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8 DEFORMATIONS OF FUCHSIAN EQUATIONS AND 0 0 LOGARITHMIC CONNECTIONS 2 n a by J 6 1 Szila´rd Szabo´ ] G A . h t Abstract. — Wegiveageometricprooftotheclassicalfactthatthedimension a ofthedeformationsofagivengenericFuchsianequationwithoutchangingthe m semi-simpleconjugacyclassofitslocalmonodromies(“numberofaccessory [ parameters”) is equal to half the dimension of the moduli space of deforma- 3 tions of the associated local system. We do this by constructing a weight 1 v Hodgestructureontheinfinitesimaldeformationsoflogarithmicconnections, 0 such that deformationsas an equationcorrespondto the (1,0)-part. This an- 3 swersaquestionofNicholasKatz,whonoticedthedimensiondoublingmen- 2 tioned above. We then show that the Hitchin map restricted to deformations 3 0 of the Fuchsian equation is a one-to-one e´tale map. Finally, we give a posi- 7 tiveanswertoaconjectureofOhtsukiaboutthemaximalnumberofapparent 0 singularitiesforaFuchsianequationwithgivensemisimplemonodromy,and / h defineaLagrangianfoliationofthemodulispaceofconnectionswhoseleaves t consistof logarithmicconnectionsthatcan be realised as Fuchsianequations a m havingapparentsingularitiesinaprescribedfiniteset. : v i X r a 1. Introduction Letp ,...,p ∈ P1 ben ≥ 2 fixed pointsintheaffinepart ofthecomplex 1 n projective line, and let p be the point at infinity. Define P to be the simple 0 effective divisor p + ···+ p in P1, and let Po = p + ···+ p . Consider 0 n 1 n thefunction (1) ψ(z) = (z −p )···(z −p ) 1 n asanidentificationbetweenOP1 andOP1(−Po)ontheaffineA1 = P1\{p0}. 2 SZILA´RDSZABO´,REŃYIINSTITUTEOFMATHEMATICS,BUDAPEST Let w = w(z) be a holomorphic function of the complex variable z, w(k) itsk-th orderdifferentialwithrespect toz and G (z) G (z) (2) w(m) − 1 w(m−1) −···− m w = 0, ψ ψm where the G are polynomials in z, be a Fuchsian differential equation. We k recall thatthismeansthatall thesolutionsw growat mostpolynomiallywith (z −p )−1 near p forany1 ≤ j ≤ n (respectively,withz near infinity). The j j left-hand side of (2) is a linear differential operator of the order m of w, that we shall denoteby L. We shall identify the equation(2) with the operator L. FuchsgaveanecessaryandsufficientconditiononthedegreesoftheG ’sfor k LtobeofFuchsian type: thedegreeofG has tobeat mostk(n−1). k Letus introducetheexpressions w = w 1 dw w = ψ 2 dz . . (3) . dm−1w w = ψm−1 . m dzm−1 Wethinkofw asalocalholomorphicfunctioninz,orinmodernterminology a local section of the structure sheaf O. Then, on the affine open A1 the function w is meromorphic with zeroes of order at least (k − 1) in P. In k other words, on the affine part A1 a vector (w ,...,w ) is a section of the 1 m holomorphicvectorbundle E˜ = OP1 ⊕OP1(−Po)⊕...⊕OP1((1−m)Po). We equip E˜ with the algebraic integrable connection with logarithmic poles at P A(z) (4) DL = d1,0 − dz, ψ(z) DEFORMATIONSOFFUCHSIANEQUATIONSAND LOGARITHMICCONNECTIONS 3 whereA(z) isthemodifiedcompanionmatrix 0 0 0 0 ...... G m 1 ψ′ 0 0 ...... G  m−1  0 1 2ψ′ 0 ...... G m−2 (5) A = . . . .  .. .. .. ..    0 0 0 ... (m−2)ψ′ G   2  0 0 0 ... 1 G +(m−1)ψ′  1  of equation (2). Here we have denoted ψ′ = dψ/dz. One readily checks that ameromorphicfunctionw onsomeopensetU ⊂ A1 withpolesatmostinP locally solves (2) if and only if the vector (w = w,w ,...,w ) is a parallel 1 2 m sectionofDLonE˜ overU forsome(hence, onlyone)vector(w2,...,wm). Remark1.1. — Instead of the formulae (3), we could have simply chosen wk = w(k), and the form of the same connection DL in this trivialisation would then be a usual companion matrix. The reason for our choice for the extension (3) is that it gives rise to a logarithmic lattice. In fact, the two points of view are equivalent, so that a connection in modified companion formisalsoinduced byanequation. By assumption, the integrable connection DL is regular at infinity as well. Therefore, according to a theorem of N. Katz (Thm. II.1.12 [Del70]), there existsalatticeforthemeromorphicbundle N = OP1(∗P)⊕...⊕OP1(∗P) at infinity with respect to which DL is a logarithmicconnection, i.e. its local form in any holomorphic trivialisation contains 1-forms with at most first- order poles. Such a lattice at infinity can be obtained similarly to the case of the other singularities. For this purpose, let ζ = z−1 be a local coordinate at infinity. Recall that the first component w = w of (3) is supposed to be 1 a section of OP1. Then, a logarithmic lattice (w˜1,...,w˜m) at infinity can be obtainedbytheformulae w˜ = w 1 dw w˜ = ζ 2 dζ . . (6) . dm−1w w˜ = ζm−1 , m dζm−1 4 SZILA´RDSZABO´,REŃYIINSTITUTEOFMATHEMATICS,BUDAPEST and the form of the connection is then again a modified companion matrix withentries in thelast columnequal to thecoefficients oftheequationmulti- pliedby an appropriatepowerofζ. Now,as djw djw ζj = (−z)j dζj dzj and forlargez onehas ψ(z) ≈ zn, wededucethatthetrivialisations(3)and (6)arelinkedon C∗ by thematrix diag(1,−zn−1,...,(−1)m−1z(m−1)(n−1)). It follows that the connection DL extends logarithmically onto the holomor- phicbundle (7) EL = OP1 ⊕OP1(∞−Po)⊕...⊕OP1((m−1)(∞−Po)) over P1. Hence, we get a bijective correspondence between local solutions of (2) and local parallel sections of the logarithmic connection DL on the holomorphicbundleEL. Thefollowingfundamentalresultispartoffolklore; however,wegiveaproofhereforlack ofappropriatereference. Proposition1.2. — If the logarithmic connections induced by the Fuchsian equationsL ,L aregauge-equivalent,thenL = L . 1 2 1 2 Proof. — Suppose there exists a gauge transformation g ∈ End(EL ,EL ) 1 2 mapping DL into DL . In the trivialisations (7) of EL and EL , g can be 1 2 1 2 written as a matrix whose entry in the k-th row and l-th column is a global holomorphicsectionofthesheafO((k−l)(n−1)). Itfollowsthatthematrix ofg islowertriangular,andthattheentriesonthediagonalareglobalsections of the trivial holomorphicline bundleoverP1, hence constants. For j = 1,2 letus writeontheaffinepart C ofP1 away from infinitytheexpressions A (z) DL = d1,0 − j dz. j ψ(z) It isthena well-knownfact that theaction ofg onDL is 1 g−1A (z)g g ·(DL ) = d1,0 − 1 dz −g−1d1,0g. 1 ψ(z) Itfollowsfromtheaboveobservationsthatg−1d1,0g isstrictlylowertriagular. In particular,theentries onand abovethediagonalin thematrices A and A 1 2 mustagree. DEFORMATIONSOFFUCHSIANEQUATIONSAND LOGARITHMICCONNECTIONS 5 Letus first considerthecasem = 2. Here, onehas g 0 g = 11 , (cid:18)g g (cid:19) 21 22 where g is a global section of O(n − 1) and g ,g are constants with 21 11 22 g g 6= 0, andtheinverseofthismatrixis 11 22 1 g 0 g−1 = 22 . g g (cid:18)−g g (cid:19) 11 22 21 11 Furthermore, thematrices oftheequationsare 0 1 A = , j (cid:18)Gj Gj +ψ′(cid:19) 2 1 whereGj,Gj arethecoefficientsofL . Straightforwardmatrixmultiplication 2 1 j yields 1 g g g2 g−1A (z)g = 21 22 22 . 1 g g (cid:18) ∗ ∗ (cid:19) 11 22 By theabove, thetermsin thefirst rowmustbe equal to 0 and 1 respectively. Weinferthatg = 0 and g = g ,hence g isamultipleoftheidentity. 21 22 11 We now come to the general case. As the computations are more involved butofthesamespirit,weonlysketch theproof. Thematrixg isequal to g 0 ... 0 11 g g ... 0  21 22  g = ... ... ... ,   g g ... g   m1 m2 mm and itsinverseisoftheform g ···g 0 ... 0 22 mm g−1 = 1  ∗.. g11g33···gmm ..... 0.. . g g ···g . . . 11 22 mm    ∗ ∗ ... g ···g   11 m−1,m−1 6 SZILA´RDSZABO´,REŃYIINSTITUTEOFMATHEMATICS,BUDAPEST Onehas 1 g−1A (z)g = · 1 g g ···g 11 22 mm g ···g 0 ... 0 22 mm .  ∗ g g ···g ... ..  11 33 mm .. ..  . . 0     ∗ ∗ ... g ···g   11 m−1,m−1 g g 0 ... 0 21 22 . g ψ′ +g g ψ′ +g g ... ..  21 31 22 32 33 .. .. .. .  . . . 0     ∗ g ψ′ +g g   m−1,m−1 m,m−1 mm  ∗ ... ∗ ∗    As the entries on and above the diagonal in this product have to be equal to those of the modified companion matrix A (z), we deduce as before that 2 g = g = ··· = g and g = g = ··· = g = 0. It follows 11 22 mm 21 32 m,m−1 that right below the diagonal all the entries of the matrix g−1d1,0g vanish. Considering now the first sub-diagonal in the product above, we deduce that g = g = ··· = g = 0. It follows that on the second sub-diagonal 31 42 m,m−2 of the matrix g−1d1,0g all the entries vanish. Continuing this argument, we eventually obtain that all the g for k > l must vanish. This concludes the kl proof. Consider now the residue res(p,DL) of DL on EL at each of the singular points p ∈ P; it is a well-defined endomorphism of the fiber of EL at p. Denotebyµj,...,µj theeigenvaluesoftheresidueat p . 1 m j Remark1.3. — Notice in particular that deg(EL) = (1 − n)m(m − 1)/2, in accordance with the classical Fuchs’ relation, which states that the sum µj oftheeigenvaluesoftheresiduesofDLinallsingularities(including j,k k iPnfinity)isequalto (n−1)m(m−1)/2. Throughoutthepaper, wewillassumethegenericity conditions: Condition1.4. — Theeigenvaluesµj,...,µj oftheresidueoftheintegrable 1 m connectionDLineach singularitypj do notdifferbyintegers. (In particular, theyaredistinct.) Wecallthisthenon-resonancecondition. Furthermore,for DEFORMATIONSOFFUCHSIANEQUATIONSAND LOGARITHMICCONNECTIONS 7 any 1 ≤ k < m there exists no choice of k-tuples of eigenvalues µj ,...,µj l1j lkj atallsingularpointssuchthat m k µj ∈ Z. j=0 r=1 lrj P P Remark1.5. — The second condition implies that any logarithmic integrable connection (E,D) with these residues is stable in the usual sense: any D-invariant subbundle of E has slope smaller than E. Indeed, by the residue theorem for such a logarithmic connection there exist no non-trivial D-invariant subbundles at all. Stability is needed at two instances: for the definition of the moduli space M of integrable connections containing DL (see Remark 1.6), and to be able to apply non-Abelian Hodge theory in Section3. Weareinterested inthefollowingtwo numbers: (i). thedimensioneofthespaceEofdeformationsofthepolynomialsin(2) so that all residues of the associated integrable connection DL remain in thesameconjugacyclass (ii). the dimension c of the moduli space M of S-equivalence classes of (semi-)stable integrable connections (E,D) logarithmic in P over a vector bundle E of degree d = (1 − n)m(m − 1)/2, with fixed con- jugacyclassesofall itsresidues. Remark1.6. — For the definition of the moduli space M, see for example Sections 6-8 of [BB04]. An alternative way would be to define it geometri- cally as a symplectic leaf of the coarse moduli scheme of stable logarithmic connections (with arbitrary residues) constructed in Theorem 3.5 of [Nit93]. On the other hand, the space E is well-known to be an affine space, see e.g. [Inc26]. It is immediate that e ≤ m, for the space of the Fuchsian deformations (i) is contained in the space of integrable connections (ii) having the right mon- odromy, and two integrable connections induced by different Fuchsian equations cannot be gauge-equivalent,so this inclusion map is injective. In short, we will call deformations leaving invariant the conjugacy classes of all the residues locally isomonodromic. Using Fuchs’ condition, the number e was computedbyForsythin[For02],pp. 127-128. Intheintroductionofhisbook [Kat96], N. Katzcomputedc, andnoticed that (8) c = 2e. In fact, bothsidesofthisequationturnout tobe (9) 2−2m2 +m(m−1)(n+1). 8 SZILA´RDSZABO´,REŃYIINSTITUTEOFMATHEMATICS,BUDAPEST He also asked whether a geometric reason underlies this equality, more pre- cisely, whether a weight 1 Hodge structure can be found on the tangent to themodulispaceofintegrableconnections,whose(1,0)-partwouldgivepre- ciselythelocallyisomonodromicdeformationsofFuchsianequations. Wewilldefinesuch aHodgestructureinSection 2: Theorem 1.7. — Let DL be an integrable connection (4)-(5) induced by a Fuchsian equation (2) satisfying Condition 1.4. Then there exists a natural weight 1 Hodge structure on the tangent at DL to the moduli space M of integrableconnections T M = H1,0 ⊕H0,1 DL with the property that its part of type (1,0) is the tangent of the space E of locallyisomonodromicdeformationsoftheFuchsianequation: T E = H1,0. DL TheHodgestructurecomesfromahypercohomologylongexactsequence, and is well-defined on the tangent space at all elements of the moduli space. A similarexact sequencealready appears in Proposition4.1 of[Nit93]. InSections 3 and4, weshow: Theorem 1.8. — TheHitchinmaprestrictedtoEisabijection. Furthermore, E is an algebraic subvariety which is Lagrangian with respect to the natural holomorphicsymplecticstructureof thedeRham modulispace. For precise definitions, see Section 3. The Lagrangian property is also provedindependentlybyJ. Aidan[Aid07]. Finally, in Corollary 4.3 we use a result of Ohtsuki to determine the exact number of apparent singularities of the Fuchsian equation associated to a generic logarithmic connection, and deduce that M is foliated in Lagrangian subspaces: Theorem 1.9. — The smallest number N such that any logarithmicconnection (E,D) ∈ M corresponds to a Fuchsianequation with at most N apparent singularitiesis equal to e. The subspaces of M consisting of connections that can be realised by a Fuchsianequation with fixed locus of apparent sin- gularitiesdefineaLagrangianfoliationofM. ThefirststatementofthistheoremisprovedindependentlybyB.Dubrovin and M.Mazzocco in [DM07]. DEFORMATIONSOFFUCHSIANEQUATIONSAND LOGARITHMICCONNECTIONS 9 2. The Hodgestructure 2.1. Construction. — Let (E,D) be an arbitrary element of M. Let us denote by End (E) the sheaf of locally isomonodromic endomorphisms, iso which are by definition theendomorphismsϕ whose valueϕ(p ) at p lies in j j theadjointorbitoftheresidueoftheconnectionintheLiealgebragl(m). By Condition 1.4, this residue is regular diagonal in a suitable trivialisation; the locally isomonodromic endomorphisms are the ones whose value at p is an j off-diagonal matrix in this basis. The infinitesimal deformations of the integrable connection D (without changing the eigenvalues of the residues) are thendescribedbythefirsthypercohomologyH1(D)ofthetwo-termcomplex (10) End(E) −→D Ω1(P)⊗End (E) iso (seeSection 12of[Biq97]). Denoting by Hi the i-th hypercohomology of a complex, the hypercoho- mologylongexact sequencefor(10)reads 0 −→ H0(D) −→ H0(End(E)) −H−−0(−D→) H0(Ω1(P)⊗End (E)) −→ iso −→ H1(D) −→ H1(End(E)) −H−−1(−D→) H1(Ω1(P)⊗End (E)) −→ iso (11) −→ H2(D) −→ 0. ThemapsHi(D)areinducedbyDonthecorrespondingcohomologyspaces. Setting (12) C = coker(H0(D)) (13) K = ker(H1(D)), there follows a short exact sequence for the space of infinitesimal deformations: 0 −→ C −→ H1(D) −→ K −→ 0. The term C roughly corresponds to infinitesimal modifications of the (1,0)- part of the integrable connection while keeping the holomorphic structure fixed, whereas the term K corresponds to infinitesimal modifications of the holomorphicstructure. Lemma2.1. — WehaveK∨ ∼= C. Proof. — Let us first compute the dual of complex (10). The dual of the sheafEnd(E)isclearlyEnd(E)itself,sotheelementofdegree1inthedualis Ω1⊗End(E). AsEnd (E)isthesheafofendomorphismsofEvanishingon iso 10 SZILA´RDSZABO´,REŃYIINSTITUTEOFMATHEMATICS,BUDAPEST thediagonal in a local diagonalisingtrivialisationofD near thesingularities, andthedualofthevanishingconditionishavingasimplepole,itfollowsthat theelementHEnd(E)ofdegree0inthedualcomplexfitsintotheshortexact sequence 0 −→ HEnd(E) −→ End(E) −→ Im −→ 0, p Mp∈P where Im stands for the skyscraper sheaf supported at p, with stalk equal to p theoff-diagonalpartofEnd(E) inadiagonalisingtrivialisation,andthemap p toIm isevaluationfollowedbyprojectiontotheoff-diagonalpart. Thisstalk p is just the image at p of the adjoint action of the residue of the connection on endomorphisms, so it is intrinsically defined. We will call HEnd(E) the Hecke transform of End(E) along the image of the residue. One also checks immediatelythatthetermsofdegree1of(10)anditsdualfittoasimilarshort exact sequence: 0 −→ Ω1 ⊗End(E) −→ Ω1(P)⊗End (E) −→ Im −→ 0, iso p Mp∈P where themap to Im is taking residueat p followedby projection to theim- p ageoftheresidueofD. Inotherwords,thedualofthecomplex(10)islinked to(10)byaHecketransformationalongtheimageoftheresidue. Thehyper- cohomology long exact sequence (11) is the long exact sequence associated toashortexactsequenceoftwo-termcomplexes. Allnon-zerotermsofthese two-termcomplexesfor(10)anditsdualarerelatedbyHecketransformation alongtheimageoftheresidue. AstheresidueofDactsbyanisomorphismon itsimage,itfollowsthattheconnectingmorphismofthelongexactsequence of the skyscraper sheaves is an isomorphism. As the hypercohomology long exact sequenceisfunctorial,wededucethat thespaces coker(H0(HEnd(E)) −H−0−(−D→) H0(Ω1 ⊗End(E))) and coker(H0(End(E)) −H−−0(−D→) H0(Ω1(P)⊗End (E))) iso areisomorphic. Weconcludeby Verdierduality. Hence, Lemma 2.1 together with (11) exhibits H1(D) as the extension of twovectorspaces ofthesamedimension: 0 −→ C −→ H1(D) −→ K −→ 0.