Isomonodromic tau function on the space of admissible covers

ISOMONODROMIC T A U FUNCTION ON THE SP A CE O F ADMISSIBLE COVERS A. KOK OTOV 1 , D. K OROTKIN 1 , AND P . ZOGRAF 2 Abstra ct. The isomonodromic t au function of the F uchsia n differential equations associated to F rob enius struct ures on Hurwitz spaces can b e view ed as a section of a line bu n dle on the space of admissible cov ers. W e study th e asymptotic b eha vior of the tau function near the b oun dary of this space and compute its divisor. This yields an explicit form ula for the pullback of the Ho dge class to the sp ace of admissible cov ers in terms of the c lasses of compactification divisors. Contents 1. In tro duction 1 2. Isomono dromic tau fu n ction 2 2.1. Hurwitz spaces 2 2.2. Definition of the tau function 3 2.3. T au function as a section of a line bu ndle 5 3. Divisor of the tau f unction 8 3.1. The space of admissible co vers 8 3.2. Asymptotics of the tau function near the b oundary 9 3.3. Relations b et w een the divisors 12 References 13 1. Introduction The space of admissib le co vers is a natural compactificaion of th e Hurwitz space of smo oth branc hed co vers of the complex pr o jectiv e lin e P 1 , or, equiv- alen tly , meromorphic fu nctions on complex algebraic curve s, of giv en degree and gen us. This space w as first in tro duced by J. Harr is and D. Mumford and app eared to b e quite useful in computin g th e Ko d aira dimension of th e mo duli space of stable curves [6]. Lately this space has attracted a ma jor atten tion, mainly in connection with Gromo v-Witten theory , quan tu m co- homology , Hu rwitz num b ers, Ho dge integ rals, etc. (The literature on this sub ject is abund an t, and it is not p ossible to giv e ev en a very br ief review here.) On the other hand, Hurwitz sp aces app ear natur ally in relationship with the Riemann-Hilb ert problem, and carry a natural F rob enius s tr ucture [3]. The tau f unction for th e corresp ondin g isomono dr omic deformations can b e AK and DK w ere supp orted in part by N SERC; DK was also supp orted by F QRN T and CURC; PZ was partially supp orted by the RFBR gran t 08-01-00379-a and by the Presiden t of Russian F ederation grant N Sh-2460.2008. 1. 1 2 A. KOK OTOV 1 , D. KOR O TKIN 1 , AND P . ZOGRAF 2 written explicitly in terms of the theta function and the prime form on the co vering complex cur ve [8]. In this pap er we study the asymp totic b ehavio r of the isomono dromic tau function near the b oundary of the Hurw itz sp ace giv en b y nod al admissi- ble co v ers, and explicitly compu te its divisor. More p recisely , a p o w er of the tau f unction corrected by a p o wer the V andermonde determinan t of the critical v alues of the br an ched co v er d escends to a holomorphic section of (the pu llb ac k of ) the Hodge bundle on the Hu rwitz space. Moreo ver this section extends to a meromorph ic section of the Ho d ge bu ndle on the com- pactification of the Hurw itz space by admissible co v ers. This allo w s us to express (the pullbac k of ) the Ho dge class on th e sp ace of admiss ib le cov ers as a linear com b ination of b oun dary divisors (in small genera this also giv es a n on-trivial relation b et w een the b oundary divisors). The pap er is organized as follo ws. In Section 2 w e defin e the isomon- o dromic tau function, give an explicit formula f or it (Theorem 1), study its transformation prop erties and in terprete it as a holomorphic sectio n of a line bu ndle on the Hurwitz space. Section 3 conta ins the main results of the p ap er: an asymptotic form ula for the tau fu nction near the b oundary of the space of admissible cov ers (Theorem 2), and a f orm ula for the Ho dge class in terms of the classes of b ound ary divisors (Theorem 3). The sp ecial cases of the latter include a form ula of Cornalba-Harris for the Ho d ge class on the hyp erelliptic lo cus [2], and a relation of Lando-Zv onkine b et w een the compactificatio n divisors in Hurwitz spaces of gen us 0 branc hed co v ers [10]. 2. Isomonodromic t au f unction 2.1. Hurwitz spaces. Let C b e a smo oth complex algebraic curve of genus g , and let f b e a mer omorp hic function on C of degree d > 0. W e can think of f as a h olomorhic b ranc hed co v er f : C → P 1 o v er the pro jectiv e line P 1 . W e call a mer omorp hic fu nction (or a b ranc hed co v er) ge neric if it h as only simple critical v alues (branc h p oin ts). F or a generic f the n umber of b r anc h p oin ts is n = 2 g + 2 d − 2, w e d enote them by z 1 , . . . , z n ∈ P 1 and alwa ys assume th at they are or der e d . Tw o m eromorphic fu nctions f 1 : C 1 → P 1 and f 2 : C 2 → P 1 are called str ongly e quivalent (or simply e qu ivalent ), if there exists an isomorphism h : C 1 → C 2 suc h that f 1 = f 2 ◦ h , and we akly e quivalent , if there exist isomorphisms h : C 1 → C 2 and γ : P 1 → P 1 suc h that γ ◦ f 1 = f 2 ◦ h . In addition to that we will also consider an equiv alence relation for mero- morhic fu nctions on T orelli mark ed cur ves. A T or el li marking is a c hoice of symplectic basis α = { a i , b i } g i =1 in the fir st homology group H 1 ( C ) of C . A curve C together with a symplectic basis α w ill b e denoted b y C α . W e sa y that t w o meromorphic functions on T orelli mark ed curve s are T or el li e quivalent , if for T orelli m ark ed cur v es C α 1 1 , C α 2 2 there exist an isomorphism h : C 1 → C 2 suc h that f 1 = f 2 ◦ h and h ∗ ( α 1 ) = α 2 elemen t w ise. F or an y fixed g ≥ 0 and d > 0 consid er the space of all generic mero- morphic fu nctions of d egree d on all smo oth gen us g curve s. Denote by H g ,d , ˜ H g ,d , ˇ H g ,d the mo duli spaces (called Hurwitz sp ac es ) defin ed b y the w eak, strong and T orelli equiv alence relations resp ectiv ely (the latter re- quires the curv es to b e T orelli mark ed). All three spaces are non-compact T AU FUNCTION AND ADM ISSIBLE COVERS 3 complex manifolds. The last tw o s paces ha v e d imension n = 2 g + 2 d − 2 and the b r anc h p oin ts z 1 , . . . , z n pro vide a system of lo cal co ordinates for b oth of them. The group P S L (2 , C ) acts freely on ˜ H g ,d and ˇ H g ,d b y lin- ear fractional t ransformations: for γ =  a b c d  ∈ P S L (2 , C ) we h a v e γ ◦ f = af + b cf + d , so that, in particular, H g ,d = ˜ H g ,d /P S L (2 , C ). In addition, the symplectic group S p (2 g , Z ) acts on ˇ H g ,d b y c hanging T orelli marking, and ˜ H g ,d = ˇ H g ,d /S p (2 g, Z ). The actions of P S L (2 , C ) and S p (2 g , Z ) on ˇ H g ,d clearly comm u te. In the sequel we will also deal with meromorph ic functions (branched co vers) that ha ve one fixed v alue, either regular at z = ∞ , or degenerate critical of t yp e µ = [ m 1 , . . . , m r ] at any z ∈ P 1 ( m i > 0 are the ramification degrees of the p oin ts in f − 1 ( z ) , m 1 + · · · + m r = d ), with all other branch p oin ts b eing simple and fin ite (the num b er of these critical v alues is n ( µ ) = 2 g + d + r − 2). Th e Hu r witz spaces of such fun ctions defined mo d ulo the w eak (while k eeping z fixed), strong and T orelli equiv alence relations w e denote b y H g ,d ( z , µ ) , ˜ H g ,d ( z , µ ) and ˇ H g ,d ( z , µ ) resp ectiv ely . The d imension of the last t w o ones is n ( µ ) = 2 g + d + r − 2, and the s imple b r anc h p oints z 1 , . . . , z n ( µ ) serv e as lo cal co ordinates for them as we ll. In particular, ˜ H g ,d ( ∞ , 1 d ) and ˇ H g ,d ( ∞ , 1 d ) are op en dense su bsets of the Hurwitz sp aces ˜ H g ,d and ˇ H g ,d resp ectiv ely . 2.2. Definition of the tau function. F or a T orelli mark ed curv e C α , denote b y B ( x, y ) the Ber gman bidiffer ential , that is, the unique symmetric meromorphic b id ifferen tial on C × C with a quad r atic p ole of biresidu e 1 on the diagonal and zero a -p erio ds (the details on meromorphic bidifferent ials and the asso ciated pro jectiv e connections can b e found , e.g., in [4] or [13]). The b -p erio ds of the Bergman bidifferential B ( x, y ) ω i = Z b i B ( · , y ) dy (2.1) are the normalize d holomo rphic differ entials on C α , that is, Z a j ω i = δ ij , Z b j ω i = Ω ij , i, j = 1 , . . . , g , (2.2) where the matrix Ω = { Ω ij } g i,j =1 is the p erio d matrix of C α . In terms of local parameters ζ ( x ) , ζ ( y ) near the diagonal { x = y } ∈ C × C , the bidifferen tial B ( x, y ) has the follo w in g Lauren t series expasion in ζ ( y ) at the p oin t ζ ( x ) B ( x, y ) =  1 ( ζ ( x ) − ζ ( y )) 2 + S B ( ζ ( x )) 6 + O (( ζ ( x ) − ζ ( y )) 2 )  dζ ( x ) dζ ( y ) , (2.3) where S B is a pr o j ectiv e connection on C called the Ber gman pr oje ctive c onne ction . The latter means that S B transforms under the c hange ζ = ζ ( w ) of the local p arameter b y the rule S B ( w ) = S B ( ζ ( w )) ζ ′ ( w ) 2 + S ζ , wh ere S ζ = ζ ′′′ ζ ′ − 3 2  ζ ′′ ζ ′  2 is the Schwarzian derivative of ζ ( w ) with resp ect to w . 4 A. KOK OTOV 1 , D. KOR O TKIN 1 , AND P . ZOGRAF 2 No w consider the Sch warzia n der iv ativ e S f = f ′′′ f ′ − 3 2  f ′′ f ′  2 of a mero- morphic fu n ction f : C → P 1 with resp ect to a lo cal parameter ζ on C . This is a meromorphic pro jectiv e conn ection on C , so that th e differen ce S B − S f is a meromorp hic quadr atic differentia l. T ak e th e tr ivial line bundle on the Hurwitz sp ace ˇ H g ,d ( z , µ ) and consider the connection d B = d + 4 n ( µ ) X i =1  Res x i S B − S f d f  dz i , (2.4) where the su m is tak en o ver all simp le fin ite branch p oin ts z i of f , and x i ∈ C are the corresp onding critical p oin ts. Rauc h’s formulas imp ly that this connection is flat (cf. [8]). The tau fu nction τ ( C α , f ) is lo cally defined as a h orizon tal (co v arian t constan t) section of the tr ivial line b undle on ˇ H g ,d ( z , µ ) with resp ect to d B : 1 ∂ log τ ( C α , f ) ∂ z i = − 4 Res x i S B − S f d f , i = 1 , . . . , n . (2.5) Let us no w recall an explicit form ula for the tau function τ ( C α , f ) de- riv ed in [8]. T ak e a nonsin gular o dd theta c haracteristic δ and consider the corresp onding th eta f unction θ [ δ ]( v ; Ω), where v = ( v 1 , . . . , v g ) ∈ C g . Put ω δ = g X i =1 ∂ θ [ δ ] ∂ v i (0; Ω) ω i . All zero es of the h olomorph ic 1-differentia l ω δ ha v e ev en multipliciti es, and √ ω δ is a w ell-defined holomorphic spinor on C . F ollo wing F a y [4], consider the pr ime form 2 E ( x, y ) = θ [ δ ] R y x ω 1 , . . . , R y x ω g ; Ω  √ ω δ ( x ) √ ω δ ( y ) . (2.6) T o make the in tegrals uniqu ely defined, w e fix 2 g simple closed lo ops in the homology classes a i , b i that cu t C int o a connected domain, and pick the in tegration p aths that do n ot in tersect the cuts. Th e sign of th e square ro ot is chosen so that E ( x, y ) = ζ ( y ) − ζ ( x ) √ dζ ( x ) √ dζ ( y ) (1 + O (( ζ ( y ) − ζ ( x )) 2 )) as y → x , where ζ is a local parameter suc h th at dζ = ω δ . W e in tro duce lo cal coord inates on C that w e call natur al (or distingui she d ) with resp ect to f . Consider the divisor ( d f ) = P k d k p k , p k ∈ C , d k ∈ Z , d k 6 = 0 , of the meromorp hic differential d f . W e take ζ = f ( x ) as a lo cal co ordinate on C − S k p k , and ζ k = ( ( f ( x ) − f ( p k )) 1 d k +1 if d k > 0, f ( x ) 1 d k +1 if d k < 0, (2.7) 1 This tau function is the 24th p ow er of the Bergman tau fun ct ion studied in [8 ] and the -48th p o w er of the isomono dromic tau function of the F rob enius manifol d structure on Hurwitz space introduced by Du b ro vin [3]. Our present defin ition is more appropriate in the context of admissible cov ers. 2 The p rime form E ( x, y ) is the canonical section of th e line bundle on C × C asso ciated with the diagonal divisor { x = y } ⊂ C × C . T AU FUNCTION AND ADM ISSIBLE COVERS 5 near p k ∈ C . In terms of these co ordinates we ha v e E ( x, y ) = E ( ζ ( x ) ,ζ ( y )) √ dζ ( x ) √ dζ ( y ) , and w e d efi ne E ( ζ , p k ) = lim y → p k E ( ζ ( x ) , ζ ( y )) s dζ k dζ ( y ) , E ( p k , p l ) = lim x → p k y → p l E ( ζ ( x ) , ζ ( y )) s dζ k dζ ( x ) s dζ l dζ ( y ) . Let A x b e the Ab el map w ith the basep oin t x , and let K x = ( K x 1 , . . . , K x g ) b e the v ector of Riemann constant s K x i = 1 2 + 1 2 Ω ii − X j 6 = i Z a i  ω i ( y ) Z y x ω j  dy (2.8) (as ab o v e, we assume that the in tegration p aths d o n ot intersect the cuts on C ). Th en we hav e A x (( d f )) + 2 K x = Ω Z + Z ′ for some Z, Z ′ ∈ Z g . No w put τ ( C α , f ) =   P g i =1 ω i ( ζ ) ∂ ∂ v i  g θ ( v ; Ω)    v = K ζ  16 e 4 π √ − 1 h Ω Z +4 K ζ ,Z i W ( ζ ) 16 Q k 0, ǫ 1 d k +1 f ( x ) 1 d k +1 if d k < 0. T AU FUNCTION AND ADM ISSIBLE COVERS 7 Moreo ver, d k = 1 f or all zero es of d f , and and d k = − m i − 1 , i = 1 , . . . , r , for the p oles of d f . S u bstituting these parameters ζ ǫ k in to (2.9), we get Eq. (2.14).  Lemma 3. On the Hurwitz sp ac e ˇ H g ,d ( ∞ , µ ) we have the identity n ( µ ) X i =1 z i Res x i S B − S f d f = − 3 n ( µ ) 4 + d 2 − 1 2 r X i =1 1 m i . (2.15) Pr o of. The homogeneit y prop ert y (2.14) implies that n ( µ ) X i =1 z i ∂ ∂ z i log τ ( C α , f ) = 3 n ( µ ) − 2 d + 2 r X i =1 1 m i . This immediately yields (2.15) due to the defin ition (2 .5 ) of the tau function.  The b eha vior of the tau function under the change of T orelli marking of C is d escrib ed in the follo wing lemma: Lemma 4. L et two c anonic al b ases α = { a i , b i } g i =1 and α ′ = { a ′ i , b ′ i } g i =1 in H 1 ( C ) b e r e late d by α ′ = σ α , wher e σ =  D C B A  ∈ S p (2 g, Z ) . (2.16) Then τ ( C α ′ , f ) τ ( C α , f ) = d et( C Ω + D ) 24 . (2.17) wher e Ω is the p erio d matrix of the T or el li marke d Riemann sutfac e C α . Pr o of. T o establish this transformation prop ert y , we use the explicit form ula (2.9). According to Lemma 6 of [9], wh en d f has at least one simple zero one can alw ays c ho ose the cut system on C in suc h a wa y that Z = Z ′ = 0 in (2.9). The change of b asis α ′ = σ α results in the follo wing transformation of the prime form E ( x, y ): E ′ ( x, y ) = E ( x, y ) e √ − 1 π v ( C Ω+ D ) − 1 C v t (2.18) (cf. [5], Eq. (1.20 )); here v = ( R y x ω 1 , . . . , R y x ω g ). F or the expr ession C ( x ) = 1 W ( x ) g X i =1 ω i ( x ) ∂ ∂ v i ! g θ ( v ; Ω)      v = K x it is sho wn in [5 ], E q . (1.23), that C ′ ( x ) = δ (det( C Ω + D )) 3 / 2 e √ − 1 π K x ( C Ω+ D ) − 1 C ( K x ) t C ( x ) , (2.19) where δ is a ro ot of unit y of eighth d egree, and K x is the ve ctor of Rie- mann constan ts (2.8). Subs tituting these formulae in to (2.9), w e obtain the statemen t the lemma.  Denote b y λ the Hod ge line bun dle on the Hurwitz space H g ,d ; the fib er of λ o v er th e p oint represented by a pair ( C , f ) is isomorp hic to det Ω 1 C = ∧ g Ω 1 C , where Ω 1 C is the space of holomorphic 1-forms (ab elian different ials) on C . The line bund le λ has a lo cal holomorphic section giv en by ω 1 ∧ · · · ∧ ω g , 8 A. KOK OTOV 1 , D. KOR O TKIN 1 , AND P . ZOGRAF 2 where ω 1 , . . . , ω g is the basis of normalized ab elian d ifferentials on a T orelli mark ed cu rv e C α . Under the c h ange of markin g α ′ = σ α with σ ∈ S p (2 g , Z ) giv en by (2.16), this section transforms by th e rule ω ′ 1 ∧ · · · ∧ ω ′ g = d et( C Ω + D ) − 1 ω 1 ∧ · · · ∧ ω g . Com bining this with Lemmas 1 and 4 w e obtain Lemma 5. The function ˇ η = τ n − 1 V − 6 on ˇ H g ,d desc ends to a nowher e vanishing holomorph ic se ction η of the line bu nd le λ 24( n − 1) on H g ,d . 3. Divisor of the t au fun ction 3.1. The space of admissible co vers. The sp ace of admissible co v ers H g ,d is a natural compactification of the Hur witz space H g ,d that was in tro duced in [6 ]. An admissible c over is a degree d regular map f : C → R of a connected gen us g no dal curv e C onto a rational n o d al cu rv e R that is simply br anc hed ov er n = 2 g + 2 d − 2 d istin ct p oin ts on the s mo oth part of R and maps n o d es to no des with the same r amification indices for the t wo branc hes at eac h no de. The sp ace of (we ak equiv alence classes of ) admiss ible co vers H g ,d has r elativ ely simple lo cal structure, though it is not a normal algebraic v ariet y and therefore not an orbifold. Ho wev er, a normalization of H g ,d is smo oth, cf. [1], [7]. The space H g ,d comes with t w o n atural morph isms. Th e first one is the br anch map β : H g ,d → M 0 ,n , (3.1) that extends the n atur al co v ering H g ,d → M 0 ,n that maps f to the configu- ration ( z 1 , . . . , z n ) of its ordered branc h p oin ts considered up to the diagonal action of P S L (2 , C ). T he second one is the for getful map π : H g ,d → M g , (3.2) that extends the natur al p ro jection H g ,d → M g sending the equ iv alence class of the branc hed co v er f : C → P 1 to the isomorphism class of the co vering curv e C . The description of the b ou n dary H g ,d − H g ,d is straigh tforw ard. Since w e are interested only in the b ound ary d ivisors , it is sufficien t to consider admissible co vers o v er the base R consisting of t w o irred ucible comp on ents R 1 and R 2 in tersecting at a single no d e p . The ramification t yp e of the co ver f : C → R o ve r the no de p we will denote b y µ = [ m 1 , . . . , m r ], wher e r is th e num b er of no des of C an d m i is the ramification in d ex at the i - th n o de, m 1 + · · · + m r = d . Let us denote b y k and n − k the n umber of br anc h p oints on R 1 \ { p } and R 2 \ { p } resp ectiv ely; we assume that 2 ≤ k ≤ g + d − 1. Let D k b e the divisor in M 0 ,n parametrizing r educible curv es with comp onen ts of typ e (0 , k + 1) and (0 , n − k + 1), and denote by ∆ k = β − 1 ( D k ) the preimage of D k in H g ,d with resp ect to the b ranc h map (3.1). The b oun dary divisor ∆ k is the u nion of divisors ∆ ( k ) µ o v er the s et of all p ossible ramification t yp es µ ov er the no d e p ∈ R . Note th at the divisors ∆ ( k ) µ are generally reducible ev en f or a fi xed partition of b ranc h p oin ts on R and a fixed µ . The lo cal structure of H g ,d near the divisors ∆ ( k ) µ w as describ ed in [7]: in the direction transv ersal to ∆ ( k ) µ with µ = [ m 1 , . . . , m r ] , it lo oks lik e the T AU FUNCTION AND ADM ISSIBLE COVERS 9 (singular) curv e ζ m 1 1 = · · · = ζ m r r near the origin in C r . Therefore, for an y irredu cible comp onent of ∆ ( k ) µ there are m 1 ...m r m (where m = l . c . m . { m 1 , . . . , m r } is the least common multiple of m 1 , . . . , m r ) br anc hes of H g ,d in tersecting at it, whereas ev ery suc h br anc h is an m -fold co v er of a n eigh b orho o d of D k in M 0 ,n ramified o v er D k with ramification in dex m . 3.2. Asymptotics of the tau function near the b oundary . Let f : C → P 1 b e a holomorphic branc hed cov er with only simp le br an ch p oin ts z 1 , . . . , z n ∈ P 1 , n = 2 g + 2 d − 2, and let γ i 7→ s i b e the m on o d rom y reprsen- tation, where γ i are non-intersect ing simple loops ab out z i with s ome b ase p oin t z 0 , and s 1 , . . . , s n are transp ositions in the symmetric group S d of d elemen ts such that s 1 . . . s n = 1. Denote b y f ǫ : C ǫ → P 1 the branc hed co ver with branc h p oin ts ǫz 1 , . . . , ǫ z k , z k +1 , . . . , z n ∈ P 1 and the same mon- o drom y as f , where we assume that z i 6 = ∞ for i = 1 , . . . , k and z i 6 = 0 for i = k + 1 , . . . , n (2 ≤ k ≤ g + d − 1 as ab o v e). At the limit ǫ → 0 the m ap f approac h es an admissible co v er f 0 : C 0 → R , wh ere C 0 is a gen u s g no dal curv e, and R = P 1 (1) ∪ P 1 (2) / {∞ , 0 } is the t w o comp onent rational cur v e with one no de p = {∞ , 0 } ( ∞ ∈ P 1 (1) is ident ified with 0 ∈ P 1 (2) ). Th e curv e C 0 splits int o t w o (not necessarily connected) comp onen ts C (1) 0 and C (2) 0 lying o v er P 1 (1) and P 1 (2) resp ectiv ely . Th e restriction f (1) 0 : C (1) 0 → P 1 (1) (resp. f (2)) 0 : C (2) 0 → P 1 (2) ) is simply branc hed ov er z 1 , . . . , z k ∈ P 1 (1) (resp. o ver z k +1 , . . . , z n ∈ P 1 (2) ). 4 Moreo ver, C (1) 0 (resp. C (2) 0 ) is connected if and only if the group generated by s 1 , . . . , s k (resp. b y s k +1 , . . . , s n ) acts transitiv ely on the set of d elements. The ramification t yp e o v er the n o d e p coincides with the t yp e of the p ermutati on s 1 . . . s k ∈ S d and, as ab o ve, w e denote it b y µ = [ m 1 , . . . , m r ]. W e will need a canonical homology basis for the f amily of curves C ǫ that is compatible with th e limiting no dal curve C 0 . Denote b y ℓ the simple lo op on P 1 that shrinks to the nod e as ǫ → 0, and b y ℓ 1 , . . . , ℓ r its preimage s in C ǫ (w e omit the dep end ence of these lo ops on the p arameter ǫ ). Cho ose some canonical bases α 1 and α 2 on the curves C (1) 0 and C (2) 0 resp ectiv ely; w e can p ull them bac k to C ǫ in su c h a wa y , that they do n ot int ersect the lo ops ℓ 1 , . . . , ℓ r . Denote b y [ ℓ i ] ∈ H 1 ( C ǫ ) the homology class of the lo op ℓ i , and pu t q = rank { [ ℓ 1 ] , . . . , [ ℓ r ] } , that is, the rank of the linear span of the classes [ ℓ 1 ] , . . . , [ ℓ r ] in H 1 ( C ǫ ). An elemen tary top ologica l consideration sho ws that g = g 1 + g 2 + q , w h ere g 1 (resp. g 2 ) is the sum of genera of the connected comp onen ts of C (1) 0 (resp. C (2) 0 ). Without loss of generalit y , w e can assum e that [ ℓ 1 ] , . . . , [ ℓ q ] are linear ind ep endent, and ad d ℓ 1 , . . . , ℓ q to the un ion of α 1 and α 2 as a -cycles, wh ile the corresp on d ing b -cycles can b e c hosen as lifts of paths connecting branc h p oints in differen t comp onents of P 1 − ℓ . W e denote th us obtained b asis on C ǫ b y α . The main tec hnical result of this pap er is 4 This is b ecause the functions f ǫ and ǫ − 1 f ǫ represent t he same p oint in the Hurwitz space H g,n . 10 A. KOK OTOV 1 , D. KOR O TKIN 1 , AND P . ZOGRAF 2 Theorem 2. The isomono dr omic tau function has the asympto tics τ ( C α ǫ , f ǫ ) = ǫ 3 k − 2 d +2 P r i =1 1 /m i τ ( C (1) ,α 1 0 , f (1) 0 ) τ ( C (2) ,α 2 0 , f (2) 0 ) (1 + o (1)) , (3.3) as ǫ → 0 , wher e the tau fu nction for a disc onne cte d br anche d c over is u n- dersto o d as the pr o duct of tau f unctions for its c onne cte d c omp onents. T o pro v e th is theorem w e will need an auxiliary lemma. T ogether with f ǫ : C α ǫ → P 1 consider the branched co v er f ǫ /ǫ : C α ǫ → P 1 with branc h p oin ts z 1 , . . . , z k , ǫ − 1 z k +1 , . . . , ǫ − 1 z n ∈ P 1 and th e same mono dromy as f . Denote the Bergman bid ifferen tials on the T orelli marked cur v es C α ǫ , C (1) ,α 1 0 and C (2) ,α 2 0 b y B ǫ , B (1) and B (2) resp ectiv ely . W e w an t to see w h at happ ens at the limit ǫ → 0. W e can alwa ys assume that | z i | < 1 /δ , i = 1 , . . . , k , and | z i | > δ, i = k +1 , . . . , n , for some δ ∈ (0 , 1). F or small enough ǫ co nsider t w o op en subsets D (1) ǫ = { x ∈ C ǫ | | f ǫ ( x ) | < ǫ/δ } and D (2) ǫ = { x ∈ C ǫ | | f ǫ ( x ) | > δ } of the curv e C ǫ . Note that the complemen t C ǫ − D (1) ǫ ∪ D (2) ǫ is the u n ion of r disjoin t cylinders aroun d the lo op s ℓ 1 , . . . , ℓ r . F or eac h ǫ the su bset D (1) ǫ (resp. D (2) ǫ ) is natur ally isomorphic to the su bset D (1) 0 = n x ∈ C (1) 0    | f (1) 0 ( x ) | < 1 δ o (resp. D (2) 0 = n x ∈ C (2) 0    | f (2) 0 ( x ) | > δ o ). As ǫ → 0, w e ha v e f ǫ ( x ) /ǫ → f (1) 0 ( x ) , x ∈ D (1) 0 and f ǫ ( x ) → f (2) 0 ( x ) , x ∈ D (2) 0 . Lemma 6. In the limit ǫ → 0 B ǫ ( x, y ) d f ǫ ( x ) d f ǫ ( y ) − → B (1) ( x, y ) d f (1) 0 ( x ) d f (1) 0 ( y ) , x, y ∈ D (1) 0 , and ǫ 2 B ǫ ( x, y ) d f ǫ ( x ) d f ǫ ( y ) − → B (2) ( x, y ) d f (2) 0 ( x ) d f (2) 0 ( y ) , x, y ∈ D (2) 0 uniformly on D (1) 0 and D (2) 0 whenever x 6 = y . Remark 2. Th is lemma extends [4 ], Corollary 3.8, that treats the p inc hing of a single non-separating lo op. Pr o of. According to our c hoice of the homology basis α on C ǫ , the integrals of B ǫ along a -cycles coming f r om C (1) 0 and C (2) 0 are iden tically 0. Moreo v er, the int egrals of B ǫ along the r v anishing cycles ℓ 1 , . . . , ℓ r tend to 0 as ǫ → 0. Therefore, r ep eating the argumen t of [4], Corollary 3.8, we see that the bidifferent ial B ǫ tends to B (1) on D (1) 0 and to B (2) 0 on D (2) 0 , as stated.  Denote by S B ǫ , S B (1) and S B (2) the pr o j ective connections corresp onding to the bidifferentials B ǫ , B (1) and B (2) resp ectiv ely . F rom the ab ov e lemma w e immediately get T AU FUNCTION AND ADM ISSIBLE COVERS 11 Corollary 1. The c o efficients of the Ber gman pr oje ctive c onne ction (2.4) have the f ol lowing asymtotics as ǫ → 0 : S B ǫ ( x ) − S f ǫ ( x ) d f ǫ ( x ) 2 − → S B (1) ( x ) − S f (1) 0 ( x ) d f (1) 0 ( x ) 2 , x ∈ D (1) 0 (3.4) ǫ 2 S B ǫ ( x ) − S f ǫ /ǫ ( x ) d f ǫ ( x ) 2 − → S B (2) ( x ) − S f (2) 0 ( x ) d f (2) 0 ( x ) 2 , x ∈ D (2) 0 . (3.5) Pr o of of The or em 2. Denote by x ǫ 1 , . . . , x ǫ n ∈ C ǫ the ramification p oints corresp onding to th e simple branch p oin ts ǫz 1 , . . . , ǫ z k , z k +1 , . . . , z n ∈ P 1 . By definition of τ ( C α ǫ , f ǫ ), cf. (2.5), we hav e ∂ ∂ ( ǫz i ) log τ ( C α ǫ , f ǫ ) = − 4 Res x ǫ i S ǫ B − S f ǫ d f ǫ , i = 1 , . . . , k , (3.6) ∂ ∂ z i log τ ( C α ǫ , f ǫ ) = − 4 Res x ǫ i S ǫ B − S f ǫ d f ǫ , i = k + 1 , . . . , n. (3.7) F rom (3.6) we see that for i = 1 , . . . , k ∂ ∂ z i log τ ( C α ǫ , f ǫ ) = − 4 Res x ǫ i S ǫ B − S f ǫ /ǫ d f ǫ /ǫ . No w C orollary 1 implies th at, as ǫ → 0, τ ( C α ǫ , f ǫ ) = c ( ǫ ) τ ( C (1) ,α 1 0 , f (1) 0 ) τ ( C (2) ,α 2 0 , f (2) 0 )(1 + o (1)) (3.8) where c ( ǫ ) is a fun ction of ǫ indep enden t of z 1 , . . . , z n . T o explicitly compute c ( ǫ ) we use Eq. (3.6): ǫ ∂ ∂ ǫ log τ ( C α ǫ , f ǫ ) = − 4 k X i =1 z i Res x ǫ i S ǫ B − S f ǫ d f ǫ . (3.9) F rom (3.4) we get lim ǫ → 0  ǫ ∂ ∂ ǫ log τ ( C α ǫ , f ǫ )  = − 4 k X i =1 z i Res x i S (1) B − S f (1) 0 d f (1) 0 , (3.10) where the righ t-hand side is ev aluated on the co v er f (1) 0 : C (1) 0 → P 1 (1) . Due to (2.15) we can r ewrite the righ t hand side of the last form ula in terms of k , d and the ramification type µ = [ m 1 , . . . , m r ] o v er the no de at ∞ ∈ P 1 (1) as follo ws: lim ǫ → 0  ǫ ∂ ∂ ǫ log τ ( C α ǫ , f ǫ )  = 3 k − 2 d + 2 r X i =1 1 m i . (3.11) Th us, c ( ǫ ) = ǫ 3 k − 2 d − 2 P r i =1 1 /m i , wh ic h yields (3.3). Remark 3. Asym p totic b eh avior of the tau function as ǫ → 0 can in prin - ciple b e deriv ed from Theorem 2.4.13 an d E q .(2.4.9) of [1 2], where it wa s describ ed in terms of traces of squares of the r esidues of the asso ciated F u c h- sian system in a rather general situation. How eve r, our appr oac h is more straigh tforw ard and suits b etter for the situation w e consider here. 12 A. KOK OTOV 1 , D. KOR O TKIN 1 , AND P . ZOGRAF 2 Corollary 2. The (mer omorp hic) se ction η of the line bu nd le λ 24( n − 1) on H g ,d (with λ b eing the pul lb ack of the Ho dge line bund le on M g ) has the fol lowing asymptotics as ǫ → 0 : η ( C α ǫ , f ǫ ) = ǫ 3 k ( n − k ) − 2( n − 1)( d − P r i =1 1 /m i ) × η ( C (1) ,α 1 0 , f (1) 0 ) η ( C (2) ,α 2 0 , f (2) 0 )(1 + o (1)) . (3.12) 3.3. Relations b et w een the divisors. Here w e discuss some explicit rela- tions b et w een the d ivisor classes in the r ational Picard group Pic( H g ,n ) ⊗ Q that follo w from the ab o v e analysis. Slight ly abus in g notation, w e use the same symb ols for line bu n dles and d ivisors on H g ,d as for their classes in Pic( H g ,n ) ⊗ Q . It will also b e conv enient to understand th e b oun dary divisors ∆ ( k ) µ in th e orbifold sense, that is, as the w eigh ted sums of their irreducible comp onen ts with weig hts 1 | Aut( f ) | , where Au t( f ) is the automorphism group of a generic ad m issible co v er f parametrized b y the irreducible comp onen t; suc h a “wei gh ted” divisor we den ote by δ ( k ) µ . Then we hav e Theorem 3. F or the Ho dge class λ ∈ Pic( H g ,n ) ⊗ Q the fol lowing formula holds: λ = g + d − 1 X k =2 X µ =[ m 1 ,...,m r ] r Y i =1 m i k ( n − k ) 8( n − 1) − 1 12 d − r X i =1 1 m i !! δ ( k ) µ . (3.13) Pr o of. As it was mentioned in the end of Section 3.1, w e can tak e ǫ 1 /m , m = l . c . m . { m 1 , . . . , m r } , as a transv ersal lo cal parameter on eac h of the m 1 ...m r m branc hes of H g ,n near eac h irreducible comp onen t of ∆ ( k ) µ . Plugging it into (3.12) and taking the action of Aut( f ) in to acco unt, we pr o ve the theorem.  W e fin ish with sev eral commen ts concerning the sp ecial cases of the ab ov e theorem. F or d = 2, Eq. (3.13) tak es the form λ = [( g +1) / 2] X i =1 i ( g + 1 − i ) 4 g + 2 δ (2 i ) [1 2 ] + [ g / 2] X j = 1 j ( g − j ) 2 g + 1 δ (2 j +1) [2] . (3.14) This we ll-kno wn formula expresses th e Ho dge class on the closure of the hy- p erelliptic lo cus in M g in terms of the b oundary strata (cf. [2], Prop osition (4.7)). The only d ifference is that our coefficient at δ (2) [1 2 ] is t wice that of [2]. This is b ecause the divisor δ (2) [1 2 ] parametrizes admissible co v ers conta ining an irreducible genus 0 comp onent with t w o no des and t wo critical p oints that has a n on-trivial automorphism group of order 2 and gets con tracted un der the forge tful map π : H g , 2 → M g . (In other w ord s , w e ha v e δ (2) [1 2 ] = 1 2 π − 1 ( δ 0 ), where δ 0 is the b oun dary divisor of irred ucible curves in M g .) F or g = 0 one has λ = 0, so that Eq. (3.13) reads d − 1 X k =2 X µ =[ m 1 ,...,m r ] r Y i =1 m i k (2 d − 2 − k ) 8(2 d − 3) − 1 12 d − r X i =1 1 m i !! δ ( k ) µ = 0 . (3.15) T AU FUNCTION AND ADM ISSIBLE COVERS 13 Let us compare this form u la with the results of [10]. Put M 0 ,d = H 0 ,d /S 2 d − 2 , where the symmetric group S 2 d − 2 acts by inte rc hanging the 2 d − 2 simple branc h p oin ts, and denote by M 0 ,d the compactification of M 0 ,d b y means of the stable maps. Consider the natural map φ : H 0 ,d → M 0 ,d , and put C d = φ  ∆ (2) [3 1 d − 3 ]  , M d = φ  ∆ (2) [2 2 1 d − 4 ]  , ∆ d = φ  ∆ (2) [1 d ]  . The strata C d , M d , ∆ d are the d ivisors in M 0 ,d , whereas φ (∆ ( k ) µ ) has co di- mension ≥ 2 in M 0 ,d for k ≥ 3. According to [10], one has the relation ( d − 6) C d − 3 M d + 3( d − 2)∆ d = 0 in Pic( M 0 ,d ) ⊗ Q , and an easy chec k sho ws that this is consistent with (3.15). F or g = 1 one has λ = 1 12 {∞} on M 1 , where {∞} = M 1 − M 1 is the (one p oin t) b oundary divisor. The preimage π − 1 ( {∞} ) ⊂ H 1 ,d − H 1 ,d with resp ect to the forgetful map (3.2) is the b oundary divisor parametrizing no dal adm issible co ve rs with g ( C (1) ) = g ( C (2) ) = 0. T h erefore, (3.13 ) giv es a non-trivial relation b et ween the b oundary divisors of H 1 ,d . It would b e instructiv e to compare this relatio n w ith the results of [14]. F or g = 2 one has λ = 1 10 δ 0 + 1 5 δ 1 on M 2 , where δ 0 (resp. δ 1 ) is the divisor of irred ucible (resp. redu cible) stable no dal curves (cf. [11]). Th e preimage π − 1 ( δ 1 ) (resp. π − 1 ( δ 0 )) in H 2 ,d − H 2 ,d parametrizes admissible co v ers w ith g ( C (1) ) = g ( C (2) ) = 1 (resp. with g ( C (1) ) + g ( C (2) ) = 1, where the sin gle irreducible gen us 1 comp onent in tersects an irredu cible genus 0 comp on ent at exactly t w o no des). In th is case we also ha ve a non trivial relatio n b et w een the b oundary d ivisors of H 2 ,d . Ac kno w ledgemen ts. W e started this work du ring our visit to the Max- Planc k-Institut f ¨ ur Mathemati k in Bonn, and gratefully ac k n o wledge its hospitalit y and su pp ort; PZ also ac knowledge s his gratitude to IPMU, Uni- v ersit y of T oky o at Kashiwa , where this work wa s finalized. DK thanks M. Mazzocco, and PZ thanks M. Kazarian, D. Orlo v and D. Zv onkine for stim ulating discuss ions . Our sp ecial thanks are to G. v an der Geer for p oin t- ing out a wr ong sign in the main form ula (3.13), and to the referee for the v aluable remarks. Referen ces [1] D. 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