Tau function and moduli of differentials
The tau function on the moduli space of generic holomorphic 1-differentials on complex algebraic curves is interpreted as a section of a line bundle on the projectivized Hodge bundle over the moduli space of stable curves. The asymptotics of the tau …
Authors: Dmitry Korotkin, Peter Zograf
T A U FUNCTION AND MODULI OF DIFFERENTIALS D. KOR OTKIN AND P . ZOGRAF Abstra ct. The tau function on the moduli space of generic holomor- phic 1-differen tials on complex algebraic curves is in terpreted as a sec- tion of a line bund le on the pro jectivized Ho dge bundle o ver the moduli space of stable curves. The asymptotics of the tau function near the b oundary o f the moduli space of generic 1-differenti als is computed, and an explicit expression for the pullback of the Ho dge class on the pro jec- tivized Hodge bundle in terms o f the taut ological class and the classes of b oundary divisors is derived. This exp ression is used to clarify the geometric meaning of the Kon tsevich-Zoric h formula for the sum of the Lyapuno v ex p onents associated with the T ei chm¨ uller flow on the Ho dge bundle. 1. Introduction Mo duli spaces of holo morphic 1-differenti als on complex algebraic curv es arise in v arious areas of mathematics from algebraic g eometry to completely in tegrable systems, holomorphic dyn amics and ergo dic theory . Notably , they admit an ergo d ic S L (2 , R ) action th at can b e desrib ed as follo ws. T ak e the Ho dge bundle on the mo duli space M g of complex algebraic cur ves (the fib ers of this bund le are the spaces of holomorphic 1-differen tials on the corresp ondin g cur v e), and consider it as a real analytic space. The group S L (2 , R ) acts b y lin ear transformations on the r eal and imaginary parts of a holomorphic 1-form. The dyn amics of this action h as b een extensiv ely studied by man y auth ors. It is closely related to billiards in rational p olygons and to inte rv al exc h ange maps, and its inv ariants admit a nice geometric in terpretation (cf. th e pioneering wo rk [11] for m ore details). The isomono d romic tau function on Hurwitz sp aces has a str aigh tforward analogue on mo duli spaces of holomorphic differenti als. I t can b e explicitly written in terms of the theta fu nction and the prime form on th e und er - lying cur v e and plays an imp ortant role in the h olomorphic factorization of determinats of flat Lap lacians [9]. Here we follo w the appr oac h of [10] to study the asymptotic b eha vior of this tau fun ction and to compute its divisor. This allo ws u s to exp ress the p u llbac k of the Ho d ge class on the pro jectivized Ho dge bun dle as a linear com bination of the ta utological class and the classes of b oundary divisors. The obtained expression allo ws us to in terp ret geo metrically the Kon tsevic h-Zoric h formula f or the su m of the Ly apunov exp onents of the diag ( e t , e − t )-actio n ( t ∈ R ) on the Hod ge bu ndle o ver M g . DK wa s partially supp orted by NSERC , FQRNT and CURC; PZ was partially sup - p orted by the RFBR grant 08-01-00379-a and by the President of Russian F ederation gran t NSh - 2460.200 8.1. 1 2 D. KOR OTKIN AND P . ZOGRAF A few wo rds ab out the stru cture of this pap er. Sectio n 2 cont ains some preliminaries on the mo duli space o f generic 1-different ials. In S ection 3 w e define the tau fu nction, giv e an explicit formula for it (Theorem 1), study its trans f ormation prop erties and in terpret it as a holomorph ic section of a line bu ndle on the p r o jectivize d Hod ge bun dle o v er the mo duli space M g . Section 4 con tains the main results of the p ap er: asym p totic f orm ulae for the tau function near the b oun dary comp onents (Th eorem 2), and a formula for the p ullbac k to the pr o jectiviz ed Ho dge line bun d le expressing it as a linear com b ination of the tautologica l class and the classes of b ound ary divisors (Theorem 3 ). In S ection 5 w e discus s the Kon tsevich-Zoric h formula for the sum of the Ly apunov exp onent s. 2. Sp aces of hol omorphic 1-differen tials Let C b e a smo oth complex algebraic curve of gen u s g , and let ω b e a non-zero holomorphic 1-differen tial on C . W e call a holomorphic different ial generic if it has exactly 2 g − 2 simple zeroes. Two p airs ( C 1 , ω 1 ) and ( C 2 , ω 2 ) are called e quivalent if there exists an isomorphism h : C 1 → C 2 suc h that ω 1 = h ∗ ( ω 2 ). The mo duli space of generic pairs ( C, ω ) defined mo dulo this relation we denote by ˜ H 0 g . Additionally we w ill consider an equiv alence relation for holomorphic 1-differen tials on T orelli mark ed curv es. A T or el li marking is a c hoice of symplectic basis α = { a i , b i } g i =1 in the first h omology group H 1 ( C ) of C . A cu r v e C together with a symplectic b asis α w ill b e denoted by C α . W e say that tw o p airs ( C α 1 1 , ω 1 ) and ( C α 2 2 , ω 2 ) are T or el li e quivalent if there exists an isomorphism h : C 1 → C 2 suc h that ω 1 = h ∗ ( ω 2 ) and h ∗ ( α 1 ) = α 2 elemen twise. The mo d uli space of pairs ( C α , ω ) mo dulo th e T orelli equiv alence we d enote b y ˇ H 0 g . The space ˇ H 0 g is a sm o oth non-compact complex manifold of dimension 4 g − 3. The sym plectic group S p (2 g , Z ) acts on ˇ H 0 g b y changing T orelli marking, and ˜ H 0 g = ˇ H 0 g /S p (2 g, Z ). Both ˜ H 0 g and ˇ H 0 g enjo y a natural action of C ∗ (b y m u ltiplication of ω ) that comm utes with the action of S p (2 g , Z ). In the sequel we will also deal with holomorphic 1-different ials with de- generate zeroes. Let ω ha ve r zero es of multipliciti es m 1 , . . . , m r with m 1 + · · · + m r = 2 g − 2. W e call µ = ( m 1 − 1 , . . . , m r − 1 ) the de ge n- er acy typ e of ω (w e ma y omit all zero en tries of µ ). The mo du li space of holomorphic 1 -different ials of a fix ed degeneracy t yp e µ defined mod ulo the ab o v e equiv alence (resp. T orelli equiv alence) w e denote by ˜ H µ g (resp. b y ˇ H µ g ). According to [12], these spaces are connected when ω has at least one simple zero (otherwise they ma y ha ve up to 3 connected comp onents). Ev- erything said in the p revious p aragraph ab out the action of S p (2 g , Z ) and C ∗ applies to the space s ˇ H µ g and ˜ H µ g as w ell. The dimension o f these spaces is 4 g − 3 − | µ | , where | µ | = P r k =1 ( µ k − 1) is th e total degeneracy . W e d escrib e a natur al completion of the space ˜ H 0 g . Let M g b e the mo d- uli space of smo oth gen us g curves, and let M g b e its Deligne-Mumford compactificatio n. T he b oundary M g − M g is the un ion of [ g / 2] + 1 irre- ducible divisors ∆ 0 , ∆ 1 , . . . , ∆ [ g / 2] , where ∆ 0 is the (closure of the) s et of irre- ducible curves of arithmetic gen u s g w ith one no d e, and ∆ j , j = 1 , . . . , [ g / 2] , parametrizes reducible curves with comp onent s of genus j and g − j . Denote T A U FUNCTION AND MODULI OF DIFFERENTIALS 3 b y E g → M g the Ho dge bund le, where the fib er E g | C o ver a p oin t repre- sen ted by a curv e C is giv en b y Ω 1 C , the space of holomorphic 1-forms on C , up to the actio n of Aut( C ). The Ho d ge bund le extends n aturally to a bund le E g → M g (understo o d in the sense of orbifolds or a lgebraic stac ks). The fib er of E g o ver a p oin t represented by a r educible curv e C = C 1 ∪ C 2 is giv en by Ω 1 C 1 ⊕ Ω 1 C 2 , whereas o ver an irredu cible curv e it is given by the vecto r space Ω 1 C ′ ; p,q of mer omorp hic 1-differential s on the n ormalization C ′ → C , g ( C ′ ) = g − 1 , with at most sim p le p oles at the preimages p, q of the no de of C with opp osite residues. W e hav e a sequen ce of in clus ions ˜ H 0 g ֒ → E g ֒ → E g , suc h that the image of ˜ H 0 g is an op en dense sub set in E g . Note that we ha v e a similar inclusion ˜ H µ g ֒ → E g for an y degeneracy type µ . Denote by H g = P ( E g ) the pro jectivizatio n of the Ho dge bundle on M g . T he space H g is a smooth compact complex orb if old (smooth Delig ne- Mumford stac k) of dimension 4 g − 4, and the factor H g = ˜ H 0 g / C ∗ is naturally included in H g as an op en dense subset. The complement H g − H g is the union of [ g / 2] + 2 divisors: H g − H g = D deg ∪ D 0 ∪ · · · ∪ D [ g / 2] . (2.1) Here the divisor D deg = H 1 g is the closure in H g of the lo cu s ˜ H 1 g / C ∗ of degenerate 1-differen tials co nsidered u p to a constant factor, and D j = π ∗ (∆ j ) , j = 0 , . . . , [ g / 2] , are th e p ullbac ks of the b ound ary divisors ∆ j ⊂ M g via the natural pr o jection π : H g → M g . Let L → H g b e the tautological line b undle on H g asso ciated with the pro jection ( E g − M g ) → P ( E g ) = H g , and put ψ = c 1 ( L ) ∈ Pic( M g ) ⊗ Q . Denote b y λ = π ∗ ( c 1 ( E g )) the Ho d ge class in Pic( H g ) ⊗ Q , that is, th e pullbac k of the class c 1 ( E g ) ∈ Pic( M g ) ⊗ Q v ia the pro jection π : H g → M g . W e also p ut δ i = [ D i ] for i 6 = 1 and δ 1 = 1 2 [ D 1 ] in Pic( H g ) ⊗ Q . Lemma 1. The r ational Pic ar d gr oup Pic( H g ) ⊗ Q of the sp ac e H g is fr e ely gener ate d over Q by the classes ψ , λ, δ 0 , . . . , δ [ g / 2] . Pr o of. By a result of [1 ], the rational P icard group Pic( M g ) ⊗ Q is freely generated by the classes λ 1 , ∆ 0 , . . . , ∆ [ g / 2] . W e us e the well- known f act that for a rank n complex vecto r bundle E → M on a sm o oth complex v ariety M one has C H ∗ ( P ( E )) ∼ = C H ∗ ( M )[ ψ ] / ( ψ n + c 1 ( E ) ψ n − 1 + · · · + c n ( E )) , where ψ is the firs t Cher n class of the tautological line bun dle on P ( E ) (cf. [8], Example 8.3.4; h ere C H ∗ stands for th e Chow ring). In particular, Pic( P ( E )) ∼ = Pic( M ) ⊕ Z ψ . The tec hniqu es of e.g. [3] allo w to extend this statemen t (with rational co efficien ts) to the Ho d ge bund le E g → M g . It then yields Pic( H g ) ⊗ Q ∼ = (Pic( M g ) ⊗ Q ) ⊕ Q ψ , i.e. Pic( H g ) is freely generated b y th e classes ψ , λ, δ 0 , . . . , δ [ g / 2] . 3. T au fun ction The aim of this p ap er is to establish a non-trivial relation b et we en the classes ψ , λ, δ 0 , . . . , δ [ g / 2] and δ deg = [ D deg ] in Pic( H g ) ⊗ Q by explicitly computing the divisor of the tau function of [9]. 4 D. KOR OTKIN AND P . ZOGRAF F or a T orelli marked cur v e C α , denote b y B ( x, y ) the Ber gman bidiffer- ential , that is, the unique s y m metric m eromorphic b idifferen tial on C × C with a quadratic p ole of b iresidue 1 o n the diagonal and zero a -per io ds. Its b -p erio ds ω i = Z b i B ( · , y ) dy (3.1) are the normalize d holomorphic 1-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 , (3.2) where the matrix Ω = { Ω ij } g i,j =1 is the p erio d matrix of C α . In terms of lo cal parameters ζ ( x ) , ζ ( y ) n ear the diagonal { x = y } ∈ C × C , the bidifferentia l B ( x, y ) has the expansion B ( x, y ) = 1 ( ζ ( x ) − ζ ( y )) 2 + S B ( ζ ( x )) 6 + O (( ζ ( x ) − ζ ( y )) 2 ) dζ ( x ) dζ ( y ) , (3.3) where S B is a p ro jectiv e connection on C calle d the Ber gman pr oje ctive c onne ction . Consider the non-linear differentia l op erator S ω = ω ′′ ω − 3 2 ω ′ ω 2 (that is, the S c h w arzian deriv ativ e of the ab elian integral R x ω with resp ect to a lo cal parameter ζ on C ). F or a holomorphic 1-differential ω , S ω is a meromorphic pro jectiv e connection on C , so that the difference S B − S ω is a meromorphic quadratic differential . S u pp ose that ω has r zero es x 1 , . . . , x r of multiplicit ies m 1 , . . . , m r ; its degeneracy t yp e is µ = ( m 1 − 1 , . . . , m r − 1) (this includ es the case µ = 0). T ak e the trivial line bu ndle on the space ˇ H µ g and consider the connection d B = d + 2 π √ − 1 2 g + r − 1 X i =1 Z s i S B − S ω ω dz i . (3.4) Here s i = − b i , s i + g = a i for i = 1 , . . . , g , and s 2 g + k is a small circle ab out x k for k = 1 , . . . , r − 1, whereas z i = R a i ω , z i + g = R b i ω for i = 1 , . . . , g , and z 2 g + k = R x k x 2 g − 2 ω f or k = 1 , . . . , r − 1 ( z 1 , . . . , z 2 g + r − 1 serv e as lo cal complex co ordinates on ˇ H µ g , cf. [11]). As it is sh o wn in [9], this connection is flat. The tau fu n ction τ µ = τ µ ( C α , ω ) is locally defined as a horizon tal (cov ariant constan t) section of the trivial lin e bun dle on ˇ H µ g with r esp ect to d B , that is, 1 d B log τ µ = 0 . (3.5) Let u s no w recall an explicit formula for the tau fun ction τ µ deriv ed in [9]. T ak e a nonsingular o dd theta c haracteristic δ and consider the corresp onding theta function 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 . 1 This tau function is the 24-th p ow er of the Bergman tau function studied in [9]. T A U FUNCTION AND MODULI OF DIFFERENTIALS 5 All zero es of th e holomorphic 1-differen tial ω δ ha v e ev en multiplicitie s, and √ ω δ is a w ell-defined holomorphic spinor on C . F ollo wing [6], consider the prime form 2 E ( x, y ) = θ [ δ ] R y x ω 1 , . . . , R y x ω g ; Ω √ ω δ ( x ) √ ω δ ( y ) . (3.6) T o mak e the integ rals uniqu ely defined, we fi x 2 g simple closed loops in the homology classes a i , b i that cut C into a connected domain, and pick th e in tegration p aths that d o not in tersect the cuts. Th e sign of the square ro ot is c hosen 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 lo cal parameter such that dζ = ω δ . W e in tro duce lo cal coord inates on C that w e call natur al (or distin- guishe d ) with resp ect to ω . W e tak e ζ ( x ) = R x x 1 ω as a lo cal co ord inate on C − { x 1 , . . . , x r } , and c ho ose ζ k near x k ∈ C in such a wa y that ω = d ( ζ m k +1 k ) = ( m k + 1) ζ m k k dζ k , k = 1 , . . . , r . I n terms of these coord inates w e ha v e E ( x, y ) = E ( ζ ( x ) ,ζ ( y )) √ dζ ( x ) √ dζ ( y ) , and w e defin e E ( ζ , x k ) = lim y → x k E ( ζ ( x ) , ζ ( y )) s dζ k dζ ( y ) , E ( x k , x l ) = lim x → x k y → x l E ( ζ ( x ) , ζ ( y )) s dζ k dζ ( x ) s dζ l dζ ( y ) . Let A x b e th e Ab el map with the basep oin t x , and let K x = ( K x 1 , . . . , K x g ) b e the v ector of Riemann constan ts 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 (3.7) (as ab o ve, w e assume that the in tegration p aths do not intersect the cuts on C ). Then w e ha ve A x (( ω )) + 2 K x = Ω Z + Z ′ for some Z , Z ′ ∈ Z g . Now put τ µ ( C α , ω ) = 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
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