Integrable systems and Torelli theorems for the moduli spaces of parabolic bundles and parabolic Higgs bundles
aa r X i v : . [ m a t h . AG ] N ov INTEGRABLE SYSTEMS AND TORELLI THEOREMS FOR THEMODULI SPACES OF PARABOLIC BUNDLES AND PARABOLICHIGGS BUNDLES
INDRANIL BISWAS, TOM ´AS L. G ´OMEZ, AND MARINA LOGARES
Abstract.
We prove a Torelli theorem for the moduli space of semistable parabolicHiggs bundles over a smooth complex projective algebraic curve under the assumptionthat the parabolic weight system is generic. When the genus is at least two, usingthis result we also prove a Torelli theorem for the moduli space of semistable parabolicbundles of rank at least two with generic parabolic weights. The key input in the proofsis a method of [Hu]. Introduction
The classical theorem by R. Torelli [CRS] says that a smooth complex algebraic curveis determined by the isomorphism class of its polarized Jacobian up to isomorphism.Similar theorems in many contexts have been worked out, e.g., for moduli spaces ofstable vector bundles [Tj, NR, MN] and moduli spaces of stable Higgs bundles [BG]. Asfar as moduli spaces of parabolic or parabolic Higgs bundles with fixed determinant (seedefinition below) are concerned, a set of Torelli theorems were proved [BBB, BHK, Seb,GL]. Here we deal with the non-fixed determinant situation.In [Hu], Hurtubise investigated algebraically completely integrable systems satisfyingcertain conditions. His main result is to extract an algebraic surface out of an integrablesystem. We observe that a moduli space of parabolic Higgs bundles is an example ofthe model of completely integrable systems studied in [Hu].The above mentioned assumption that the determinant is not fixed stems from the factthat in the set-up of [Hu] the Lagrangians in the fibers are required to be Jacobians, whilefixing the determinant amounts to making the fibers Prym varieties. To consider themoduli spaces with fixed determinant with our techniques, we would need an analogueof our main tool, namely Theorem 1.11 of [Hu], but for an integrable system in whichthe fibers are Prym varieties instead of Jacobians. This is planned for future work.
Mathematics Subject Classification.
Primary: 14D22 Secondary: 14D20.
Key words and phrases.
Parabolic bundles, Higgs field, Torelli theorem.The authors want to thank the support by MINECO: ICMAT Severo Ochoa project SEV-2011-0087 and MTM2010/17389. We acknowledge the support of the grant 612534 MODULI within the7th European Union Framework Programme. The second and third author also wants to thank TataInstitute for Fundamental Research (Mumbai) where this work was finished. The third author wasalso partially supported by FCT (Portugal) with European Regional Development Fund (COMPETE),national funds through the projects PTDC/MAT/098770/2008 and PTDC/MAT/099275/2008. Thefirst author is supported by J. C. Bose Fellowship.
We will prove the following theorems.
Theorem 1.1 (Main Theorem) . Let X and X ′ be smooth projective curves with genus g and parabolic points D and D ′ respectively. Let M X ( d, r, α ) (respectively, M X ′ ( d, r, α ) )be the moduli space of stable parabolic Higgs bundles over X (respectively, X ′ ) endowedwith the usual C ∗ action (cf. (2) ) and the determinant line bundle L (respectively, L ′ ) (cf. (6) ). If there is a C ∗ -equivariant isomorphism between M X ( d, r, α ) and M X ′ ( d, r, α ) ,such the the pullback of the N´eron-Severi class N S ( L ′ ) is N S ( L ) , then there exists anisomorphism between X and X ′ inducing a bijection between the parabolic points D and D ′ , whenever the following conditions on the genus and the rank are satisfied, • if g = 2 then r ≥ , • if g = 3 then r ≥ , • if g ≥ then r ≥ . Since the moduli space of stable parabolic bundles sits inside the moduli space ofstable parabolic Higgs bundles, in all cases where its codimension is greater than two,we get the following extension of the Torelli theorem for the moduli space of stableparabolic bundles given in [BBB].
Theorem 1.2.
Let X and X ′ be smooth projective curves with genus g and parabolicpoints D and D ′ respectively. Let M X ( d, r, α ) be the moduli space of stable parabolic bun-dles over X (respectively, M X ′ ( d, r, α ) ), and let L (respectively, L ′ ) be the determinantline bundle (cf. (6) ). If there is an isomorphism between M X ( d, r, α ) and M X ′ ( d, r, α ) such that the pullback of N S ( L ′ ) is N S ( L ) , then there exists an isomorphism between X and X ′ inducing a bijection between the parabolic points D and D ′ , whenever thefollowing conditions on the genus and the rank are satisfied, • if g = 2 then r ≥ , • if g = 3 then r ≥ , • if g ≥ then r ≥ . Preliminaries
Let X be an irreducible smooth projective algebraic curve over C . The holomorphiccotangent bundle of X will be denoted by K . Let { p , · · · , p n } be a set of distinct parabolic points in X and let D = p + . . . + p n the corresponding reduced effective divi-sor. A parabolic bundle on X with parabolic structure over D consists of a holomorphicvector bundle E equipped with a weighted flag over each parabolic point p ∈ D , that isa filtration of subspaces E | p = E p, ⊃ · · · ⊃ E p,r ( p ) ⊃ E p,r ( p )+1 = 0together with a system of parabolic weights ≤ α ( p ) < · · · < α r ( p ) ( p ) < . NTEGRABLE SYSTEMS AND TORELLI THEOREMS 3
The parabolic degree and parabolic slope of E are defined as follows:pardeg ( E ) := deg( E ) + X p ∈ D r ( p ) X i =1 α i ( p ) · m i ( p ) par µ ( E ) := pardeg ( E )rk ( E ) , where m i ( p ) := dim( E p,i /E p,i +1 ) is the multiplicity of the parabolic weight α i ( p ). Theparabolic bundle is called stable (respectively, semistable ) if for all subbundles 0 = V ( E , par µ ( V ) < par µ ( E ) (respectively, par µ ( V ) ≤ par µ ( E )) (1)where V has the induced parabolic structure. Given rank and degree the system ofparabolic weights is called generic if every semistable parabolic bundle is stable. Wenote that the semistability condition describes hyperplanes (or walls ) in the space ofweights. Hence the genericity condition means that the parabolic weights lie in theinterior of the chambers defined by the walls.We denote by M X ( d, r, α ) the moduli space of stable parabolic bundles over X withdegree d , rank r and generic weights α . This moduli space is a smooth projective varietywith dim M X ( d, r, α ) = r ( g −
1) + 1 + 12 X p ∈ D r ( p ) X i =1 ( r − m i ( p ) ) . For notational convenience we assume that the flag is full that is, m i ( p ) = 1 for all p and i , so r ( p ) = r for all p , but all the results generalize to non full flags case. Henceforth,we will only consider full flags. Therefore,dim M X ( d, r, α ) = r ( g −
1) + 1 + 12 nr ( r − . An endomorphism of a parabolic bundle E is called non-strongly parabolic if, for all p ∈ D and i , ϕ ( E p,i ) ⊂ E x,i , and it is called strongly parabolic if ϕ ( E p,i ) ⊂ E p,i +1 . The sheaves of non-strongly and strongly parabolic endomorphisms are denoted byParEnd ( E ) and SParEnd ( E ) respectively.A parabolic Higgs bundle is a pair ( E, Φ) where E is a parabolic bundle andΦ : E −→ E ⊗ K ( D ) = E ⊗ K ⊗ O X ( D )is a strongly parabolic homomorphism, i.e.,Φ( E x,i ) ⊂ E x,i +1 ⊗ K ( D ) x for each point x ∈ D and all i . A parabolic Higgs bundle is stable (respectively, semistable ) if the inequality (1) is satisfied for those V with Φ( V ) ⊂ V ⊗ K ( D ).Let M X ( d, r, α ) denote the moduli space of stable parabolic Higgs bundles with degree d , rank r and generic weights α . It is a smooth quasiprojective variety that satisfydim M X ( d, r, α ) = 2 r ( g −
1) + 2 + nr ( r −
1) = 2 · dim M X ( d, r, α ) I. BISWAS, T. L. G ´OMEZ, AND M. LOGARES (recall that the quasiparabolic flags are full).For any E ∈ M X ( r, d, α ) the tangent space at E , T E M X ( r, d, α ), is H (ParEnd ( E )).Also, the parabolic version of Serre duality gives an isomorphism H (ParEnd ( E )) ∗ ∼ = H (SParEnd ( E ) ⊗ K ( D )) . Therefore, the total space of the cotangent bundle T ∗ M X ( r, d, α ) is a Zariski open subsetof M X ( r, d, α ).The moduli space of parabolic Higgs bundles is endowed with a C ∗ action, where t ∈ C ∗ acts as scalar multiplication on the Higgs field( E, Φ) ( E, t · Φ) (2)The total space of the cotangent bundle T ∗ M X ( r, d, α ) also has a canonical C ∗ actiongiven by scalar multiplication on the fibers. Both actions are compatible, in the sensethat the inclusion of the cotangent in the moduli space of Higgs bundles is C ∗ equivariant.3. The Hitchin system
Let K ( D ) denote the total space of the line bundle K ( D ) over X , and let γ : K ( D ) −→ X be the natural projection. Let e x ∈ H ( K ( D ) , γ ∗ K ( D ))be the tautological section whose evaluation at any point z is z itself. The characteristicpolynomial of a Higgs field Φ isdet( e x · Id − γ ∗ Φ) = e x r + e s e x r − + e s e x r − + · · · + e s r . (3)The sections e s i , descent to X , meaning there are sections s i ∈ H ( X, K i ( iD )) such that e s i = γ ∗ s i . Since Φ is strongly parabolic its residue at each parabolic point is nilpotent,and hence s i ∈ H ( X, K i (( i − D )). Therefore, there is a morphism, called the Hitchinmap , H : M X ( d, r, α ) −→ U := r M i =1 H ( X, K i (( i − D )) . (4)This morphism is proper [Hi], and it induces an isomorphism on globally defined al-gebraic functions, i.e., the lower arrow in the following commutative diagram is anisomorphism M X ( d, r, α ) a (cid:15) (cid:15) H / / U Spec Γ( M X ( d, r, α )) ∼ = / / Spec Γ( U ) (5)The variety M X ( d, r, α ) has a natural holomorphic symplectic structure, and the Hitchinmap defines an algebraically complete integrable system, in particular, the fibers of H are Lagrangians (these is explained in [GL]).When the parabolic set is empty ( n = 0), Hausel proved that the nilpotent cone H − (0) coincides with the downwards Morse flow on M X ( d, r, α ) giving a deformation NTEGRABLE SYSTEMS AND TORELLI THEOREMS 5 retraction of M X ( d, r, α ) to H − (0) [Hau, Theorem 5.2]. The proof in [Hau] can betranslated into the parabolic situation word by word.The fiber of H over a point u ∈ U is canonically isomorphic to the Jacobian of acurve called the spectral curve ; we now recall its construction.Given a point u = ( s , · · · , s r ) ∈ U consider the curve X u ⊂ K ( D ) defined by theequation e x r + s e x r − + s e x r − + · · · + s r = 0(compare it with (3)). Note that when X u is reduced, the projection ρ := γ | X u : X u −→ X is a ramified covering of X of degree r which is completely ramified over the parabolicpoints. Denote by R u the ramification divisor on X u . Denote by S the family of spectralcurves over U . Proposition 3.1.
For any u ∈ U such that the corresponding spectral curve X u issmooth, the fiber H − ( u ) is identified with Pic d + r ( r − g − n ) / ( X u ) .Proof. It follows from the proof of Proposition 3.6. in [BNR]. (cid:3)
Let E be a universal bundle on X × M X ( d, r, α ) and let q be the projection to M X ( d, r, α ). Fix a point x ∈ X of the curve. Let χ = χ ( E ) (since we have fixedthe rank and degree, this does not depend on the particular E chosen, and can be cal-culated by the Riemann-Roch formula). There is a line bundle L x defined as follows[KM] L x = det( Rq ∗ E ) − r ⊗ ( ∧ r E | x ×M ) χ (6)where the presence of the second factor is a normalization that guarantees that this doesnot depend on the choice of universal bundle. Note that this determinant line bundlecan also be defined for the moduli space M X ( d, r, α ) without Higgs bundle.We remark that this line bundle is invariant under the standard C ∗ action (2) and wecan choose a lift of this action.The fiber of this line bundle over a point corresponding to a Higgs bundle ( E, Φ) iscanonically isomorphic to h ( ∧ top H ( X, E )) ∗ ⊗ ( ∧ top H ( X, E )) i ⊗ r ⊗ ( ∧ E x ) χ Since the curve X is connected, the N´eron-Severi class N S ( L x ) of the line bundle doesnot depend on the choice of the point x ∈ X . Lemma 3.2.
Let u ∈ U is a point in the Hitchin space corresponding to a smooth curve,then the restriction of the line bundle L x to the fiber H − ( u ) = Pic d + r ( r − g − n ) / ( X u ) is a multiple of the principal polarization of the Jacobian J ( X u ) of the spectral curve X u Proof.
Let ( E, Φ) be a point in the moduli space M . If it is in the fiber H − ( u ), thenthere is a line bundle η on the spectral curve π : X u −→ X such that E = π ∗ η . Then I. BISWAS, T. L. G ´OMEZ, AND M. LOGARES the fiber of L x over this point is canonically isomorphic to h ( ∧ top H ( X, π ∗ η )) ∗ ⊗ ( ∧ top H ( X, π ∗ η )) i ⊗ r ⊗ ( ∧ ( π ∗ η ) x ) χ = h ( ∧ top H ( X s , η )) ∗ ⊗ ( ∧ top H ( X s , η )) i ⊗ r ⊗ ( ∧ η π − ( x ) ) χ This is the fiber of a line bundle defining a multiple of a principal polarization of theJacobian. The last factor is just a normalization, and the N´eron-Severi class of the linebundle does not depend on the choice of the point. (cid:3)
In [Hu], Hurtubise considers (local) integrable systems H : J −→ U , where U is an open subset of C m and J is a 2 m -dimensional symplectic variety withholomorphic symplectic form Ω, such that the fibers of H are Lagrangian. Furthermore,suppose there is a family of curves H ′ : S −→ U such that for each u ∈ U , the fiber J u = H − ( u ) is isomorphic to the Jacobian of S u = H ′− ( u ). To define the Abel map I : S −→ J we need a section of H ′ . This can be done locally on U . Under the assumption that I ∗ Ω ∧ I ∗ Ω = 0 (7)Hurtubise proves that for the embedding I the variety S is coisotropic, and the quo-tienting of S by the null foliation results a surface Q . The form I ∗ Ω descends to Q ,and the descended form on Q , which we will denote by ω , is a holomorphic symplecticform [Hu, Theorem 1.11]. He also proves that, choosing a different Abel map I ′ with I ′∗ Ω ∧ I ′∗ Ω = 0, we have I ∗ Ω = I ′∗ Ω when m ≥
3, so that the surface Q depends onlyon S and it is independent of the Abel map. We summarize: Theorem 3.3 ([Hu, Theorem 1.11 (i) and (ii)]) . For an integrable system H : J −→ U ⊂ C m , with maps H ′ : S −→ U , and I : S −→ J , as described above, there is an invariantsurface Q which only depends on S and not on the Abel map I , whenever m ≥ . In [Hu, Example 4.3] he shows that all these conditions are satisfied for the usualmoduli space of Higgs bundles (i.e., no parabolic points), but restricted to the opensubset U of the Hitchin space U corresponding to smooth spectral curves H : M X | U −→ U .
Let q be the projection q : S −→ K sending each point on a spectral curve to the totalspace of the cotangent bundle and let ω the natural symplectic form on the cotangent.Hurtubise shows that I ∗ Ω = q ∗ ω It follows that the surface Q is K . NTEGRABLE SYSTEMS AND TORELLI THEOREMS 7
The conditions of the theorem also hold for the moduli space of strongly parabolicHiggs bundles equipped with the Hitchin map, and in this case the surface Q is theimage of S −→ K ( D ). Note that all spectral curves go through zero on the fibers overthe parabolic points, because the eigenvalues of the residues are zero. Therefore, weobtain the following Corollary, which will be our main tool in the proof of the MainTheorem.Note that the integer m in the statement of Theorem 3.3 is the genus of the spectralcurve, which is equal to dim M X ( d, r, α ) and hence, under the assumptions on genus andrank of Theorems 1.1 and 1.2 we always have m ≥ Corollary 3.4.
Let H : M X ( d, r, α ) | U −→ U . be the restriction of the Hitchin map on the moduli space of parabolic Higgs bundles withgeneric weights α to the open set U corresponding to nonsingular curves (cf. Lemma (4.1) ). Then this integrable system satisfies the conditions of the Theorem of Hurtubiseand the surface Q is the image of K in K ( D ) under the injective morphism of sheaves K −→ K ( D ) . Proof of the Theorems
Let h : T ∗ M X ( d, r, α ) −→ U = r M i =1 H ( X, K i (( i − D ))be the restriction to the cotangent bundle of the moduli space of stable bundles of theHitchin integrable system in (4). To each point u ∈ U we associated its spectral curve X u ⊂ S . Lemma 4.1. If g ≥ , then the Zariski open subset U of U that parametrizes the smoothspectral curves is non-empty.Proof. If K r (( r − D ) has a section without multiple zeros, then the above open subset U is nonempty (cf. [BNR, Remark 3.5]). A holomorphic line bundle on X of degreeat least 2 g + 1 is very ample (cf. [Har, IV Corollary 3.2]), and hence U is non-emptywhenever r (2 g −
2) + ( r − n ≥ g + 1, and this holds when g ≥ (cid:3) Define J := H − ( U ), where H is the Hitchin map for the moduli of Higgs bundles(4) and U is the open subset in Lemma 4.1. Let H J : J −→ U be the restriction of H . Let H S : S −→ U be the total space for the family of spectral curve over U , so that the fiber of H S overany u ∈ U is the spectral curve X u . I. BISWAS, T. L. G ´OMEZ, AND M. LOGARES
As we have seen in Corollary 3.4, the surface Q given by the Theorem of Hurtubisein this setting is the image of K in K ( D ) under the injective morphism of sheaves K −→ K ( D ). In particular, Q is singular.The moduli space of parabolic Higgs bundles is known to be a K¨ahler manifold pro-vided with a C ∗ action whose restriction to an S action preserves the K¨ahler structure τ : C ∗ × M X ( r, d, α ) −→ M X ( r, d, α ) (8)( t , ( E ,
Φ)) ( E , t Φ) . This C ∗ action is compatible with scalar multiplication in the fibers of the cotangentbundle T ∗ M X ( r, d, α ) under the inclusion of this cotangent bundle in the moduli ofparabolic Higgs bundles. It induces a C ∗ action on S : C ∗ × S −→ S (9)( t , x ∈ X u ) ( tx ∈ X t · u ) , where t · ( s , · · · , s r ) = ( ts , t s , · · · , t r s r ) (see (3)), and the multiplication tx is definedusing the embedding of the spectral curve X u in the total space of K ( D ). This actionof C ∗ on S evidently produces an action of C ∗ on the quotient surface Q . Let Q C ∗ ⊂ Q be the fixed point locus for the above C ∗ action on Q . Lemma 4.2.
The subset Q C ∗ is the zero section of the fibration K ( D ) −→ X .Proof. Since the natural inclusion
K ֒ → K ( D ) of O X –modules commutes with themultiplicative action of C ∗ , the surface Q , which is the image of the total space of K in K ( D ), is preserved by the action of C ∗ on K ( D ). Therefore, the action of C ∗ on K ( D )produces an action of C ∗ on Q . This action of C ∗ on Q coincides with the action on Q induced by (9). The lemma follows from this. (cid:3) Corollary 4.3.
The curve X coincides with Q C ∗ . Proposition 4.4.
The set of parabolic points coincides with the subset of Q C ∗ throughwhich every spectral cover pass.Proof. Since the residue of Φ on the parabolic points is nilpotent, all spectral curves X u totally ramify over the parabolic points, and they intersect the fibre over the parabolicpoints at zero.Conversely, let x ∈ X be a point which is not parabolic. There exists a section s r ∈ H ( K r (( r − D )) which does not vanish at x since this linear systems is base pointfree (recall that we are assuming g ≥ x when considered as a section of H ( K r ( rD )) because x is not a parabolic point.Therefore, the spectral curve e x r + s r = 0 on K ( D ) intersects the fibre over x away fromzero, and the spectral curve e x r = 0 intersects it only at zero, so there is no point overthe fibre of x through which every spectral cover passes. (cid:3) NTEGRABLE SYSTEMS AND TORELLI THEOREMS 9
Proof of Theorem 1.1.
We are given the moduli space M as an abstract algebraicvariety with a holomorphic symplectic form, a line bundle L and a algebraic C ∗ actionon M with a linearization on L . Looking at global functions on M α : M −→
Spec Γ( M )we obtain a morphism α which is isomorphic to the Hitchin fibration (cf. (5)) and thefibers are Lagrangians with respect to the given holomorphic symplectic form. The sub-set U ⊂ Spec Γ( M ) of points corresponding to smooth spectral curves can be recoveredas the points whose fibers are abelian varieties. Let β be the restriction of α over U J (cid:31) (cid:127) / / β (cid:15) (cid:15) M α (cid:15) (cid:15) U (cid:31) (cid:127) / / Spec Γ M The line bundle L restricts to a (multiple of) a principal polarization on these abelianvarieties, and then the classical Torelli theorem gives us a family of curves S −→ U , suchthat the fiber J u over u ∈ U is the Jacobian of S u . Locally on U there is an Abel-Jacobimap I : S −→ J .The C ∗ action on M restricts to a C ∗ action on the family of Jacobians J . Thisfamily of Jacobians has a family of principal polarizations given by the line bundle L .The C ∗ action has a lift to L hence we have an action on the family of principal polarizedJacobians.By the proof given by Weil of the Torelli theorem [We, Hauptsatz, p. 35], an isomor-phism ψ : ( J u , θ u ) −→ ( J u ′ , θ u ′ ) of principal polarized Jacobians induces an isomorphism f : S u −→ S ′ u of the corresponding curves, and this provides an action of C ∗ on the familyof curves S .Now we apply Corollary 3.4 to obtain a surface Q as a quotient of S . The actionon S we have just defined clearly coincides with the action given in (9), therefore byCorollary 4.3 we recover X , and by Proposition 4.4 we recover the parabolic points D ,thus proving our main theorem.4.2. Proof of Theorem 1.2.
We are given the moduli space as a smooth algebraicvariety M with a line bundle L . We consider the total space of the cotangent bundle T ∗ M . This has a canonical holomorphic symplectic structure, and a C ∗ given by scalarmultiplication on the fibers. The pullback of the line bundle to T ∗ M is trivial along thefibers, so there is a canonical lift of the C ∗ action to the pullback of the line bundle L to T ∗ M .We claim that the generic fiber of the morphism given by global sections h : T ∗ M −→ Spec(Γ( T ∗ M ))is an open subset of an abelian variety. Indeed, we know that M is the moduli space forsome algebraic curve X (which we want to find), so we know that T ∗ M is an open subsetof a moduli space of parabolic Higgs bundles M , and by Corollary 5.11 we know thatthe codimension of the complement of this open set is at least two. Therefore, global section on T ∗ M extend uniquely to global sections on M and the morphism h is therestriction of the morphism of global sections of some moduli space of Higgs bundles M T ∗ M h / / (cid:127) _ (cid:15) (cid:15) Spec Γ( T ∗ M ) M H / / Spec Γ( M )The compactification of the fiber over u to an abelian variety is unique, because birationalabelian varieties are isomorphic. Therefore, the isomorphism class of J := H − ( U ) isuniquely defined by the isomorphism class of M , and does not depend on the choice of M .Since the codimension of the complement of the inclusion T ∗ M | U ⊂ J is at least two,all the structure that we have on T ∗ M extends uniquely to J , namely the determinantline bundle L , the C ∗ action with the lift to L and the holomorphic symplectic form.Therefore we can now use the same arguments as in the proof of the main theorem torecover the curve X and the parabolic points.5. Codimension computation
In this section we compute the codimension of the complement of T ∗ M ( d, r, α ) inside M ( d, r, α ) fiber-wise following the arguments in [BGL, Section 5]. This complement is V = { ( E, Φ) ∈ M ( d, r, α ) | E is not stable } . Recall from (8) that the moduli space of parabolic Higgs bundles is known to be aK¨ahler manifold provided with a C ∗ action, whose restriction to a S action preservesthe K¨ahler structure.This action provide us with two stratifications of the moduli space. The first one isthe Bia lynicki-Birula stratification consisting of subsets of M ( d, r, α ) such U + λ := { p ∈ M X ( d, r, α ); lim t → tp ∈ F λ } and U − λ := { p ∈ M X ( d, r, α ); lim t →∞ tp ∈ F λ } where F λ are the disjoint connected components of the fixed pointed set F for the C ∗ -action on M ( d, r, α ).The second one is known as the Morse stratification and comes from the restrictionof the C ∗ -action to an S -action. The last also preserves the K¨ahler form, hence it giveus a circle Hamiltonian action on M ( d, r, α ) with associated moment map, µ : M X ( r, d, α ) −→ R ( E, Φ) Φ || which is proper, bounded below and has a finite number of critical submanifolds. Sothis map is a Morse-Bott map. NTEGRABLE SYSTEMS AND TORELLI THEOREMS 11
For any component F λ , we recall the definition of the upwards Morse strata, e U + λ , andthe downwards Morse strata, e U − λ , that is e U + λ := { p ∈ M X ( d, r, α ); lim t →−∞ ψ t ( p ) ∈ F λ } and e U − λ := { p ∈ M X ( d, r, α ); lim t → + ∞ ψ t ( p ) ∈ F λ } Recall that this stratifications were proven to be equal U + = e U + and U − = e U − byKirwan in [Ki, Theorem 6.16].The union N = S λ e U − λ is known as downwards Morse flow .The inverse over the 0 point of the Hitchin map H − (0) is called nilpotent cone , andit coincides with the downwards Morse flow, i.e. N = H − (0) [GGM, Theorem 3.13].The following proposition takes the same steps as Proposition 5.1 in [BGL] providedthat in a family of parabolic bundles the Harder-Narasimhan type increases under spe-cialization. Proposition 5.1.
Let V be the complement of the cotangent bundle of M X ( d, r, α ) in M X ( d, r, α ) and let V ′ be the Bia lynicki-Birula flow which does not converge to M X ( d, r, α ) , that is V = { ( E, Φ) ∈ M X ( d, r, α ) : E is not stable } and V ′ = { ( E, Φ) : lim t → ( E, t Φ) / ∈ M X ( d, r, α ) } . Then V ′ = V . Proof.
Let ( E, Φ) / ∈ V that is E is stable, then lim t → ( E, t
Φ) = ( E, ∈ M X ( d, r, α ), soit proves that V ′ ⊂ V . To prove the converse, take ( E, Φ) where E is not stable. Thereexists a Harder-Narasimhan filtration for E , that is E = E m ⊃ E m − ⊃ · · · ⊃ E ⊃ . Following Atiyah and Bott [AB] we define the type of the Harder-Narasimhan filtration,as the following vector ( µ , . . . , µ r ) where µ i = deg( F i ) / rk ( F i ) and F i = E i /E i +1 .In a family of parabolic bundles Nitsure [Ni, Proposition 1.10] proved that the Harder-Narasimhan type increases under specialization. Hence, as the Hitchin map is proper,and its composition with the map given by the C ∗ -action, is such that lim t → h ( E, t
Φ) = 0then τ extends to a morphism e τ : C × M ( d, r, α ) −→ M ( d, r, α ) . This maps gives, by pullback, that a family over C ∗ of non semistable parabolic bundlesspecializes to a non semistable parabolic bundle. That is, for E , the universal bundle over M ( d, r, α ) × X when pullback ( e τ × X ) ∗ ( E ) gives a family of bundles over X parametrizedby C . If ( e τ × X ) ∗ ( E ) | t × X = E for t = 0, is not semistable, then the specializationresult says that ( e τ × X ) ∗ ( E ) | × X is not semistable. This completes the proof. (cid:3) The following facts are recovered from the literature on parabolic Higgs bundles.Let ( E, Φ) be a fixed point for the circle action, we have an isomorphism ( E, Φ) ∼ =( E, e iθ Φ) for θ ∈ [0 , π ) yielding the following commutative diagram. E Φ / / ψ θ (cid:15) (cid:15) E ⊗ K ( D ) ψ θ ⊗ K ( D ) (cid:15) (cid:15) E e iθ Φ / / E ⊗ K ( D ) . Proposition 5.2 ([Si, Theorem 8]) . If ( E, Φ) belongs to a critical subvariety F λ for thecircle action on M X ( d, r, α ) then E splits E = m M l =0 E l and Φ ∈ H (SParHom ( E l , E l +1 ) ⊗ K ( D )) . The parabolic Higgs bundle in this case ( E, Φ) is called Hodge bundle.The deformation theory of the moduli space of parabolic Higgs bundles was workedout in [Yo]. It is given by the following complex of bundles, C • ( E ) : ParEnd ( E ) Φ:=[ · , Φ] −→ SParEnd ( E ) ⊗ K ( D ) . The tangent space of the moduli space M X ( d, r, α ) at a stable point ( E, Φ) is thenthe first cohomology group H ( C • ( E )) of this complex. Hence for a fixed point ( E, Φ)of the C ∗ -action, the decomposition in Proposition 5.2 induces a decomposition of thedeformation complex and of the tangent space at the fixed point. That is, we define C k := M j − i = k ParHom ( E i , E j ) and b C k +1 := M j − i = k SParHom ( E i , E j )so then C • ( E ) k : C k Φ k −→ b C k +1 ⊗ K ( D ) , and C • ( E ) = k = m M k = − m − C • ( E ) k . For this deformation complex, there is a long exact sequence0 → H ( C • ( E ) k ) → H ( M j − i = k ParHom ( E i , E j ) → H ( M j − i = k SParHom ( E i , E j ) ⊗ K ( D )) → H ( C • ( E ) k ) → H ( M j − i = k ParHom ( E i , E j ) → H ( M j − i = k SParHom ( E i , E j ) ⊗ K ( D )) → H ( C • ( E ) k ) → . Proposition 5.3 ([GGM, Proposition 3.9]) . NTEGRABLE SYSTEMS AND TORELLI THEOREMS 13 • There is a natural isomorphism H ( C • ( E ) k ) ≃ H ( C • ( E ) − k − ) ∗ and hence a natural isomorphism T ( E, Φ) M k ≃ ( T ( E, Φ) M − k ) ∗ • If ( E, Φ) is stable, then we have H ( C • ( E ) k ) = (cid:26) C if k = 00 otherwise, and H ( C • ( E ) k ) = (cid:26) C if k = − otherwise. (cid:3) Hence,
Lemma 5.4. dim H ( C • ( E ) k ) = (cid:26) − χ ( C • ( E ) k ) if k = 0 − χ ( C • ( E ) k ) otherwise. (cid:3) Theorem 5.5 ([GGM, Theorem 3.8]) . The function µ : M X ( r, d, α ) −→ R defined by µ ( E, Φ) = k Φ k is a perfect Bott–Morse function. A parabolic Higgs bundle representsa critical point of µ if and only if it is a parabolic complex variation of Hodge structure,i.e. E = L mk =0 E k with Φ k = Φ | E k : E l −→ E k +1 ⊗ K ( D ) strongly parabolic (where Φ = 0 if and only if m = 0 ). The tangent space to M X ( r, d, α ) at a critical point ( E, Φ) decomposes as T ( E, Φ) M X ( r, d, α ) = m +1 M k = − m T ( E, Φ) M X ( r, d, α ) k where the eigenvalue k subspace of the Hessian of µ is T ( E, Φ) M X ( r, d, α ) k ∼ = H ( C • ( E ) k ) . (cid:3) For a critical point ( E, Φ) of µ we denote by T ( E, Φ) M X ( r, d, α ) < the subspace of thetangent space on which the Hessian of µ has negative eigenvalues. The real dimensionof this subspace is called the Morse index at the point ( E, Φ).
Proposition 5.6.
The codimension of the complement of T ∗ M X ( d, r, α ) in M X ( d, r, α ) is equal to half of the minima of the Morse indexes at points ( E, Φ) ∈ F λ for λ = 0 .Proof. The complement of T ∗ M X ( d, r, α ) is equal to V which is also equal to V ′ fromProposition 5.1. Bott–Morse theory give us that V ′ = S λ =0 U + λ , so we conclude thatcodim( V ) = min λ =0 codim U + λ . From Bott–Morse theory we also know that the dimension of the upwards Morse flow issuch that dim U + λ + dim T ( E, Φ) M X ( d, r, α ) < = dim M X ( d, r, α ) , where T E M X ( d, r, α ) < is the negative eigenspace for the Hessian of the perfect Bott–Morse function µ for an E ∈ U + λ . As the Morse index µ λ = 2 dim T E M X ( d, r, α ) < ,codim U + λ = 12 µ λ . Our statement is then codim( V ) = min λ =0 µ λ that is, codim( V ) = min λ =0 dim T E M X ( d, r, α ) < (cid:3) Lemma 5.7. T ( E, Φ) M X ( d, r, α ) < = X k> − χ ( C • ( E ) k . (cid:3) So, we need to bound the Euler characteristic for any k . Proposition 5.8. − χ ( C • ( E ) k ) ≥ ( g − C k − rk ( b C k +1 )) Proof.
Recall that χ ( C • ( E ) k ) = dim H ( C k ) − dim H ( C k ) − dim H ( b C k +1 ⊗ K ( D )) + dim H ( b C k +1 ⊗ K ( D ))= deg( C k ) − deg( b C k +1 ) − rk ( b C k +1 ) deg( K ( D )) + (rk ( C k ) − rk ( b C k +1 ))(1 − g ) . (10) We first bound deg( C k ) − deg( b C k +1 ). Consider the following short exact sequences ofbundles, 0 −→ ker(Φ k ) −→ C k −→ im (Φ k ) −→ −→ im (Φ k ) −→ b C k +1 ⊗ K ( D ) −→ coker (Φ k ) −→ . thendeg( C k ) − deg( b C k +1 ) = deg(ker(Φ k )) + deg( K ( D )) rk ( b C k +1 ) − deg(coker (Φ k )) . (11)The ker(Φ k ) ⊂ ParEnd ( E ) is a subbundle of the bundle of parabolic endomorphismsof E , which we claim is semistable whenever E is stable (see Lemma 5.9). Hencepardeg (ker(Φ k )) ≤ k )) ≤ C k ) − deg( b C k +1 ) ≤ deg( K ( D )) rk ( b C k +1 ) − deg(coker (Φ k )) . NTEGRABLE SYSTEMS AND TORELLI THEOREMS 15
We also get that − deg(coker (Φ k )) ≤ (2 − g )(rk ( b C k +1 ) − rk (Φ k )) , (12)so that deg( C k ) − deg( b C k +1 ) ≤ n rk ( b C k +1 ) + (2 g −
2) rk (Φ k ) (13)in the following way.Note that for any two parabolic bundles E , F , then ParHom ( E, F ) ∗ = SParHom ( F, E ) ⊗O ( D ). So then C ∗ k = ( L j − i = k ParHom ( E i , E j )) ∗ = L j − i = k SParHom ( E j , E i ) ⊗ O ( D ) = b C k ⊗ O ( D ).Consider the adjoint mapΦ tk : ( b C k +1 ⊗ K ( D )) ∗ −→ ( C k ) ∗ (14)then ker(Φ tk ) ֒ → ( b C k +1 ⊗ K ( D )) ∗ ∼ = C − − k ⊗ K − . Dualizing again we get a surjective homomorphism, b C k +1 ⊗ K ( D ) −→ (ker(Φ tk )) ∗ Define the homomorphism f : coker (Φ k ) −→ ker(Φ tk ) ∗ which makes the following diagram commutative0 / / im (Φ k ) / / b C k +1 ⊗ K ( D ) / / coker (Φ k ) / / f (cid:15) (cid:15) / / (im (Φ tk )) ∗ / / b C k +1 ⊗ K ( D ) / / (ker(Φ tk )) ∗ / / f is surjective and ker( f ) is a torsion subsheaf. Hence,0 −→ ker( f ) −→ coker (Φ k ) −→ (ker(Φ tk )) ∗ −→ , and deg(coker (Φ k )) ≥ deg(ker(Φ tk ) ∗ ) . As ker(Φ tk ) is a sub bundle of C k +1 ⊗ K , − deg(coker (Φ k )) ≤ deg(ker(Φ tk )) (15)Note that there are isomorphisms, making the following diagram commutative( b C k +1 ⊗ K ( D )) ∗∼ = (cid:15) (cid:15) Φ tk / / ( C k ) ∗∼ = (cid:15) (cid:15) C − − k ⊗ K − − − k ⊗ K − / / b C − k ⊗ O ( D ) , therefore Φ tk ∼ = Φ − − k ⊗ K − so ker(Φ tk ) = ker(Φ − − k ) ⊗ K − and deg(ker(Φ tk )) = deg(ker(Φ − − k )) + (2 − g ) rk (ker(Φ − − k ))Notice that rk (Φ − − k ) = rk (Φ tk ) = rk (Φ k ) and rk ( b C k +1 ) = rk (( b C k +1 ) ∗ ) = rk ( C − − k ).Then rk (ker(Φ − − k )) = rk ( b C k +1 ) − rk (Φ k ), so that, equation (15) becomes − deg(coker Φ k ) ≤ deg(ker(Φ − − k )) + (2 − g )(rk ( b C k +1 ) − rk (Φ k ))and finally, by stability (see Lemma 5.9), − deg(coker Φ k ) ≤ (2 − g )(rk ( b C k +1 ) − rk (Φ k )) . This provides equation (12).Putting together equations (12) and (13) we get χ ( C • ( E ) k ) ≤ (1 − g )(rk ( C k ) − rk ( b C k +1 )) , hence, − χ ( C • ( E ) k ) ≥ ( g − C k ) − rk ( b C k +1 )) , as we wanted. (cid:3) Lemma 5.9.
Let ( E, Φ) be a stable parabolic Higgs bundle, then (ParEnd ( E ) , ad (Φ)) is semistable.Proof. The proof follows the arguments in [GLM, Proposition 6.7] adapted to the para-bolic situation. That is, the vector bundle ParEnd ( E ) has a natural parabolic structureinduced by the parabolic structure of E . In fact ParEnd ( E ) as a parabolic bundle isthe parabolic tensor product of the parabolic bundle E and the parabolic dual of E (see [Yo]), and hence its parabolic degree is 0. With respect to this parabolic structure(ParEnd ( E ) , ad (Φ)), where ad (Φ) : ParEnd ( E ) → SParEnd ( E ) ⊗ K ( D ), is, again,a parabolic Higgs bundle. Now, the stability of ( E, Φ) implies the polystability of(ParEnd ( E ) , ad (Φ)). (cid:3) Proposition 5.10. codim( V ) ≥
12 ( r − g − . Proof.
Propositions 5.6 and 5.8 givecodim( V ) = min F λ (X k> − χ ( C • ( E ) k ) ) ≥ min F λ (X k> ( g − C k − rk ( b C k +1 )) ) = rk ( C )( g − , where rk ( C ) = rk ( ⊕ j − i =1 ParHom ( E i , E j ) so if we denote r i = rk ( E i the rank of eachpiece is rk (ParHom ( E i , E j ) = r i r j , so then rk ( C ) = ( r r + · · · + r m − r m ) which isdefinitely rk ( C ) ≥ ( r − (cid:3) NTEGRABLE SYSTEMS AND TORELLI THEOREMS 17
Corollary 5.11.
For g = 2 and r ≥ or g = 3 and r ≥ g ≥ and r ≥ , and u ageneric point in U , the codimension of the fiber h − ( u ) in h − ( u ) is greater or equal to2.Remark . The codimension does not depend on the number of marked points, asin [BGL] it did not depend on the degree of the line bundle L , which was twisting theHiggs bundle and in this case is K ( D ).We also obtain the following. Corollary 5.13.
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