of dim(VHS L − ( X, (1 , , ( d , d − d ))) iscomputed similarly using (7.13). Then, (6.5) becomes[ M Λ L (3 , d )] = L − d L +9 − g [ M (3 , d )]+ L − d L +5 − g ⌊ d − dL ⌋ X d = ⌊ d ⌋ +1 [VHS L − ((1 , , ( d , d − d ))]+ L − d L +5 − g ⌊ d − dL ⌋ X d = ⌊ d ⌋ +1 [VHS L − ((2 , , ( d , d − d ))]+ L − d L +3 − g X ( d ,d ) ∈ ∆ − dL ( d ) [VHS L − ((1 , , , ( d , d , d − d − d ))] , and we obtain the result from (7.9)–(7.16). (cid:3) E -polynomials of M Λ L ( r, d ) for r = 2 , . We now apply the E -polynomial map (4.1) to theformulas of the preceding result.If X is again our smooth projective curve of genus g , it is well-known that E (Jac( X )) = (1 + u ) g (1 + v ) g , thus E ( λ n ([ X ])) = coeff x (1 + ux ) g (1 + vx ) g (1 − x )(1 − uvx ) x n . In addition, from (7.3) and (7.4), we have that, for any k , E ( P X ( L k )) = g X i =0 E ( λ i ( h ( X )))( u k v k ) i = g X i =0 X p + q = i h p,q (Jac( X )) u p v q ( u k v k ) i = g X i =0 X p + q = i h p,q (Jac( X ))( u k +1 v k ) p ( u k v k +1 ) q = E (Jac( X ))( u k +1 v k , u k v k +1 ) = (1 + u k +1 v k ) g ( u k v k +1 ) g . In particular,(7.17) E ( P X ( L )) = (1 + u v ) g (1 + uv ) g and E ( P X ( L )) = (1 + u v ) g (1 + u v ) g , Corollary 7.9. Let X be smooth projective curve of genus g and let L be a rank Lie algebroidon X . Write d L = deg( L ) and suppose that d L < − g .(1) Let d be odd. Then, E ( M Λ L (2 , uv ) − d L +4 − g E ( M (2 , d ))+ ( uv ) − d L +2 − g (1 + u ) g (1 + v ) g coeff x (cid:18) (1 + ux ) g (1 + vx ) g x d L +1 (1 − x )(1 − x )(1 − uvx ) (cid:19) , where E ( M (2 , d )) = (1 + u ) g (1 + v ) g (1 + u v ) g (1 + uv ) g − ( uv ) g (1 + u ) g (1 + v ) g ( uv − uv ) − . IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 45 (2) Let d be coprime with . Then, E ( M Λ L (3 , uv ) − d L +9 − g E ( M (3 , d ))+ (1 + u ) g (1 + v ) g ( uv ) − d L +6 − g uv − · coeff x (1 + ux ) g (1 + vx ) g x d L +2 (1 − x )(1 − uvx )(1 − ( uv ) x )(1 − uvx ) − (1 + u ) g (1 + v ) g ( uv ) − d L +6 − g uv − · coeff x (1 + ux ) g (1 + vx ) g x d L +2 (1 − x )(1 − uvx )( uv − x )(( uv ) − x )+ (1 + u ) g (1 + v ) g ( uv ) − d L +5 − g uv − · coeff x (1 + ux ) g (1 + vx ) g x d L +1 (1 − x )(1 − uvx )(1 − ( uv ) x )(1 − uvx ) − (1 + u ) g (1 + v ) g ( uv ) − d L +7 − g uv − · coeff x (1 + ux ) g (1 + vx ) g x d L +1 (1 − x )(1 − uvx )( uv − x )(( uv ) − x )+ (1 + u ) g (1 + v ) g ( uv ) − d L +3 − g ·· coeff x y (1 + ux ) g (1 + vx ) g (1 + uy ) g (1 + vy ) g x d L +2 y d L +1 ( x − d L − y − d L )( y − d L − x − d L )(1 − x )(1 − uvx )(1 − y )(1 − uvy )( x − y )( y − x ) , where E ( M (3 , d )) = (1 + u ) g (1 + v ) g ( uv − uv ) − (( uv ) − (cid:18) ( uv ) g − (1 + uv + ( uv ) )(1 + u ) g (1 + v ) g − ( uv ) g − (1 + uv ) (1 + u ) g (1 + v ) g (1 + u v ) g (1 + uv ) g + (1 + u v ) g (1 + uv ) g (1 + u v ) g (1 + u v ) g (cid:19) . Proof. This follows from Corollary 7.7 and from computations which are now standard. We leave thedetails for the reader, who may see for example [Got94], [GPHS14], [Ben10] (especially Theorems3.1.4 and 3.5.7) and [S´an14] for techniques on these computations. Note however that the ones in[Ben10] contain some slight inaccuracies (so that the final results stated there, in Theorems 3.1.4and 3.5.7, are not correct). The formulas for the E -polynomials of the appropriate powers of the λ -operations of [ X ] + L and of [ X ] L + 1, which appear in Corollary 7.7, may be found in equation(3.3) of page 39 of [S´an14] and the ones concerning P X ( L ) and P X ( L ) are given in (7.17). (cid:3) Remark 7.10. The E -polynomials of the moduli space of vector bundles of rank and andcoprime degree d were first computed recursively in [EK00, Theorem 1] . The one for rank wasalso explicitly obtained“ with different techniques, in Theorem 1.2 of [Mu˜n08] (even though theformula there has a minor inaccuracy on a sign and the one on Theorem 7.1 – not in Theorem 1.2– has an extra (1 + u ) g (1 + v ) g term which should not be there). Motives of moduli spaces of L -connections with fixed determinant So far we have considered moduli spaces of flat L -connections (or Λ-modules) with fixed rank anddegree, but the techniques presented in the previous sections also allow us to obtain analogues of theprevious results for moduli spaces of flat L -connections with fixed determinant. In this section wewill define the moduli space of flat L -connections with fixed determinant and show several resultsproving the invariance of its motivic class and E -polynomial regarding the Lie algebroid structure,providing the necessary changes in the previously exposed arguments to treat the fixed determinantscenario.8.1. Twisted Higgs bundles with fixed determinant. Let ξ be an algebraic line bundle over X of degree d coprime with r . Let M L ( r, ξ ) ⊂ M L ( r, d ) be the moduli space of traceless L -twisted Higgs bundles with fixed determinant ξ , i.e., the moduli space of pairs ( E, ϕ ) with deg( E ) ∼ = ξ and ϕ ∈ H (End ( E ) ⊗ L ), where End ( E ) denotes the endomorphisms of E with trace 0.By [BGL11, Theorem 1.2], if g ≥ L ) > g − M L ( r, ξ ) is a smooth irreducible C ∗ -invariant closed subvariety of M L ( r, d ), so it is a smooth semiprojective variety. Hence thedigression held about the structure of the Bialynicki-Birula stratification of M L ( r, d ) applies to M L ( r, ξ ) as well. The fixed-point locus of the C ∗ -action on M L ( r, ξ ) clearly corresponds to theintersection of the fixed-point locus of the C ∗ -action on M L ( r, d ) with M L ( r, ξ ). In section 6 wedescribed a decomposition of the fixed-point locus as M L ( r, d ) C ∗ = [ ( r,d ) ∈ ∆ L | r | = r, | d | = d VHS L ( r, d ) , thus we have a decomposition M L ( r, ξ ) C ∗ = [ ( r,d ) ∈ ∆ L | r | = r, | d | = d VHS L ( r, d, ξ ) , where VHS L ( r, d, ξ ) = VHS L ( r, d ) ∩ M L ( r, ξ ). By construction all variations of Hodge structurehave traceless Higgs fields, soVHS L ( r, d, ξ ) = ( ( E • , ϕ • ) ∈ VHS L ( r, d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k O i =1 det( E i ) ∼ = ξ ) . On the other hand, we can consider algebraic chains with fixed “total determinant” in the followingsense. For r and d such that | r | = r and | d | = d , defineHC α ( r, d, ξ ) = ( ( E • , ϕ • ) ∈ HC α ( r, d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k O i =1 det( E i ) ∼ = ξ ) . Lemma 8.1. Given r = ( r , . . . , r k ) and a degree d line bundle ξ , consider the line bundle ξ L = ξ ⊗ L ⊗ ( P ki =1 ( i − k ) r i ) . Then the isomorphism described in Corollary 6.2 induces an isomorphism VHS L ( r, d, ξ ) ∼ = HC α L ( r, d L , ξ L ) . Proof. Given ( E • , ϕ • ) ∈ VHS L ( r, d ), the underlying bundles of its corresponding algebraic chainare ˜ E i = E i ⊗ L i − k . So if ( E • , ϕ • ) ∈ VHS L ( r, d, ξ ), then k O i =1 det( ˜ E i ) = k O i =1 (cid:16) det( E i ) ⊗ L ( i − k ) r i (cid:17) = k O i =1 det( E i ) ! ⊗ L ⊗ ( P ki =1 ( i − k ) r i ) ∼ = ξ L . The converse is analogous. (cid:3) Lemma 8.2. Fix r = ( r , . . . , r k ) and d = ( d , . . . , d k ) . Let ξ and ξ ′ be two line bundles over X ofdegree P ki =1 d i . Then HC α ( r, d, ξ ) ∼ = HC α ( r, d, ξ ′ ) . Proof. Let r = P ki =1 r i . As ξ an ξ ′ have the same degree, ξ ′ ⊗ ξ − has degree zero, so there exists aline bundle ψ such that ψ ⊗ r ∼ = ξ ′ ⊗ ξ − . Given ( E • , ϕ • ) ∈ HC α ( r, d, ξ ), consider the algebraic chain( E • ⊗ ψ, ϕ • ⊗ Id ψ ) . IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 47 As ψ has degree zero, tensoring by ψ gives an α -slope-preserving correspondence between subchainsof ( E • , ϕ • ) and those of ( E • ⊗ ψ, ϕ • ⊗ Id ψ ). Thus, ( E • ⊗ ψ, ϕ • ⊗ Id ψ ) is α -(semi)stable and clearly k O i =1 det( E • ⊗ ψ ) = k O i =1 (det( E i ) ⊗ ψ r i ) = k O i =1 det( E i ) ! ⊗ ψ P ri =1 r i ∼ = ξ ⊗ ψ r ∼ = ξ ′ , and then tensorization by ψ yields the desired isomorphism. (cid:3) Corollary 8.3. Let ξ and ξ ′ be line bundles on X of degree d . Let L and L ′ be line bundles with deg( L ) = deg( L ′ ) . Then VHS L ( r, d, ξ ) ∼ = VHS L ′ ( r, d, ξ ′ ) Proof. This follows from Lemmas 8.1 and 8.2, using the fact that α L = α L ′ , d L = d L ′ . (cid:3) Finally, consider the Bialynicki-Birula docomposition of M L ( r, ξ ) M L ( r, ξ ) = [ ( r,d ) ∈ ∆ L | r | = r, | d | = d U + r,d,ξ where, clearly, U + r,d,ξ = U r,d,ξ ∩ M L ( r, ξ ). We know that U + r,d,ξ → VHS L ( r, d, ξ ) is an affine bundleof rank N + L,r,d,ξ and, therefore, we have the analogue of equation (6.5),(8.1) [ M L ( r, ξ )] = X ( r,d ) ∈ ∆ L | r | = r, | d | = d L N + L,r,d,ξ [VHS L ( r, d, ξ )] , which can be used in an analogous way to prove the following invariance property of the motivicclass of the moduli space of L -twisted Higgs bundles with fixed determinant. Theorem 8.4. Let X be a smooth projective curve of genus g ≥ . Let L and L ′ be line bundlesover X such that deg( L ) = deg( L ′ ) > g − . Assume that ξ and ξ ′ are line bundles of degree d coprime with the rank r . Then the motives of the corresponding moduli spaces [ M L ( r, ξ )] and [ M L ′ ( r, ξ ′ )] are equal in K ( V ar C ) . Moreover, if d ′′ is any integer coprime with r , and ξ ′′ is anyline bundle of degree d ′′ , then E ( M L ( r, ξ )) = E ( M L ′ ( r, ξ ′′ )) .Proof. The argument is completely analogous to the one which lead to Theorem 6.7. One justhas to use the corresponding fixed determinant versions of the objects involved. Note that theappropriate deformation complex is C • ( E, ϕ ) : End ( E ) [ − ,ϕ ] −→ End ( E ) ⊗ L , which decomposesas C • ( E, ϕ ) = L k − l = − k +1 C • ,l ( E, ϕ ), just like C • ( E, ϕ ) in (6.6), but C • ,l ( E, ϕ ) = C • l ( E, ϕ ), so theMorse index for the fixed determinant case equals the non-fixed determinant case. Moreover, theequality of the E -polynomials is also precisely the same argument, but here one has to refer to[MS20b, Theorem 0.5] (see also [GWZ20, Corollary 7.17]), instead of the references stated in theproof of Theorem 6.7. The details are left to the reader. (cid:3) Moduli spaces of L -connections with fixed determinant. Let L be any Lie algebroid.Let E be a rank r vector bundle with determinant ξ = det( E ) = Λ r E and let ∇ L : E → E ⊗ Ω L be an integrable L -connection on E . Then ∇ L induces a maptr( ∇ L ) : ξ −→ ξ ⊗ Ω L , defined as follows. For local sections s , . . . , s r of E ,tr( ∇ L )( s ∧ . . . ∧ s r ) = r X i =1 s ∧ . . . ∧ ∇ L ( s i ) ∧ . . . ∧ s r . Observe that if v , . . . , v r is a local trivializing basis of E over some open subset of X , then if wewrite ∇ L in that basis as ∇ L = d L + G with G = ( g ij ), we gettr( ∇ L )( v ∧ . . . ∧ v r ) = r X i =1 v ∧ . . . ∧ Gv i ∧ . . . ∧ v r = r X i =1 v ∧ . . . ∧ g ii v i ∧ . . . ∧ v r = tr( G ) v ∧ . . . ∧ v r , justifying the notation “tr( ∇ L )”. Lemma 8.5. Let X be a smooth projective curve and let L be a Lie algebroid on X . Let ( E, ∇ L ) be an integrable L -connection with det( E ) ∼ = ξ . Then tr( ∇ L ) is an integrable L -connection on ξ .Proof. It is clear by construction that tr( ∇ L ) is C -linear, so we need to prove that it satisfies theLeibniz rule and that it is integrable. Let s , . . . s r be local sections E and f a local algebraicfunction on X . Then, for each j = 1 , . . . , r , we havetr( ∇ L )( s ∧ . . . ∧ f s j ∧ . . . ∧ s r ) = X i = j s ∧ . . . ∧ ∇ L ( s i ) ∧ . . . ∧ f s j ∧ . . . ∧ s r + s ∧ . . . ∧ f ∇ L ( s j ) ∧ . . . ∧ s r + s ∧ . . . ∧ s j ⊗ d L ( f ) ∧ . . . ∧ s r = f r X i =1 s ∧ . . . ∧ ∇ L ( s i ) ∧ . . . ∧ s r + s ∧ . . . ∧ s r ⊗ d L ( f )= f tr( ∇ L )( s ∧ . . . ∧ s r ) + s ∧ . . . ∧ s r ⊗ d L ( f ) , so ( ξ, tr( ∇ L )) is an L -connection.Let us now prove that it is integrable. Suppose L = ( V, [ · , · ] , δ ), with rk( V ) = k . We will proveit via a local representation of tr( ∇ L ). Let U ⊂ X be an open subset such that E and V are trivialbundles over U . Write ∇ L locally over U as ∇ L = d L + G , where G is an V ∗ -valued r × r matrix.Let w , . . . , w k be a trivializing basis of V ∗ over U . Then we can write G = k X i =1 G i ⊗ w i where G i is an O X ( U )-valued matrix. Now, we have that, over U , translates into(8.2) ∇ L = d L + G ∧ G = d L ( G ) + k X i,j =1 G i G j ⊗ w i ∧ w j = d L ( G ) + X i Let ( E, ∇ L ) ∈ M Λ L ( r, d ). As ξ is a line bundle, ( ξ, tr( ∇ L )) is automatically stable, so ( ξ, tr( ∇ L )) ∈M Λ L (1 , d ). As the determinant construction can be clearly done in families, then it defines thefollowing map, det : M Λ L ( r, d ) −→ M Λ L (1 , d ) , det( E, ∇ L ) = (det( E ) , tr( ∇ L )) . Let ( ξ, δ ) ∈ M Λ , L (1 , d ) be an integrable L -connection of rank 1 and degree d . Define M Λ L ( r, ξ, δ ) = det − ( ξ, δ ) ⊂ M Λ L ( r, d )as the moduli space of L -connections with fixed determinant ( ξ, δ ).For example, if L = ( L, , L a line bundle, then M Λ ( L, , ( r, ξ, 0) = M L − ( r, ξ ) . If T X is the canonical Lie algebroid on X , then M Λ T X ( r, O X , 0) is the moduli space of SL( r, C )-connections on X .The determinant map extends to the L -Hodge moduli space, obtaining a mapdet : M Λ red L ( r, d ) −→ M Λ red L (1 , d ) , over C , by taking det( E, ∇ L , λ ) = (det( E ) , tr( ∇ L ) , λ ). Moreover, this map is C ∗ -equivariant foraction (3.19). For each ( ξ, δ, λ ) ∈ M Λ red L (1 , d ), define M Λ red L ( r, ξ, δ ) = det − ( C · ( ξ, δ, λ ))as the L -Hodge moduli space with fixed determinant ( ξ, δ ). Here C · ( ξ, δ, λ ) denotes the closure ofthe C ∗ -orbit of ( ξ, δ, λ ) in M Λ red L (1 , d ) which, since ξ is a line bundle, is just the set of elements ofthe form ( ξ, tδ, tλ ), with t ∈ C . Then M Λ red L ( r, ξ, δ ) is clearly a C ∗ -invariant closed subvariety ofthe L -Hodge moduli space M Λ red L ( r, d ) and if π : M Λ red L ( r, ξ, δ ) → C is the restriction of the map(3.16), we have • π − (0) ∼ = M L − ( r, ξ ); • π − (1) ∼ = M Λ L ( r, ξ, δ ); • π − ( C ∗ ) ∼ = M Λ L ( r, ξ, δ ) × C ∗ .The deformation theory for this moduli space is very similar to the deformation for the modulispace of L -connections with fixed degree computed in [Tor11, Theorem 47]. Lemma 8.6. The Zariski tangent space to the moduli space M Λ L ( r, ξ, δ ) at a point ( E, ∇ L ) isisomorphic to H ( C • ( E, ∇ L )) , where C • ( E, ∇ L ) is the complex C • ( E, ∇ L ) : End ( E ) [ − , ∇ L ] −→ End ( E ) ⊗ Ω L [ − , ∇ L ] −→ . . . [ − , ∇ L ] −→ End ( E ) ⊗ Ω rk( L ) L , and the obstruction for the deformation theory lies in H ( C • ( E, ∇ L )) .Proof. The deformations of M Λ L ( r, ξ, δ ) are precisely the deformations of M Λ L ( r, d ) which preservethe determinant and trace. Following the same notation as the one used in Lemma 5.11, let U = { U α } be a covering of X such that E is trivial over U α . Fix a trivialization of E over U and foreach α and β , let g αβ : U αβ → GL( r, C ) be the transition functions for E and let ∇ L ,α = d L + G α be the local representation of ∇ L over U α .Let ( E ′ , ∇ ′L ) be a deformation of ( E, ∇ L ) over X × Spec( C [ ε ] /ε ) such that the transition func-tions of E ′ are g ′ αβ = g αβ + εg αβ and ∇ ′L ,α = εd L + G α + εG α . By [Tor11, Theorem 47], g ′ αβ and G α correspond to a deformation of ( E, ∇ L ) in M Λ L ( r, d ) if andonly if the cocycles c ∈ C ( U , End( E )) and C ∈ C ( U , End( E ) ⊗ Ω L ) defined by c ( α ) αβ = g αβ g βα and C ( α ) α = G α satisfy ∂c = 0, ∂C = ˜ ∇ L c and ˜ ∇ L C = 0. We will prove that ( c, C ) defines a deformation of ( E, ∇ L ) in M Λ L ( r, ξ, δ ) if and only if ( c, C ) ∈ C ( U , End ( E )) × C ( U , End ( E ) ⊗ Ω L ). We have tr( ∇ ′L ) = δ = tr( ∇ L ) if and only iftr( G α ) = tr( G α ) = tr( G α ) + ε tr( G α ) , so tr( G α ) = 0 and, therefore, C ∈ C ( U , End ( E ) ⊗ Ω L ). On the other hand, det( E ′ ) = ξ if andonly ifdet( g αβ ) = det( g αβ ) = det( g αβ + εg αβ ) = det( g αβ ) + ε r X i =1 det (cid:0) ( g αβ ) | · · · | ( g αβ ) i | · · · | ( g αβ ) r (cid:1) , since ε = 0. Here, (( g αβ ) | · · · | ( g αβ ) i | · · · | ( g αβ ) r ) denotes the r × r matrix whose i -th column equalsthe i -th column of g αβ and the other columns are the corresponding ones of g αβ . Set D i = det (cid:0) ( g αβ ) | · · · | ( g αβ ) i | · · · ( g αβ ) r (cid:1) so that det( E ′ ) = ξ if and only if P ri =1 D i = 0. Let A be such that g αβ A = g αβ . By Cramer’s rule, D i = A ii det( g αβ ), thus r X i =1 D i = det( g αβ ) tr( A ) = det( g αβ ) tr( g − αβ g αβ ) = det( g αβ ) tr( c ( α ) αβ ) , and so det( E ′ ) = ξ if and only if tr( c ) = 0, i.e., if c ∈ C ( U , End ( E )).The rest of the proof is exactly the same as the one of [Tor11, Theorem 47]. (cid:3) Proposition 8.7. Let X be a smooth projective curve of genus g ≥ . Let L be a Lie algebroidwith rk( L ) = 1 and deg( L ) < − g . Take r and d is coprime. Then, for each ( ξ, δ ) ∈ M Λ L (1 , d ) ,the moduli space M Λ red L ( r, ξ, δ ) is a smooth semiprojective variety for the C ∗ -action t · ( E, ∇ L , λ ) =( E, t ∇ L , tλ ) . Furthermore, the map π : M Λ red L ( r, ξ, δ ) → C , π ( E, ∇ L , λ ) = λ is a surjective sub-mersion and M Λ L ( r, ξ, δ ) is a smooth variety of dimension deg( L )(1 − r ) .Proof. The argument is exactly the same as the one carried on in section 5.3. The only differenceis that the computation of the dimension of the tangent bundle done in Lemma 5.10 now becomesdim T ( E, ∇ L ) M Λ L ( r, ξ, δ ) = deg( L )(1 − r ) + dim (cid:0) H ( C • ( E, ∇ L )) (cid:1) . Notice that here we have to take trace-free endomorphisms, hence by point (3) of Lemma 5.9, H ( C • ( E, ∇ L )) = 0. Taking into account Lemma 8.6, the deformation theory computed in Lemma5.11 becomes now T ( E, ∇ L , M Λ red L ( r, ξ, δ ) ∼ = ( c, C, λ ε ) ∈ (cid:18) C ( U , End( E )) × C ( U , End( E ) ⊗ Ω L ) × C (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂c = 0 ∂C = ˜ ∇ L c + λ ε ω ˜ ∇ L C = − λ ε d L ( ∇ L )tr( c ) = 0tr( C ) = λ ε δ n ( ∂η, ˜ ∇ L η, (cid:12)(cid:12)(cid:12) η ∈ C ( U , End ( E )) o with dπ ([( c, C, λ ε )]) = λ ε . Then, clearlyker dπ ∼ = ( c, C, ∈ (cid:18) C ( U , End ( E )) × C ( U , End ( E ) ⊗ Ω L ) × C (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂c = 0 ∂C = ˜ ∇ L c + λ ε ω ˜ ∇ L C = − λ ε d L ( ∇ L ) n ( ∂η, ˜ ∇ L η, (cid:12)(cid:12)(cid:12) η ∈ C ( U , End ( E )) o ∼ = T ( E, ∇ L ) M L − ( r, ξ )and the proof proceeds exactly as in Lemma 5.13 and Theorem 5.15. (cid:3) IE ALGEBROID CONNECTIONS, TWISTED HIGGS BUNDLES AND MOTIVES OF MODULI SPACES 51 Theorem 8.8. Let L = ( L, [ · , · ] , δ ) be Lie algebroid on X such that L is a line bundle with deg( L ) < − g . If r and d are coprime, then, for each ( ξ, δ ) ∈ M Λ L (1 , d ) , we have [ M Λ L ( r, ξ, δ )] = [ M L − ( r, ξ, δ )] , [ M Λ red L ( r, ξ, δ )] = L [ M L − ( r, ξ, δ )] and we have an isomorphism of Hodge structures H • ( M Λ L ( r, ξ, δ )) ∼ = H • ( M L − ( r, ξ, δ )) In particular, E ( M Λ L ( r, ξ, δ )) = E ( M L − ( r, ξ, δ )) , E ( M Λ red L ( r, ξ, δ )) = uvE ( M L − ( r, ξ, δ )) . Moreover, both M Λ L ( r, ξ, δ ) and M Λ red L ( r, ξ, δ ) have pure mixed Hodge structures.Proof. The proof is completely analogous to that of Theorem 5.17. The details are left to thereader. (cid:3) Finally, combining this result with Theorem 8.4 and working analogously to Theorem 7.4, yieldsthe fixed-determinant version of Theorems 6.7 and 7.4. Theorem 8.9. Let X be a smooth projective curve of genus g ≥ and let L and L ′ be any Liealgebroids on X such that rk( L ) = rk( L ′ ) = 1 and deg( L ) = deg( L ′ ) < − g . Suppose that r and d are coprime. Let ( ξ, δ ) ∈ M Λ L (1 , d ) and ( ξ ′ , δ ′ ) ∈ M Λ L′ (1 , d ) . Then I ( M Λ L ( r, ξ, δ )) = I ( M Λ L′ ( r, ξ ′ , δ ′ )) where I ( X ) denotes one of the following(1) The virtual motive [ X ] ∈ ˆ K ( V ar C ) ;(2) The Voevodsky motive M ( X ) ∈ DM eff ( C , R ) for any ring R . 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