Moduli of Parabolic Higgs Bundles and Atiyah Algebroids

MODULI OF P ARABOLIC HIGGS BUNDLES AND A TIY AH ALGEBR OIDS MARINA LOGARES AND JOHAN MAR TENS Abstract. In this pap er we study the geometry of the mo duli space of (non-strongly) parab olic H igg s bundles over a Riemann surface wit h marked p o in ts. W e show that this s pace p ossesses a Poisson structure, e xtending the o ne on the dual of an Atiy ah alge- broid ov er the mo duli space of pa rabo lic vector bundles. By con- sidering the case of full flag s, we get a Grothendieck-Springer reso - lution for all other flag types, in particular for the mo duli spaces of t wisted Higg s bundles, as studied b y Markman and Bottacin and used in the recent work o f Laumon-Ngˆ o. W e discuss the Hitchin system, and demonstrate th at all these moduli spaces are in te- grable systems in the Poisson sens e. Contents 1. In tro duction 2 1.1. Remark on notatio n 4 1.2. Ac kno wledgemen ts 4 2. Mo duli spaces of P arab olic Higgs bundles 5 2.1. P arab olic vec tor bundles 5 2.2. P arab olic Higg s bundles 8 2.3. The Hitchin fibration 9 2.3.1. Hitc hin map 9 2.3.2. Sp ectral curv es 10 2.3.3. Generic fib ers 10 3. P oisson structure 13 3.1. Bac kground material 13 3.1.1. P oisson geometry 13 3.1.2. Algebroids and P oisson structures 14 3.1.3. Symplectic leav es for the dual of an a lg ebroid 16 3.2. P oisson structure on P α 17 3.2.1. The ♯ map 17 3.2.2. P oisson structure via Lie alg ebroids 18 Date : June 2, 2 018. 1991 Mathematics Subje ct Classific atio n. Prima ry 14H60; Sec o ndary 14D20 . 1 2 MARINA LOGARES A ND JOHA N MAR T ENS 3.2.3. Extension of the brack et 20 3.2.4. Symplectic leav es 21 3.2.5. Complete integrabilit y of the Hitc hin system 23 4. Morphisms b et we en mo duli spaces, Gro t hendiec k-Springer resolution 25 5. F urther remarks 27 5.1. Comparison with Botta cin-Markman 27 5.2. P arab olic vs. orbifold bundles 29 App endix A. A L evi principal bundle o v er the mo duli space of parab olic bundles 30 References 31 1. Introd uction Higgs bundles, introduced by Hitc hin in [29 , 30], ha v e emerged in the last tw o decades as a cen tral ob ject of study in geometry , with sev eral links to ph ysics and num ber theory . Ov er a smo oth compact Riemann surface the mo duli space of Higgs bundles contains as a dense op en subset the total space o f the cotangent bundle to the mo duli space o f v ector bundles. In fact the induced complex symplectic form is part of a hyper-K¨ ahler structure and extends to the whole of the mo duli space of Higgs bundles, and it is a celebrated fact that the mo duli space comes equipp ed with a algebraically completely in tegrable sys- tem, thro ugh the Hitchin map. A na t ur a l generalization o f v ector bundles arises when one endo ws the v ector bundle with a par abo lic structure [43], i.e. with c hoices of flags in the fib ers ov er certain mark ed p oints on the R iemann sur- face. One can talk of Higgs bundles in that setting as w ell, as w as first done b y Simpson in [53]. V arious c hoices can b e made for this. In o r- der to hav e the corresp onding mo duli space con tain as an op en subset the to tal space of the cotangent bundle to the mo duli space of par- ab olic v ector bundles, replicating the non-para bolic situation, sev eral authors [5 5, 38, 26, 24] restrict the pa r a bo lic Higgs bundles to those that w e shall r efer to a s strongly parab olic, meaning that the Higgs field is nilp oten t with resp ect to the flag. One can ho w ev er also demand the Hig gs field to simply r espect the parab olic structure at the mark ed p oints, and a mo duli space P α for those w as constructed b y Y ok ogaw a in [56]. The lo cus of tha t mo d- uli space where the underlying parab olic v ector bundle is stable again MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 3 forms a ve ctor bundle ov er the mo duli space of para b olic vector bundles N α . W e sho w here tha t this v ector bundle is the dual of an Atiy ah alge- broid asso ciated with a principal bundle ov er N α , the structure group for whic h is the pro duct of the Levi gr o ups g iv en b y the v arious flags at the marked p oin ts, mo dulo C ∗ to account for global endomorphisms of the bundle. As the dua l of an algebroid its total space carries a complex algebraic P oisson structure, whic h in fact extends to the whole of P α . In the part icular situation where all the flags are trivial, { 0 } ⊂ E | p , this w as already shown indep enden tly b y Bottacin [13] a nd Markman [42]. W e further study t he Hitc hin system for P α and its symplectic lea v es, sho wing that this mak es the P α for all flag t yp es in to in tegrable systems in the sense of P oisson geometry , with Casimir functions that gener- ically induce the folia tion of symplectic lea ve s, which ar e in tegrable systems in the usual symplectic sense. Though w e don’t explicitly use Lie g roupo ids, our philosoph y is very muc h that the symplectic lea v es are the co-adj o in t or bits for the group oid determined b y the principal bundle. With this in mind w e also lo o k at the for g etful morphisms b et w een suc h mo duli spaces of v arious flag types. W e show that they are P ois- son and generically finite. By lo oking at such morphisms starting from the mo duli space for f ull flags w e obtain a globa l analogue of the Gro- thendiec k-Springer resolution of Lie alg ebras, as the mo duli space fo r full flags is a regular Pois son v ariet y . As the Grothendiec k-Springer resolution pla ys a crucial role in mo dern geometric represen tation the- ory , this op ens p ersp ectiv es on generalizing classical constructions to this glo bal setting 1 . The presen tation we ha v e giv en is larg ely done in the language of Lie algebroids, but one could hav e reformulated ev erything we sa y ab out A tiy ah algebroids in terms of P oisson r eduction of cotangent bundles. The c hoice is pa rtly one of p ersonal preference, and partly due to the fact that the Atiy ah sequence o f the algebroid naturally follows from the deformation theory of parab olic v ector bundles. Similarly , though almost all of our argumen ts use h yp ercohomology , we ha v e av oided the use of deriv ed categor ies, in order to exhibit the P oisson structures more explicitly . 1 Indeed, we very recently beca me aw are of the prepr in t [57], where parab olic Higgs bundles are used to gener alize the Springer theory of W eyl gr o up r epre- sentations to a ‘global’ setting, without howev er tak ing our viewp oint of Atiy ah algebroids and g roupo ids. 4 MARINA LOGARES A ND JOHA N MAR T ENS W e remark here that our en tire construction is depending on the ex- istence o f the principal bundle o v er N α . In the app endix A w e describ e a construction o f this bundle for full flags, using previous w ork b y Hur- tubise, Jeffrey and Sjamaar [32], a nd outline a p ossible construction for ot her flag t yp es. This pap er is orga nized as follows: in section 2 w e give the necessary bac kground regarding parab olic Higgs bundles and t heir mo duli, as w ell as a description of the Hitch in fibratio n. Most of this is standard, with the p ossible exception of the observ atio n (Prop osition 2.2) that the smo othness of the sp ectral curv e implies that the Higgs field uniquely determines the parab olic structure, ev en when eigen v alues are rep eated. In section 3 w e give the required background material regarding Lie group oids and Lie alg ebroids, and pro ve the main result of this a rticle, the interpretation of the mo duli space as a partial compactification of the dual o f an Atiy ah alg ebroid. W e also show here that for t he induced P oisson structure the Hitc hin map is an in tegrable system. In section 4 w e remark that for nearby parab olic w eigh ts the morphisms b et w een the v arious mo duli spaces (with differen t flag structures) are P oisson, giving a Grothendiec k-Springer resolution b y means of the full flag s. In section 5 w e discuss the relationship of our w ork with the earlier results b y Bo t tacin and Markman, as w ell as further directions. App endix A discusse s a construction of t he principal bundle ov er the mo duli space of para bolic v ector bundles used in the main theorem. 1.1. Remark on notation. Unfo r tunately nomenclature con v en tions regarding parab olic Higg s bundles v ary in the literat ure. F or us a parab olic Higg s bundle will only require the Higgs field a t a mark ed p oin t to resp ect the flag there. W e will refer to the sp ecial case where the Higgs field is nilp otent with resp ect to the filtration as a str ongly parab olic Higgs bundle. 1.2. Ac kno wledgemen ts. The a uthors w ould like to thank Sergey Arkhip o v, David Ben-Zvi, Philip Boalch, Hans Bo den, Chris Bra v, R on Donagi, T om´ as Gom´ ez, Peter Gothen, T am´ as Hausel, Nigel Hitc hin, Jacques Hurtubise, Lisa Jeffrey , Ra j Meh ta, Ec khard Meinrenk en, Szi- l´ ard Szab´ o and Mic hael Thaddeus for useful con v ersations, remarks, and encourag ement, as well as the MPI Bo nn, NSERC a nd the Cen tro de Matem´ a t ica da Univ ersidade do Porto for financial supp ort. MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 5 2. Moduli sp aces of P arabolic Higgs bundles 2.1. P arab olic v ector bundles. L et X b e a compact Riemann sur- face or smo oth complex 2 pro j ective curv e of genus g with n distinct mark ed p oin ts p 1 , . . . , p n . If g = 0 w e assume n ≥ 3, if g = 1 we assume n ≥ 1. Let D b e the effectiv e reduced divisor p 1 + . . . + p n . A p ar ab olic ve ctor bund le ([43]) on X is a n algebraic rank r v ector bundle E o ve r X together with a parab olic structure, i.e. a (not necessarily full) flag fo r the fib er of E ov e r the marked p oin ts E | p = E p, 1 ⊃ · · · ⊃ E p,r ( p ) ⊃ { 0 } , together with a set of parab olic w eights 0 ≤ α 1 ( p ) < · · · < α r ( p ) ( p ) < 1 . W e denote the m ultiplicities b y m i ( p ) = dim E p,i − dim E p,i +1 , and the asso ciated graded as Gr( p ) = ⊕ i E p,i /E p,i +1 . Note t ha t the structure group of the bundle E is GL ( r ). In terms of the asso ciated frame bundle the parab olic structure corresp onds to a reduction of the structure group of this principal bundle to a certain parab olic subgroup of GL ( r ) at eac h mark ed p oin t p i . W e will denote this parab olic subgroup by P p , and its corresp onding Levi gr o up by L p , with Lie algebra l p . F or the sak e of con v en tion w e will fix as a Borel subgroup in GL ( r ) the low er triangular matrices, and all parab olic sub- groups are take n t o contain this Borel. W e will further need linear endomorphisms 3 Φ of a parab olic v ector bundle whic h are either parab olic - meaning that at the fib er o v er a marked po in t p w e hav e Φ | p ( E p,i ) ⊂ E p,i - or str on g l y p ar ab olic - meaning that Φ | p ( E p,i ) ⊂ E p,i +1 . W e denote the shea v es of par- ab olic resp ective ly strongly pa rab o lic endomorphisms as P ar E nd ( E ) and S P ar E nd ( E ). The relev a nce of the parab olic weigh ts α comes from the notion of p ar ab olic de gr e e of a bundle, denoted by p de g : p deg( E ) = deg( E ) + X p ∈ D X i m i ( p ) α i ( p ) , 2 Presumably all the results b elow ho ld ov er arbitrar y algebraic ally clos e d fields. Neither Y o k ogawa’s construction of the mo duli spaces we shall use , nor any of our work, req uires the ground field to b e C . W e do rely cr ucially howev er on the res ults of [4], which ass umes the characteristic to be zer o. 3 A pr io ri we don’t require morphisms b etw een v ector bundles to hav e co ns tan t rank. 6 MARINA LOGARES A ND JOHA N MAR T ENS whic h satisfies the Gauss-Chern form ula for connections with logarith- mic singularities ( see Prop osition 2.9 in [7 ]). The α also o ccur in the celebrated Meh ta-Seshadri t heorem ([43]) that establishes a correspon- dence b et w een stable parab olic bundles and unitary represen tations of the fundamental g roup of the punctured surface X \ D , where they determine t he holonomy around the punctures. Ev ery algebraic subbundle F of E is naturally giv en the structure of a parab olic bundle as w ell, by simply inters ecting F | p with the elemen ts o f the fla g of E | p , discarding any suc h subspace of F | p that coincides with a previous o ne, and endow ing it with the lar g est of the corresp onding parab olic w eights: for F p,i = F | p ∩ E p,j α F i ( p ) = max j { α j ; F | p ∩ E p,j = F p,i } . W e sa y that a parab olic v ector bundle is stable if for eac h prop er subbundle F we hav e that (1) p deg( F ) rk ( F ) < p deg( E ) rk ( E ) . Semi-stabilit y is defined similarly , b y a skin g for w eak inequalit y . The w eigh ts are called gene ri c when stability and semi-stabilit y coincide. Note that the term generic is used in the sense t ha t the set of non- generic we ights has p ositiv e co dimension. There exists a mo duli space for semi-stable para b olic v ector bun- dles [43], whic h w e shall denote b y N α . This is a normal pro jectiv e v ariet y of dimension dim N α = ( g − 1) r 2 + 1 + X p ∈ D 1 2 r 2 − X i m i ( p ) 2 ! and when the w eigh ts are generic it is non-singular. F rom no w on w e will assume genericit y of w eigh ts, eve n though in the non- generic case all of what w e say can still b e carried through when restricted to the stable lo cus of N α . W e will need a further lemma that states that stability implies sim- plicit y . This is completely standard in the non-parab olic setting ( see e.g. [45, Corollary to Prop osition 4.3]) but a para bolic vers ion do es not seem to hav e app eared in the literat ure, so w e include it here fo r completeness . MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 7 Lemma 2.1. If E is a n α -stable p ar ab olic ve ctor bund l e over X , then H 0 ( X , P ar E nd ( E )) = C and H 0 ( X , S P ar E nd ( E ) ) = { 0 } . Pr o of. First note t ha t giv en a rank r algebraic vec tor bundle ov er a complete non- singular v ariet y , t he r -th p o w er of an y endomorphism of this bundle necess arily has constan t rank, since the co efficien ts of the characteristic p olynomial are regular functions and hence constan t and the r -th p o w er has no nilp oten t Jor da n blo c ks. Therefore w e can talk of the k ernel of this r -th p ow e r of the endomorphism as algebraic subbundles. No w, give n a parab olic endomorphism e of a para bolic v ector bundle E , e is zero or an isomorphism if and only if e r is. W e shall therefore consider f = e r . The subbundle k er( f ) has a canonical induced pa r a- b olic structure. The same is true for im( f ) , whic h w e also think of a s a subbundle of E . The parab olic we ights that ker( f ) and im( f ) inherit as subbundles of E , when coun ted with m ultiplicities, are complemen tary to each other with resp ect to the pa r a bo lic we ights of E . Assume now that f is neither zero nor an isomorphism, so b oth k er( f ) and im( f ) are prop er subbundles o f E . Using the stability w e ha v e that (2) p deg (k er( f )) rk (k er( f )) < p deg( E ) rk ( E ) and p deg (im ( f )) rk (im( f )) < p deg( E ) rk ( E ) . Using the complemen tarit y o f the parab olic we ights of k er( f ) and im( f ) how ev er w e also ha v e that (3) p deg( E ) rk ( E ) = p deg (ker( f )) + p deg (im( f )) rk (k er( f )) + rk (im( f )) . One easily sees that the combination of (2) and ( 3 ) w ould give p deg (ker( f )) rk (ker( f )) < p deg (im( f )) rk (im( f )) and p deg (im( f )) rk (im( f )) < p deg (ker( f )) rk (k er( f )) , hence f is either zero o r an isomorphism. If f is an isomor phism, one just has to tak e a p oin t x ∈ X a nd consider an eigen v alue λ of f x : E x → E x . Next consider the para bo lic endomorphism of E giv en b y ( f − λ Id E ), b y the same reasoning as b efore this is an isomorphism or it is zero. Hence H 0 ( P ar E nd ( E )) = C . In t he case of a strongly parab olic endomorphism, as for any mark ed p oin t p ∈ D zero is one of the eigen v alues of f p , one gets similarly H 0 ( S P ar E nd ( E )) = { 0 } .  8 MARINA LOGARES A ND JOHA N MAR T ENS Finally , notice that P ar E ndE is naturally dual to S P ar E nd ( E )( D ), and vice vers a S P ar E nd ( E ) is dual to P ar E nd ( E )( D ). Throughout the pap er w e shall oft en use Serre dualit y f or the h yp ercohomology of a complex o n a curv e, so we recall its statemen t: for a b ounded complex C of lo cally free sheav es on X of the form 0 → C 0 → . . . → C m → 0 w e hav e the natura l duality H i ( C ) ∗ ∼ = H 1 − i + m ( C ∗ ⊗ K ) , where C ∗ ⊗ K is the complex 0 → ( C m ) ∗ ⊗ K → . . . → ( C 0 ) ∗ ⊗ K → 0 , again considered in degrees 0 t hro ugh m . 2.2. P arab olic Higgs bundles. A p ar ab o lic Higg s bund le ([53 ]) is a parab olic v ector bundle tog ether with a Higgs field Φ, a bundle mor- phism Φ : E → E ⊗ K ( D ) , where K is the canonical bundle of X , whic h preserv es the pa rab o lic structure at each mark ed p oin t: Φ | p ( E p,i ) ⊂ E p,i ⊗ K ( D ) | p , i.e. Φ ∈ H 0 ( X , P ar E nd ( E ) ⊗ K ( D )). In ke eping with the notation in tro duced ab o v e we refer to t he Higgs bundle a s str ongly parab olic if the Higgs field is actually nilp oten t with resp ect to the filtration, i.e. if Φ | p ( E p,i ) ⊂ E p,i +1 ⊗ K ( D ) | p . Similar t o v ector bundles a Higgs bundle is (semi) stable if the slop e condition p deg( F ) rk ( F ) < (=) p deg( E ) rk ( E ) holds, restricted no w to all prop er subbundles F preserv ed b y the Higgs field, i.e. with Φ( F ) ⊂ F ⊗ K ( D ). Denote b y P α the mo duli space of α - semi-stable para bolic Higgs bundles o f degree d and rank r , whic h w as constructed by Y ok ogaw a in [56] and further discussed in [11]. This space is a normal, quasi- pro j ective v ariet y of dimension (4) dim P α = (2 g − 2 + n ) r 2 + 1 . Observ e that this dimens ion is indep enden t of the flag- t yp e a t t he mark ed p oin ts, in con trast to the dimension of N α . Indeed, there is MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 9 a natural partial orderings on the flag t yp es, and if P ˜ α is the corre- sp onding mo duli space for a finer flag type and the w eigh ts ˜ α and α are close enough such that semi-stabilit y is preserv ed b y the forg etful functor, then there is a fo rgetful morphism (5) P ˜ α → P α whic h is generically finite - see also prop osition 2.2 b elo w. 2.3. The Hit c hin fibration. 2.3.1. Hitchin ma p. Just as fo r ordinary Higgs bundles [29, 30], the parab olic Higgs bundles fo r m an in tegrable system by means of the Hitchin map , defined as follo ws. Giv en a vec tor bundle E , any in v ari- an t, homogeneous degree i p olynomial naturally defines a map H 0 (End( E ) ⊗ K ( D )) → H 0 ( K ( D ) i ) . No w, t a k e the elemen tary symmetric p olynomials as a ho mogeneous ba- sis of p olynomials o n gl ( r ) in v arian t under the adjo in t actio n o f GL ( r ), then the corresp onding maps a i com bine to giv e the Hitchin m ap h α : P α → H where the v ector space H is the Hitchin space H = H 0 ( X , K ( D )) ⊕ H 0 ( X , K ( D ) 2 ) ⊕ · · · ⊕ H 0 ( X , K ( D ) r ) . The comp onen ts of h α are defined as follows : for an y parab olic Higgs bundle ( E , Φ) and for an y x ∈ X , let k ∈ K ( D ) | x . Then we ha v e that det( k . Id E | x − Φ | x ) = k r + a 1 (Φ)( x ) k r − 1 + · · · + a r − 1 (Φ)( x ) k + a r (Φ)( x ) , and h α ( E , Φ) is given b y ( a 1 (Φ) , . . . , a r (Φ)). In [5 6], § 5, it is shown that h α is prop er, and in f a ct pro jectiv e. Notice that h α is blind to the p ar ab o lic structur e at each mark ed p oin t, as it only dep ends o n Φ and the line bundle K ( D ). Indeed, supp ose a giv en v ector bundle E and Higgs field Φ : E → E ⊗ K ( D ) can b e equipped with tw o distinct parab olic structures compat ible with Φ, so as to obta in differen t stable parab olic Higgs bundles, p ossibly but not necessarily of differen t flagt yp e. If w e denote these t w o para bo lic v ector bundles b y F and e F , with ( F , Φ) ∈ P α and ( e F , Φ) ∈ P ˜ α , t hen necessarily h α ( F , Φ) = h ˜ α ( e F , Φ). 10 MARINA LOGARES A ND JOHA N MAR T ENS 2.3.2. Sp e ct r al curves. F or eac h elemen t s = ( s 1 , . . . , s r ) o f H o ne can define a sp ectral curv e X s in S , the total space of K ( D ), as follo ws: pull bac k K ( D ) to S and denote it s canonical section as λ . Then X s is the zero-lo cus of λ r + s 1 λ r − 1 + · · · + s r , a (p ossibly ramified) co v ering of X . As usual, by a Bertini ar g umen t, for a generic elemen t in H the corresp onding X s is smo oth. The gen us of X s can b e giv en using the adjunction formu la: 2 g ( X s ) − 2 = deg ( K X s ) = K X s .X s = ( K S + X s ) .X s = K S .X s + X 2 s = r c 1 ( O ( − D )) + r 2 X 2 = − r n + r 2 (2 g − 2 + n ) and hence (6) g ( X s ) = r 2 ( g − 1) + r n ( r − 1 ) 2 + 1 . The eigen v alues o f Φ | x for x ∈ X control the ramification of X s o v er x , e.g. if all eigenv alues are 0 , then X s is completely ramified o v er x , if all are differen t then X s is unramified o ve r x . W e denote the cov ering b y ρ : X s → X , with ramification divisor R on X s . 2.3.3. Generic fib ers. Now , if X s is smo oth, pull back E to X s b y ρ . W e canonically get a line bundle L on X s , suc h that L ( − R ) sits inside this pull bac k (see e.g. [4 ],Prop.3.6). Aw a y from a ramification p oint the fib er of L is g iv en b y an eigenspace o f ρ ∗ E , exactly corresp onding to the eigen v alue of Φ giv en by that p oin t of X s , in fa ct one has that (7) 0 → L ( − R ) → ρ ∗ E ρ ∗ Φ − λ Id − → ρ ∗ ( E ⊗ K ( D )) → L ⊗ ρ ∗ K ( D ) → 0 is exact, see [4],Remark 3.7. F urthermore we ha v e ρ ∗ L = E , and the Higgs field Φ is also easily reco v ered: multiplic atio n by the cano nical section λ of ρ ∗ K ( D ) descends to a mo r phism Φ : ρ ∗ L → ρ ∗ L ⊗ K ( D ) = ρ ∗ ( L ⊗ ρ ∗ K ( D )) . In order to obtain the degree o f L , apply Grot hendiec k-Riemann- Ro c h to the morphism ρ : X s → X , and then inte gra t e b oth sides. This give s deg( E ) + r 1 2 deg( T X ) = deg ( L ) + 1 2 deg( T X s ) MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 11 and hence deg( L ) = deg ( E ) + r 1 2 deg( T X ) − 1 2 deg( T X S ) = d + r (1 − g ) + r 2 ( g − 1) + r n ( r − 1) 2 = d + r (1 − r )(1 − g − n 2 ) . Moreo v er, the smo othness of t he sp ectral curv e g uaran tees that only finitely man y para bo lic structures are compatible with the Higgs field, ev en if the Higgs field ha s rep eated eigen v a lues at the p oin ts of D . Indeed, we ha v e Prop osition 2.2. If the sp e ctr a l curve X s is smo oth, then h − 1 α ( s ) c on- sists of dis joint c opies of the Jac obian of X s , one fo r every p artition of the eigenvalues along the multiplicities o f the flagtyp e. With ‘partition a lo ng the m ultiplicities’ w e mean here a partition of the set of eigenv alues in to subsets , the sizes o f whic h a r e the v arious m ultiplicities. Pr o of. W e w ant to sho w that with each line bundle L on X s of degree d + r (1 − r )(1 − g ) w e get a para bolic Higgs bundle of degree d . As stated ab o v e, the push forw ard ρ ∗ L determines a v ector bundle E on X [4], and multiplication by λ on L descends to a Higg s field. What remains is to construct a flag of the desired ty p e at the marked p oin ts. If p is in D , denote the eigen v alues of Φ | p b y σ 1 , . . . , σ r . Cho ose a partitio n of the σ i according to the m ultiplicities m i ( p ) and relab el the σ i suc h that t he partitio n is giv en b y: { σ 1 , . . . , σ m 1 ( p ) } , { σ m 1 ( p )+1 , . . . , σ m 1 ( p )+ m 2 ( p ) } , . . . , { σ r − m r ( p ) , . . . , σ r } . No w, c ho ose a Za riski op en set W around p suc h that L is trivial o v er its in v erse imag e ρ − 1 ( W ), and K ( D ) is trivial o ve r W . Observ e that w e can alwa ys do this: tak e an y rational section of L , then b y the indep endence of v aluation theorem (see e.g. [20], page 19) w e can c ho ose a rational f unction suc h that the pro duct of the tw o has no p oles or zero es on ρ − 1 ( p ). This new ra tional section trivializes L on the complemen t of its divisor, whic h then easily g iv es the desired W . Restricting to W , ρ ∗ L is give n a s an O ( W )-mo dule b y O ( W )[ x ] / ( x r + s 1 x r − 1 + · · · + s r ) , with the s i ∈ O ( W ). Because o f the c hoices made, we also kno w that at p , the s j are give n by the elemen tary symmetric p olynomials in the σ i . Therefore t he fib er of ρ ∗ ( L ) ov er p is exactly giv en by C [ x ] / (( x − σ 1 ) . . . ( x − σ r )) . 12 MARINA LOGARES A ND JOHA N MAR T ENS Φ is of course given by m ultiplication by x here, and hence this deter- mines a basis: if w e put e j = Q j − 1 l =1 ( x − σ l ), with e 1 = 1, then with resp ect to this basis Φ lo oks lik e the lo w er-triang ula r matrix (8)        σ 1 1 σ 2 1 . . . . . . . . . 1 σ r        . Hence by using e 1 , . . . , e r as a n adapted basis, i.e. E p, 1 = < e 1 , . . . , e r >, · · · , E p,r ( p ) = < e r − m r ( p ) ( p ) , . . . , e r >, w e get the parab olic structure, whic h is indep enden t of our trivializa- tions of L a nd K ( D ) and the c hoice of the basis e i . F rom the matrix form (8) one can immediately see that the eigenspace fo r ev ery eigen- v alue is one-dimensional, ev en if the eigen v alue has m ultiplicit y , this is what giv es the uniqueness of the filtration. It remains to sho w that this para b olic Higgs bundle is stable: for this observ e tha t a Φ-preserv ed subbundle of E w ould necessarily cor- resp ond to a subsheaf of L (see again [4], page 174), and as X s is assumed to b e smo oth, this has to b e a lo cally free sheaf it self as w ell, necessarily of lo w er degree. Not ice that this sho ws that smo othness of the sp ectral curv e implies that there a re no subbundles preserv ed b y the Higgs field, and hence the slop e-stability condition need not ev en b e applied.  In section 3.2.4 we shall see that eac h of these Jacobians is actually con tained in a different symplectic leaf for the Poiss on structure on P α . In section 4 w e shall further study the forg etfull morphisms, men tioned ab o v e in (5), fro m mo duli spaces P ˜ α of finer fla g ty p e (e.g. full flag t yp e) to mo duli spaces P α of coarser fla g t yp e (e.g. P 0 for the minimal flag types, with all flags b eing E p, 1 ⊃ E p, 2 = { 0 } ). Note that from theorem 2.2 w e can already conclude that suc h a morphism will b e finite ov er the lo cus in H corresponding to smo oth sp ectral curves . The mo duli-space M Higgs of non-pa rab olic Higgs bundles (with Higgs fields Φ : E → E ⊗ K ) is a subv ariet y of P 0 ; in fact it is a symplec tic leaf for the P oisson structure w e shall exhibit on P 0 . All in all this MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 13 giv es us t he follow ing diagra m (with h the Hitc hin map for M Higgs ): P ˜ α   h ˜ α   ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; M Higgs   / / h ' ' O O O O O O O O O O O P 0 h α L L L L L L & & L L L L L L L i H 0 ( X , K i )   / / H . 3. Poisson structure 3.1. Bac kground material. W e will b egin b y briefly reviewing the bac kground on P oisson geometry and Lie algebroids that w e need. R e- mark that in the literature Lie algebroids and g roupo ids are usually used in a differential geometric setting. F or our purp oses, where w e will use these notio ns on the mo duli space o f pa rab o lic Higgs bundles, all structures (bundles, spaces, actions) are algebraic how ev er. There ar e t w o main differences in the holomorphic or algebraic set- ting vs. the smo oth settings: algebroids hav e to b e defined using t he sheaf of sections of the underlying ve ctor bundle rather t ha n just the global sections, and more significantly , principal bundles do not a lw a ys ha v e connections, or equiv a len tly t he corresp onding A tiy ah sequence (14) do es not alwa ys split. As w e only use Lie g roup oids and alge- broids in a non-singular (algebraic) setting, w e hav e ho w ev er k ept the differen tial geometric notions of submersion etc. F or more bac kground on Lie group oids and a lgebroids see [15, 41], whic h w e use without reference in this section. 3.1.1. Poisson ge ometry. There are ma ny w a ys of pac k aging a Pois - son structure, so just to fix conv en tions w e shall state the one most con v enien t for our purp oses: Definition 3.1. A P oisson structure on a c omplex manifold M is gi ven by a bund le m o rphism ♯ : T ∗ M → T M that is an ti-symm etric, i.e. ♯ ∗ = − ♯ , such that the Schouten-Nijenhuis br acket [ ˜ ♯, ˜ ♯ ] of the c orr esp onding bive ctor ˜ ♯ ∈ V 2 T M is zer o. No w, for an y manifold N , lo ok at the tota l space of the cotangen t bundle π : T ∗ N → N . The tangent and co-t a ngen t bundles to T ∗ N b oth fit in short exact sequences, (9) 0 → π ∗ ( T ∗ N ) → T ∗ ( T ∗ N ) → π ∗ T N → 0 14 MARINA LOGARES A ND JOHA N MAR T ENS and (10) 0 → π ∗ ( T ∗ N ) → T ( T ∗ N ) → π ∗ T N → 0 . The canonical Poiss on structure (whic h is of course ev en a symplectic structure) is defined as the unique an ti-symmetric bundle morphism ♯ T ∗ N : T ∗ ( T ∗ N ) → T ( T ∗ N ) suc h tha t in the diagram (11) 0 / / π ∗ ( T ∗ N ) I d   / / T ∗ ( T ∗ N ) ♯ T ∗ N   / / π ∗ T N − I d   / / 0 0 / / π ∗ ( T ∗ N ) / / T ( T ∗ N ) / / π ∗ T N / / 0 all squares comm ute. Definition 3.2 . A morphis m f : M 1 → M 2 b etwe en Poisson sp ac es ( M 1 , ♯ 1 ) a n d ( M 2 , ♯ 2 ) is Poisson if the squar e (12) T ∗ M 1 ♯ 1 / / T M 1 d f   T ∗ M 2 ( d f ) ∗ O O ♯ 2 / / T M 2 c ommutes. In particular, if a P oisson manifo ld M is equipp ed with an action b y a group G that preserv es the Poiss on structure, then if the quotient exists the quotien t map M → M /G is a P oisson morphism. 3.1.2. A lgebr oids and Po i s son structur es. Tw o places where P oisson structures naturally o ccur are on the quotien ts of a symplectic mani- fold b y a Hamiltonian group action - where the symplectic leav es are giv en by the v arious symplectic reductions. Another place is on the total space of the dual of a L ie algebroid (whic h includes as a special example the dual o f a Lie a lg ebra). W e will mainly b e interes ted in the sp ecial case of an A tiy ah alg ebroid, whic h is an example of b oth of these situations. Definition 3.3. A Lie a lgebroid over a (c omplex) varie ty M is a ve ctor bund le E → M such that the she af of se ctions of E is a she af of Lie algebr as fo r a br acket [ ., . ] : O ( E )( U ) × O ( E )( U ) → O ( E )( U ) , to gether with a bund le map, the anc hor , a : E → T M w h ich pr eserves the Lie br ackets on se ctions. Mor e over the fol lowing L eibniz rule has to hold , f or f ∈ O ( U ) , X , Y ∈ O ( E )( U ) : [ X , f Y ] = f [ X , Y ] + ( a ( X ) f ) Y . MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 15 Tw o natural classes of examples of Lie alg ebroids are g iven b y tan- gen t bundles T M of a manifold M (where the map a is the identit y), and Lie algebras g , regar ded as a v ector bundle ov er a p oint. One can think of tra nsitive Lie algebroids (i.e. a lgebroids with surjectiv e anc hor maps) a s interpolations b et w een these t wo. The relev ance of Lie alg ebroids for us is through the follo wing theo- rem, which in this form is due to Coura n t [18] (see also [41, 15] or [13]): Theorem 3.4. The total sp ac e of the dual ve ctor bund l e E ∗ of a Lie algebr oid E h as a na tur al Poisson s tructur e. F or the t w o previous examples men tioned ab ov e the P oisson struc- tures are giv en b y the canonical symplectic structure on the to t al space of T ∗ M , a nd the Kirillov-Kostan t-Souriau Poiss on structure on g ∗ . In the fo rmer there is one single symplectic leaf, in the latter case t he symplectic lea ve s are give n b y the co- adjoin t orbits of the Lie group G . Let a gr oup G act freely and prop erly on a manifold P , in other w ords P π → P /G is a G - principal bundle. Of course then G also acts freely and prop erly , in a Hamiltonian f a shion, on the symplectic ma nif o ld T ∗ P , and t herefore the quotien t T ∗ P / G is a Poiss on manifold. Anot her wa y to realize t he P oisson structure on T ∗ P / G is as the dual of a particular t yp e of Lie algebroid, the so-called At iyah algeb r oid , a s follow s. G acts freely on T P , and T P /G is a Lie algebroid o v er P /G . One sees this most easily b y in terpreting the sections of T P /G as G -inv ariant v ector fields on P , and t he sections o f T ( P /G ) a s G - inv ariant sections of t he bundle on P that is the quotient of T P by the tangen t spaces to the orbits: (13) 0 → T orbits P → T P → π ∗ T ( P /G ) → 0 . The anchor map a : T P /G → T ( P /G ) is just giv en b y pro jecting a n in v arian t vec tor field to the part ‘orthogona l’ to the orbits. It is clear that this satisfies the required prop erty , since functions on P / G corre- sp ond to G -in v arian t functions on P , and any tangent field alo ng the orbits annihilates an inv ariant function. All in all the ab ov e sho ws that T P /G is the extension of T ( P /G ) b y the adjoint bundle Ad( P ) = P × Ad g . The latter is, as a bundle of Lie a lgebras, o f course a L ie algebroid with trivial anc hor map, and the corresp onding Atiyah se quenc e (14) 0 → Ad( P ) → T P /G → T ( P /G ) → 0 16 MARINA LOGARES A ND JOHA N MAR T ENS preserv es all Lie brack ets o n lo cal sections. Though mainly used in a differen tial g eometric setting, the A tiyah algebroid and the corresp onding short exact sequence w ere o riginally in tro duced in [1] in the con text of the study of the existence of holo- morphic connections in complex fib er bundles. 3.1.3. Symple ctic le aves for the dual of an algebr oid. Since w e are con- cerned with A tiy ah alg ebroids w e can study the symplec tic lea v es on its dual f a irly directly , using the general fact that t he symplectic leav es of the P oisson reduction of a symplec tic manifold b y a free Hamilton- ian group action correspo nd to the v arious symplectic reductions. W e shall do this in section 3.2.4. F or the sake of completeness w e do briefly indicate here how ev er that these symplectic leav es can b e seen as co- adjoin t orbits of a Lie group oid. F or us this is mainly of philosophical relev a nce, leading to the in terpretation of P α in the case of full flags as a Gro thendiec k-Springer resolution. Just as Lie algebras g are giv en as ta ngen t spaces to Lie g roups G , Lie algebroids can come from a Lie group oid - though not ev ery algebroid in tegrates to a Lie group oid, see [19]. Definition 3.5. A Lie group oid G ⇒ M over a manifold M is a sp ac e 4 G to gether with two submersions α, β : G → M , as wel l as an ass o ci a - tive pr o duct ( g 1 , g 2 ) 7→ g 1 g 2 define d on comp o sable pairs , i.e. ( g 1 , g 2 ) s.t. β ( g 2 ) = α ( g 1 ) , such that α ( g 1 g 2 ) = α ( g 2 ) and β ( g 1 g 2 ) = β ( g 1 ) . F urthermor e an identity se ction ǫ : M → G has to b e g i v e n, such that the fol lowing ho l d for al l g ∈ G : ǫ ( β ( g )) g = g and g ǫ ( α ( g )) = g as wel l as an in v ersion ι : G → G , with ι ( g ) g = ǫ ( α ( g )) and g ι ( g ) = ǫ ( β ( g )) . The maps α and β are often referred to as resp ectiv ely the sour c e and tar get maps of the group oid, and o ne thinks of G as consisting of arro ws g from α ( g ) to β ( g ), and comp ositions, in ve rses and iden tities can b e understo o d as suc h. With ev ery Lie group oid one can naturally asso ciate a Lie algebroid structure on the normal bundle to M ∼ = ǫ ( M ) ⊂ G . W e refer to [1 5 , 41] 4 In the differential geo metric setting it is in ge ne r al not requir ed here that G is Hausdorff, but it is assumed that M is. MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 17 for furt her background material. No w, let G ⇒ M b e a Lie group oid, with asso ciated Lie a lgebroid E → M . Then there exists a c otangent g r oup oid T ∗ G ⇒ E ∗ , whic h is b oth a vector bundle ov er G and a Lie group oid ov er E ∗ . Clearly a n y group oid H ⇒ N acts on its ba se N . Of particular relev a nce for us is the fo llo wing (see e.g. [41], Prop osition 11.5 .4 and Theorem 11.5.18): Theorem 3.6. The symple ctic le aves for the Poisson structur e on the total sp ac e of the dual of a Lie algeb r oid E → M a s so c i a te d with a Lie gr oup oid G ⇒ M a r e the (c onne cte d c omp onents of ) the orbits for the action of T ∗ G ⇒ E ∗ on E ∗ . The orbits of T ∗ G ⇒ E ∗ are often referred to as t he co-adjoint orbits of the original group oid G ⇒ M . This theorem establishes a cota ngen t group oid as a part icular case of the general notion of a symple ctic gr oup oid (see [17]). The base of a symplectic gr oupo id is a lw a ys Poisson, and its symplec tic leav es are giv en b y the orbits of the symplectic group oid [17]. 3.2. P oisson structure on P α . 3.2.1. The ♯ map. The tangen t space to P α at a stable parab olic Higg s bundle ( E , Φ) is given b y the H 1 h yp ercohomology of the tw o-term complex (15) P ar E nd ( E ) [ . , Φ] − → P ar E nd ( E ) ⊗ K ( D ) . Let us now write do wn the Poiss on brack et. The dual of the complex (15), t ensored with K , is giv en by (16) S P ar E nd ( E ) − [ . , Φ] − → S P ar E nd ( E ) ⊗ K ( D ) . W e can no w inject (16) in to (15), as follows: (17) S P ar E ndE − [ . , Φ]     Id / / P ar E nd ( E ) [ . , Φ]   S P ar E nd ( E ) ⊗ K ( D )   − Id ⊗ Id K ( D ) / / P ar E nd ( E ) ⊗ K ( D ) . Using Serre duality for h yp ercohomology w e therefore get a map (18) ♯ P α : T ∗ [ E , Φ] P α ∼ = H 1 ( S P ar E nd ( E ) → S P ar E nd ( E ) ⊗ K ( D )) → H 1 ( P ar E nd ( E ) → P ar E nd ( E ) ⊗ K ( D )) ∼ = T [ E , Φ] P α . 18 MARINA LOGARES A ND JOHA N MAR T ENS Because of the c hoice of signs in (17) ♯ P α is antisy mmetric. W e no w w an t to show t ha t this determines a P oisson structure on P α 5 . One could try to do this directly b y calculating the Schouten-Nijenh uis brac k et, but it w ould b e r a ther hard and not so instructiv e, therefore w e f ollo w a differen t ro ut e b elo w. 3.2.2. Poisson structur e vi a Lie a lgebr oids. No w let P 0 α b e the op en sub v ariet y of P α consisting of those parab olic Higgs bundles ( E , Φ) whose underlying parab olic bundle is stable. As men tioned b efore, this is a v ector bundle o v er N α , t he mo duli space of parab olic ve ctor bundles, with fib er H 0 ( X , P ar E nd ( E ) ⊗ K ( D )). W e shall use the follo wing tw o pro jections: (19) N α × X η z z u u u u u u u u u ν # # H H H H H H H H H N α X No w, on N α × X w e ha v e a parab olic univ ersal bundle [12, Theorem 3.2], whic h we denote b y E . This leads to a short exact sequence of shea v es on N α × X : 0 → S P ar E nd ( E ) → P ar E nd ( E ) → Y p ∈ D l p ⊗ O ν − 1 ( p ) → 0 . Applying η ∗ to this sequence give s the exact sequence (20) 0 → η ∗ S P ar E nd ( E ) → η ∗ P ar E nd ( E ) → η ∗  Y l p ⊗ O ν − 1 ( p )  → R 1 η ∗ S P ar E nd ( E ) → R 1 η ∗ P ar E nd ( E ) → R 1 η ∗  Y l p ⊗ O ν − 1 ( p )  → 0 . As the supp ort of Q l p ⊗ O ν − 1 ( p ) has relative dimension zero with resp ect to η , the last term of this sequence is easily seen to b e zero by relativ e dimension v anishing (see e.g. [28], I I I.11.2). The first term is zero since ev ery stable bundle is simple (lemma 2.1) , and for the same reason the second term 6 is an inv ertible sheaf. Moreov er, R 1 η ∗ P ar E nd ( E ) is the tangen t sheaf to N α , hence we shall denote it as T N α . 5 W e w ould like to p oint out the similar ity b et ween the bivector (1 8 ) a nd the bivector obtained b y Bottacin in his study of Poisson str uctures o n mo duli spaces of para bolic vector bundles on algebr aic sur faces, se e [14], equation 4 .1. As Higgs bundles or iginally arose throug h a dimensio nal reduction of a structure in higher dimensions this is not surpr ising. 6 Observe that this term would also b e zero in the cas e of a semi- simple structure group, so for parab olic pr incipal Higgs bundles the r e lev a n t principal bundle ov er the mo duli space N α would just have the pro duct of the Lev i gro ups as structure group, without quo tien ting by global endomo rphisms. MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 19 Lemma 3.7. Al l o f the she aves o c curring in the se quenc e (20) ar e lo c a l ly fr e e. Pr o of. It suffices to no t ice that the corresp onding cohomology gro ups (e.g. H 1 ( S P ar E nd ( E )) ) ha v e constan t ra nk as E v aries in N α , and apply G rauert’s theorem ([28], I I I.11.2).  Denote now η ∗  Q l p ⊗ O ν − 1 ( p )  /η ∗ P ar E nd ( E ) b y Ad . This clearly is a bundle of Lie algebras. Our claim is that the short exact sequence (21) 0 → Ad → R 1 η ∗ S P ar E nd ( E ) → T N α → 0 is an A tiy ah sequenc e, as in (1 4). Lemma 3.8. Th e d ual of the ve ctor b und le P 0 α → N α is R 1 η ∗ S P ar E nd ( E ) . Pr o of. By relative Serre dualit y for t he morphism η in (19), the dual of R 1 η ∗ S P ar E nd ( E ) is giv en by η ∗ P ar E nd ( E ⊗ ν ∗ K ( D )), whic h is a lo cally free sheaf a s well, with fib ers of the asso ciated v ector bundle o v er a p oint [ E ] ∈ N α giv en by H 0 ( P ar E nd ( E ⊗ K ( D ))). Clearly the obv ious bundle morphism η ∗ ( S P ar E nd ( E ⊗ ν ∗ K ( D )) → P 0 α is an isomorphism.  W e now assume the existence of a principal bundle π : F α → N α with structure gr o up (22) L = Y p ∈ D L p ! /C , where C is the diagonal subgroup of the pro duct of the cen ters of the P p , suc h that the total space of F α can b e in terpreted a s a mo duli space of framed α -stable parab olic bundles. A framed parab olic bundle here is a para bo lic ve ctor bundle together with a fr aming of the a ssociated graded space at the mark ed p oin ts, i.e. the c hoice of an isomorphism Gr( p ) = r ( p ) M i =1 F p,i /F p,i +1 ∼ = − → M C m i ( p ) at eac h marked p oint p ∈ D . In app endix A a construction fo r F α is giv en in the case of full flag s. Theorem 3.9. The se quenc e (21) is the Atiyah se quenc e for the L - princip al b und le F α → N α . Pr o of. Since the sheaf of sections of the a djoin t bundle of a principal bundle is the direct image of the relativ e tang ent sheaf of the asso ciated pro j ection, it suffices to sho w that t he in v erse image under π of the 20 MARINA LOGARES A ND JOHA N MAR T ENS short exact sequenc e (13) is the sequence on F α determining the relativ e tangen t sheaf: 0 → T π → T F α → π ∗ T N α → 0 . T o fix notation, let us lo ok at the comm utative diagram (23) F α × X e π / / e η   N α × X η   F α π / / N α where F α is a mo duli space of framed stable parab olic bundles. The space F α × X comes equipp ed with a sheaf of framed endomorphisms (endomorphisms that preserv e the framing) o f a univ ersal bundle F P ar E nd , suc h that the tangen t sheaf to F α is giv en b y R 1 e η ∗ F P ar E nd . As it is also easy to see that F P ar E nd is equal to e π ∗ S P ar E nd ( E ), the commu- tativit y of the diagr a m (23) - and flatness of π , guarantee d b y in v oking e.g. [27], prop osition 6.1.5 - giv e that indeed R 1 e η ∗ F P ar E nd ∼ = π ∗ R 1 η ∗ S P ar E nd ( E ) , see e.g. [28 ], Prop osition I I I.9.3.  3.2.3. Extension of the br acket. Theorem 3.10. The bive ctor on P α determine d by (1 8) e x tends the Poisson structur e on P 0 α given by the Lie algebr oid on ( P 0 α ) ∗ . Pr o of. F o r an α - stable framed parab olic bundle ( E , ∼ = ) the tangent and cotangen t spaces to F α are give n r espectiv ely b y T [ E , ∼ = ] F α = H 1 ( S P ar E nd ( E )) and T ∗ [ E , ∼ = ] F α = H 0 ( P ar E nd ( E ) ⊗ K ( D )) . The tangent and the cotangen t spaces to T ∗ F α at a p oin t [ E , Φ , ∼ = ] are giv en b y the first hypercohomology groups T [ E , Φ , ∼ = ] T ∗ F α ∼ = H 1  S P ar E nd ( E ) [ . , Φ] − → P ar E n dE ⊗ K ( D )  and T ∗ [ E , Φ , ∼ = ] T ∗ F α ∼ = H 1  S P ar E nd ( E ) − [ . , Φ] − → P ar E ndE ⊗ K ( D )  . F urthermore, if w e lo ok at the short exact sequences of complexes 0 / / 0 / /   S P ar E nd ( E ) / / ± [ , , Φ]   S P ar E nd ( E ) / /   0 0 / / P ar E nd ( E ) ⊗ K ( D ) / / P ar E nd ( E ) ⊗ K ( D ) / / 0 / / 0 , MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 21 tak e their lo ng exact sequence in hypercohomology , a nd notice that H 0 ( S P ar E nd ( E )) is zero since E is stable as a parab olic v ector bundle, w e o btain the short exact sequences 0 → H 0 ( P ar E nd ( E ) ⊗ K ( D )) → H 1  S P ar E nd ( E ) ± [ . , Φ] ↓ P ar E nd ( E ) ⊗ K ( D )  → H 1 ( S P ar E nd ( E )) → 0 , corresp onding to (9) and (10). No w, using the ch ara cterization giv en in (11 ) for the cano nical Poisson structure on T ∗ F α , one can see that this is induced b y the morphism of complexes S P ar E ndE − [ . , Φ]     Id / / S P ar E nd ( E ) [ . , Φ]   P ar E nd ( E ) ⊗ K ( D )   − Id ⊗ Id K ( D ) / / P ar E nd ( E ) ⊗ K ( D ) . . Since the ma p T ∗ F α → P 0 α is a P oisson morphism, a nd using the c haracterization (1 2) and the definition (18) o f ♯ P α it suffices to notice that by these c hoices indeed the square H 1  S P ar E nd ( E ) − [ . , Φ] ↓ P ar E nd ( E ) ⊗ K ( D )  / / H 1  S P ar E nd ( E ) [ . , Φ] ↓ P ar E nd ( E ) ⊗ K ( D )    H 1  S P ar E nd ( E ) − [ . , Φ] ↓ S P ar E nd ( E ) ⊗ K ( D )  / / O O H 1  P ar E nd ( E ) [ . , Φ] ↓ P ar E nd ( E ) ⊗ K ( D )  comm utes.  As P 0 α is op en and dense in P α this establishes the Poiss on structure on all of P α . 3.2.4. Symple ctic l e aves. Give n a principal G -bundle P → M , the sym- plectic lea v es of the dual of an Atiy ah a lgebroid T P /G are simply t he symplectic reductions of the cotangent bundle T ∗ P . The lift of the ac- tion of a group G on a manif o ld N t o the total space of t he cotangent bundle T ∗ N is of course alwa ys Hamiltonia n, with a canonical momen t map give n by g ∈ g , χ ∈ T ∗ x N : µ ( χ )( g ) = χ ( ξ g ( x )) , where ξ g is the Hamiltonian vec tor field correspo nding to g ∈ g . In the particular case where the a ctio n of G is f ree, i.e. when the ma nif o ld 22 MARINA LOGARES A ND JOHA N MAR T ENS is a principal G -bundle, the momen t map can also b e understo o d b y dualizing t he sequence (13): 0 → π ∗ T ∗ ( P /G ) → T ∗ P µ → T ∗ orbits P → 0 , observing that T orbits P ∼ = P × g and T ∗ orbits P ∼ = P × g ∗ . Recalling (2 0) and the pro of of Theorem 3.9 this tells us immediately t hat on T ∗ F α the momen t map µ is giv en by the p ar ab olic r esidue , i.e. the map that, when restricted to a fib er ov er a framed parab olic bundle E , giv es H 0 ( P ar E nd ( E ) ⊗ K ( D )) → ker M p ∈ D l ∗ p → H 0 ( P ar E nd ( E )) ∗ ! whic h is the dual of the b oundary map asso ciated with the short exact sequence 0 → S P ar E nd ( E ) → P ar E nd ( E ) → M p ∈ D l p ⊗ O p → 0 . The reduction, and hence symplectic lea v es, a r e simply µ − 1 ( O ) / L , where O is a co-adjoint or bit in Lie ∗ ( L ). Notice that in the generic case, when the eigen v alues of the Higgs field a t the marked p oin ts are all distinct, there is a unique co-adjoint orbit with these eigen v alues. When eigen v alues are rep eated on a particular E p,i /E p,i +1 there will b e sev eral co-adjoint orbits. One can c hec k that this ag rees with the rank of the P oisson structure giv en b y ♯ P α at a parab olic Higg s bundle ( E , Φ). Indeed, if one recalls the definition (18) then the short exact sequence of complexes 0 / / S P ar E nd ( E ) Id / / − [ . , Φ]   P ar E nd ( E ) [ . , Φ]   / / L l p ⊗ O p / / [ . , Φ] | Gr( p )   0 0 / / S P ar E nd ( E ) ⊗ K ( D ) / / P ar E nd ( E ) ⊗ K ( D ) / / L l p ⊗ K ( D ) p / / 0 giv es rise to the long exact sequence of h yp ercohomology (at least when E is stable): 0 → C → ⊕ ker  [ . , Φ] | Gr( p )  → H 1  S P ar E nd ( E ) ↓ S P ar E nd ( E ) ⊗ K ( D )  ♯ P α − → H 1  P ar E nd ( E ) ↓ P ar E nd ( E ) ⊗ K ( D )  → ..., and hence the generic, maximal rank of ♯ P α , o ccurring when all eigen- v alues of Φ are differen t, is (24) rk P α = dim P α − nr + 1 = (2 g − 2) r 2 + nr ( r − 1) + 2 . MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 23 The eigen v alues of the Higg s field are in fact determined by the Hitc hin map, and as the latter is blind to the parab olic structure, it factors through morphisms of the form (5 ) : (25) P ˜ α   h ˜ α   @ @ @ @ @ @ @ @ P α h α / / H e / / H / H 0 , where H 0 is the subspace of H giv en by H 0 = H 0 ( X , K ) ⊕ H 0 ( X , K 2 ( D )) ⊕ · · · ⊕ H 0 ( X , K r ( D r − 1 )) . Observ e that the Hit chin map for the mo duli space of strongly parab olic Higgs bundles (whic h is a symplectic leaf of P α ) tak es v alues in H 0 . One easily sees that, roughly sp eaking, H / H 0 determines the eigenv alues of the Higgs field at the mark ed p oints p ∈ D , without ordering. In the case where ˜ α correspo nds to full flags, the connected comp onen ts (one for ev ery ordering of the eigen v alues) of the fib ers of e ◦ h ˜ α are exactly the symplectic leav es. 3.2.5. Complete inte gr ability of the Hitchin system. The comp osition e ◦ h α from (25) also plays a role in the Hitc hin system, whic h w e can discuss no w that w e ha v e the Poisson structure at our dispo sal. Recall that f or a holomorphic P oisson manifold or P oisson v ariet y of dimension 2 k + l , where the r a nk (or dimension of the generic leaf ) of the P oisson structure is 2 k , a completely in tegrable system is g iv en b y k + l Poisson-comm uting, f unctiona lly independent functions, suc h that l of them are Casimirs , i.e. they P oisson-comm ute with any func- tion. F urthermore the generic fib er of the collectiv e of these functions is required to b e an ab elian v ariet y . The connected comp onen ts of the sim ultaneous fib ers o f the Casimir functions are the closures of the top- dimensional symplectic lea v es. By Riemann-R o c h one gets (assuming that n ≥ 1) dim( H ) = (2 g − 2 + n ) ( r + 1) r 2 + r (1 − g ) and dim( H 0 ) = (2 g − 2) r ( r + 1) 2 + n r ( r − 1 ) 2 + r (1 − g ) + 1 , hence dim ( H / H 0 ) = nr − 1 . 24 MARINA LOGARES A ND JOHA N MAR T ENS Notice also from (6) that g ( X s ) = dim ( H 0 ), and from (4) that 2 dim ( H 0 ) + dim ( H / H 0 ) = dim P α . In order to sho w directly that P α equipped with h α is a completely in tegrable system, w e would ha v e to w ork with lo cal informatio n, in order to establish the v anishing of the relev a n t Poiss on brack ets. As our map ♯ P α is defined fibre-wise how ev er, w e use an alternativ e c har- acterization (follo wing [42], section 8.1): Prop osition 3.11. The c onne cte d c omp onents of a generic fib er h − 1 α ( s ) , c orr esp onding to a smo oth sp e ctr al curve X s , ar e L agr angian in sym- ple ctic le aves for the Poisson s tructur e on P α . In order to prov e this we shall need t w o lemmas. The first is the follo wing ch ara cterization of coisotropic submanifolds o f symplectic lea v es, whic h is easy to see: Lemma 3.12. L et J b e a submanifold of a symple ctic le af L o f a Pois- son manifold ( M , ♯ ) . Then J is c o-isotr opic in L if for any p oint p ∈ J we have that ♯ ( N ∗ p J ) ⊂ T p J, wher e T p J is the tangent sp ac e to J at p , N p J is the normal s p ac e at p of J in M , a nd N ∗ p J is the c onormal sp ac e at p . The second is a description of the tangent space to h − 1 α ( s ): Lemma 3.13. L et ( E , Φ) b e a Higgs b und le in one of the c omp onents of h − 1 α ( s ) , whi c h w e identify with the Jac obian J s of X s . Then we have the fol lowing sho rt exact se quenc e on X : 0 → T ( E , Φ) J s ∼ = H 1 ( ρ ∗ O X s ) → H 1  P ar E nd ( E ) [ . , Φ] ↓ P ar E nd ( E ) ⊗ K ( D )  → H 0 (( ρ ∗ K X s )( D )) → 0 Pr o of. Observ e that b y using Hurwitz’ theorem ([28], Prop osition IV.2.3) w e hav e that ρ ∗ ( K X )( R ) = K X s , a nd b y using Hurwitz’ theorem and relativ e Serre duality w e ha v e ρ ∗ ( L − 1 ( R )) = E ∗ . If w e tensor the exact sequence (7) on X s with L − 1 ( R ) and push it forw ard by ρ we obtain the exact sequence 0 → ρ ∗ ( O X s ) → P ar E nd ( E ) [ . , Φ] − → P ar E nd ( E ) ⊗ K ( D ) → ρ ∗ ( K X s )( D ) → 0 on X . Using this w e can lo ok at the short exact sequences of complexes 0 / / ρ ∗ ( O X s )   / / P ar E ndE [ . , Φ]   / / im([ . , Φ])   / / 0 0 / / 0 / / P ar E nd ( E ) ⊗ K ( D ) / / P ar E nd ( E ) ⊗ K ( D ) / / 0 MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 25 and 0 / / im([ . , Φ]) / /   im([ . , Φ]) / /   0 / /   0 0 / / im([ . , Φ]) / / P ar E nd ( E ) ⊗ K ( D ) / / ρ ∗ ( K X s )( D ) / / 0 Com bining the h yp ercohomology long exact sequences give s the desired result.  Pr o of of Pr op osition 3.11. By the discussion in section 3.2.4, it is clear that the connected comp onen ts of h − 1 α ( s ) are con tained in symplectic lea v es. No tice that by (6 ), Prop osition 2.2 and (24) w e already kno w that generically these connected comp onen ts are smo oth sub v arieties of half the dimension of the symplectic leav es, therefore it suffices to sho w that the connected comp onen ts of h − 1 α ( s ) are co-isotropic. F or this w e can now apply Lemma 3.12 to a connected comp onen t of h − 1 α ( s ) corresp onding to a smo oth sp ectral curv e X s b y observing that in the comm utativ e diagra m H 0 ( ρ ∗ K X s ( D )) ∗ ∼ = H 1 ( ρ ∗ O X s ( − D )) / /   H 1 ( ρ ∗ O X s )   H 1  S P ar E nd ( E ) − [ . , Φ] → S P ar E nd ( E ) ⊗ K ( D )  ♯ P α / /   H 1  P ar E nd ( E ) [ . , Φ] → P ar E nd ( E ) ⊗ K ( D )    H 1 ( ρ ∗ O X s ) ∗ ∼ = H 0 ( ρ ∗ K X s ) / / H 0 ( ρ ∗ K X s ( D )) the columns (given by Lemma 3.13) are exact. This ends the pro of of Prop osition 3.11.  This finally gives us: Theorem 3.14. The mo duli sp ac es P α , with the Poisson structur e intr o duc e d a b ov e an d the Hitchin map h α , form c ompletely inte gr able systems, for which the Casimi rs a r e given by e ◦ h α . 4. Morphisms betwee n moduli sp a ces, Grothendieck-Springer resolution Giv en a complex semi-simple connected Lie G roup G with Lie alge- bra g and W eyl group W G , one can construct the so- called Grothendiec k- Springer morphism. There are v arious incarna tions of this, for the 26 MARINA LOGARES A ND JOHA N MAR T ENS group, the Lie alg ebra, etc, so w e just briefly recall this here. The Grothendiec k-Springer space is defined as G S G = { ( g , b ) | g ∈ g , b ∈ G/B , g ∈ b } , where B is a Borel subgroup 7 of G . The ob vious map µ : G S G → g is widely used in geometric represen tation theory , see e.g. [16]. It is generically finite ( | W G | : 1), and provide s a resolution o f singularities of the nilp oten t cone n ⊂ g , whic h is referred to as Springer’s r esolution . After c ho osing an equiv arian t iden tification g ∼ = g ∗ w e can think of G S G as t he dual of an algebroid ov er G/B , and the map µ as the momen t- map for the induced G action. In particular G S G is a regular Poiss on manifold (i.e. a ll the symplectic lea v es hav e the same dimension), and µ will b e a Poisson morphism. Moreo v er there is the follow ing diagram, called the Gr othendie ck simultane ous r esolution : (26) G S G / /   t   g / / t / W G where t is the abstract Cartan and W G the a bstract W eyl group o f G . F or more details regarding this we refer to [16], section 3.1. Our construction giv es a similar picture for Atiy ah algebroids rather than Lie a lgebras, where the role of g is now play ed by any of the P α , but in pa r t icular can b e the mo duli space of parab olic Higgs bundles with minimal flag- t yp e (see also section 5 b elo w), and t he role of the Grothendiec k-Springer v a r iet y b y the mo duli space of para bolic Higgs bundles with full flags. Indeed, w e show b elo w easily tha t this and similar f o rgetful morphisms are P oisson. Let us lo ok at the mo duli spaces fo r tw o differen t flag t yp es on the same divisor of marked p oin ts, P ˜ α and P α , where the flag type of the latter is coarser than that of the former. W e assume that the para b olic w eigh ts α and ˜ α a r e close enough that if one forgets pa r t of the flag o n an ˜ α - stable para bo lic Higgs bundle the result is α -stable, so that w e obtain a morphism (27) P ˜ α → P α . Prop osition 4.1. T he morphism (27) is Poisson. 7 One ca n genera lize this to G/P , that is, to par a bolic subgro ups other tha n Borels. MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 27 Pr o of. Let us denote a parab olic Higgs bundle for the finer flag type as ( e E , e Φ)and its image under the forgetful morphism as ( E , Φ). Then clearly we hav e the natural inclusions of sheav es S P ar E nd ( E ) ⊂ S P ar E nd ( e E ) and P ar E nd ( e E ) ⊂ P ar E nd ( E ) . Therefore we get tha t the diagram H 1  S P ar E nd ( e E ) ↓ S P ar E nd ( e E ) ⊗ K ( D )  / / H 1  P ar E nd ( e E ) ↓ P ar E nd ( e E ) ⊗ K ( D )    H 1  S P ar E nd ( E ) ↓ S P ar E nd ( E ) ⊗ K ( D )  / / O O H 1  P ar E nd ( E ) ↓ P ar E nd ( E ) ⊗ K ( D )  comm utes.  As said ab ov e, it is particularly interesting to lo ok at the morphism (27) in the case where P ˜ α corresp onds to full flags, as then P ˜ α is a regular P oisson manifold. W e can put things together in the analogue of the Grot hendiec k simultaneous resolution (26): (28) P ˜ α   o / / h ˜ α   ? ? ? ? ? ? ? ? C r n / C   P α h α / / H / / H / H 0 ∼ = / / ( C r n / ( S r ) n ) / C . The map o : P ˜ α → C r n → C r n / C is just giv en b y the eigen v alue s of Φ at the marked p oin ts - b ecause of t he full flag s they come with an ordering. 5. Fur ther re marks 5.1. Comparison with Bottacin-Markman. A particular case, the case of minimal flags, of the ab ov e has already b een discussed in the literature, in independent w ork b y Bottacin [13] and Markman [42], though not framed in terms of pa r abo lic (Higgs) bundles or algebroids. Reviews of this w ork also app eared in [21, 22]. Bottacin and Markman study stable pairs or twis ted Higgs bundles, i.e. a v ector bundle E o v er a curv e X together with a morphism Φ : E → E ⊗ F , f or some fixed line bundle F . P airs of this kind (w orking ov er a field o f p ositive c haracteristic), their mo duli stack and the Hitc hin fibration for them also play ed a crucial role in the recen t w ork of Laumon- Ngˆ o [39]. A 28 MARINA LOGARES A ND JOHA N MAR T ENS mo duli space for these w as constructed b y Nitsure in [48], and in [13] and [42] it is sho wn that, if deg ( F ) > deg ( K ) (or F = K ) a nd once one c ho oses an effectiv e divisor D in F K − 1 , this space has a canonical P oisson structure. Once this choice is ma de, and if D is moreov er reduced, suc h a stable pair ( E , E Φ → E ⊗ K ( D )) can of course a lso b e interpre ted as a para- b olic Higgs bundle for the minimal flag t yp e E p, 1 ⊃ E p, 2 = { 0 } . In the case o f suc h minimal flag s there is only a single w eigh t at eac h mark ed p oin t, and o ne sees that it do es not con tribute to t he slop e inequal- it y (1). Therefore one cannot afford the luxury of the assumption of genericit y of t he we ights, and unless the rank and degree are coprime there are prop erly semi-stable p oints, and the mo duli space of v ector bundles is singular. Ev en o ve r the non-singular lo cus, corresp onding to the stable ve ctor bundles, there do es not exist a unive rsal bundle, but there is how ev er still a sheaf ov er the stable lo cus pla ying the role of sheaf of endomorphisms of a univ ersal bundle [13], Remark 1 .2.3, whic h is as useable as our P ar E nd ( E ) 8 . Both Bottacin a nd Markman are prima r ily fo cused on t he P ois- son structure, and mak e no mention of Lie algebroids. Nev ertheless, Bottacin uses the same philosophy of o bta ining the P oisson structure through studying the dual v ector bundle. He ev en writes do wn the definition of a Lie algebroid and pro ve s theorem 3.4 in [13], section 4 .2. He how ev er do es not iden tify the a lg ebroid a s an Atiy ah algebroid, but rather exhibits the Lie brac k et on lo cal sections explicitly on the level of co cycles and co c hains. Markman do es use the principal bundle ov er the mo duli space of bundles (using a construction of Seshadri [52]) , but phrases ev erything in terms of reduction of its cotangen t bundle. Despite this our ap- proac h is closest to Markman, a nd a careful reader migh t find sev eral similarities in o ur exp osition, in pa r ticular in section 3.2.5, for whic h w e were help ed b y [42], section 8.1. Neither Bottacin or Markman mak e the restriction that we do tha t D is a reduced divisor - i.e. they a llow the Higgs field Φ to ha v e p oles of arbitrary or der, when interpre ted as a meromorphic bundle morphism 8 Notice that in the cas e of pa r abo lic bundles with non-g eneric weight one could use the same s trategy , in fact for so me of the non-g eneric weigh ts an actual univ ersal bundle do es exist ov er the stable lo cus, see [1 2], Theorem3 .2 . MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 29 from E to E ⊗ K , though Bottacin assumes D to b e reduced in some of his pro ofs. Also for non-minimal flags this w ould b e a desirable prop- ert y , in particular in the light o f the geometric Langlands program with wild ramificatio n, and should not b e significan tly more complicated. Another ob vious generalization would b e to lo ok at other semi-simple or reductiv e structure g roups, replacing the use of sp ectral curve s with cameral cov ers. Though most of the statemen ts w e mak e can at least formally b e translated into this setting, w e hav e refra ined from w orking in this generalit y as it seems that the dust has not settled on the notion of stabilit y for parab olic principal bundles, cf. [6, 54, 3 , 2]. By w orking in t he con text of stac ks rather t ha n mo duli sc hemes t hese problems w ould of course b e av oided, a nd we intend to take up this matter in the fut ure. 5.2. P arab olic vs. orbifold bundles. In the case where all we ights are rational there is an alternativ e description of parab olic bundles in terms of orbif o ld bundles, whic h pro vided muc h of the original motiv a- tion ( see [51], a history of the g enes is o f parab olic bundles is giv en in [49]). Giv en a finite group Γ acting on a curv e Y , giving rise to the ramified co v ering p : Y → X = Y / Γ , an o rbifold bundle is a Γ- equiv ariant bundle on Y . Alternatively , in the analytic category , one can define an orbifold Riemann surface to b e a (compact) Riemann surface X with n marked p oin ts p 1 , . . . , p n on X and a p ositiv e integer α i asso ciated to eac h p i . An orbifold bundle is then determined by lo cal orbifold t r ivialisatio ns and transition func- tions, where near a mar k ed p oint p i the trivializatio ns should b e of the form D × C r /σ i × τ i , where D is a disk in C , σ i is the standard repre- sen tation of Z α i , a nd τ is an isotrop y represen tation τ : Z α i → GL r ( C ) X . These tw o definitions of orbifold bundles are equiv alen t ( under the condition that n > 2 if g = 0), see e.g. [25], page 42. An o r bif o ld bundle in this sense corresp onds to a parab olic bundle on X with rational w eigh ts. The corresp ondence ha s b een extended to b oth higher dimensions in [9] a nd principal bundles [3]. F or explicit descriptions w e refer to [25], section 5, [10], section 4, or [9], section 2c. In [46], Nasat yr and Steer discuss Higgs bundles of rank 2 in the orbifold setting, fo cusing on analytic asp ects. They define an orbifold Higgs bundle (or Higgs V -bundle) on an o rbifold Riemann surface X to b e an orbifold bundle E on X to g ether with an orbifold bundle mor- phism Φ : E → E ⊗ K , where K is the or bif o ld canonical bundle of 30 MARINA LOGARES A ND JOHA N MAR T ENS X . This definition is also used in [47 , 3]. The corresp o ndence with parab olic Higgs bundles is work ed out in [46 ], section 5 , as is the in- tegrable system. Ho w ev er, t he parab olic Higgs bundles corresponding to the orbifold Higgs bundles they obtain are all (in our terminology) strongly para b olic ( as one can observ e b y taking the residue o f equation (5c) in [46]). It would b e in teresting to discuss the matter of non-strongly para- b olic Higgs bundles from an o rbifold p erspective and see if o ne could obtain the analogous results of our theorems 3.9, 3.10, 3.14, 4.1. Pre- sumably the ana lo gue of all parab olic Higgs bundles (i.e. not neces- sarily strongly parab olic) w ould b e giv en b y lo oking at o rbifold Higgs bundles with a Higgs field E → E ⊗ L , where L is the orbifold line bundle obtained by t wisting K with (follo wing the notation of [46])the fractional divisor P i 1 α i p i . An orbifo ld v ersion of the work of Bo t tacin and Markman would then corresp o nd to our results. Appendix A. A Levi princip al bundle over the moduli sp ace of p arabolic bundles In this app endix we giv e a construction of a principal bundle F α → N α with structure group L , in the particular case of full flags at a ll mark ed p oints. In the general case, one should think of the tota l space F α of this principal bundle as a mo duli space f o r α -stable parab olic bundles, together with isomorphisms of all consecutiv e quotien ts in the flags to a fixed v ector space E i ( p ) /E i +1 ( p ) ∼ = C m i . One can think of sev eral approac hes to this problem. One approac h (for general flag t yp es) one could tak e is suggested in [31]: start fro m a suitable mo duli space o f framed ve ctor bundles (also kno wn as bundles with lev el structure, in the case where the divi- sor ov er whic h one f rames is r educed), as w as for instance constructed in [3 4 ], g eneralizing earlier work by Seshadri [52]. The structure group of the v ector bundles under consideration acts on this space b y chang- ing the framing, and w e w ould lik e to tak e a GIT-style quotient by the unip oten t radical of the parab olic subgroup. F or the Borel sub- group (leading to full flags) this is describ ed in [31], a more general approac h is giv en in [23, 36]. The v arious N α w ould t hen b e giv en b y GIT quotien ts by the Levi group of this space, with the α o ccurring as the choice of a linearization. As suc h the N α are, for generic α , geo- metric quotien ts f or the Levi g roup actions, and therefore (with some mild extra conditions) principal bundles for the Levi group, see b elo w. MODULI OF P ARABOLIC HIGGS BUND LES AND A TIY AH A LGEBR OIDS 31 Notice that the action of P GL ( r ) on a mo duli space of fr a med bundles w as discussed in [8], where it w as sho wn that the action linearizes and the G IT quotien t is the mo duli space of v ector bundles. In order to k eep the exposition from b ecoming to o technic al w e shall use a construction a lr eady done in the literature, follow ing [32]. Here a pro jectiv e v ariet y F w as constructed directly , whic h w as in terpreted as a mo duli space of fra med parab olic shea v es 9 . The construction w as inspired b y a similar construction [33] in symplectic geometry through symple ctic implosion . Th e connection b et w een symplectic implosion and non-reductiv e GIT was discussed in [37]. This v ariet y F comes with a natura l to r us action, an action whic h linearizes on a relatively ample line bundle. Using an earlier construction fo r the mo duli space of para bo lic bundles giv en b y Bhosle [5], it is sho wn that at a lineariza- tion give n b y a character α , the G IT quotien t is the mo duli space o f parab olic v ector bundles, F / / α T ∼ = N α (ev en in the case of partial flags, if o ne uses the α i with the corresp onding m ultiplicities). If α is regular, i.e. we are lo oking at full flags, this is sufficien t fo r us: the Luna slice theorem [40], see also [44 ], App endix to Chapter 1, and [35], Corollary 4.2.13, no w establishes t ha t the α - stable lo cus (the stabilit y simply corresp onds to the stabilit y of the underlying bundle) F α ⊂ F is a principal bundle whic h is lo cally trivial in the ´ eta le to polo gy . F ur- thermore, b y a result of Serre [50], as the structure gro up is a torus, it is eve n lo cally trivial in the Zariski top ology . 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Dep ar t amento de Ma te m ´ atica Pura, F aculd ade de Ci ˆ encias, Univer- sidade do Por to, Rua do Campo Alegre, 687, 4169-00 7 P or to, Por tugal Curr ent addr ess : Departamento de Matem´ aticas, CSIC, Serrano 1 21, 28 006 Madrid, Spa in E-mail addr ess : mari na.logare s@icmat.es Dep ar tment of Ma thema tics, U niversity of Toronto, 40 St. George Street, Toronto Ont ario M5S 2E 4, Canada Curr ent addr ess : Centre for Quantum Geo metry of Mo duli Spaces, Department of Ma thematical Sciences, Aarhus Universit y , Ny Munkegade 118, bldg. 1530 , DK- 8000 ˚ Arhus C, Denmar k E-mail addr ess : jmar tens@imf. au.dk