Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces 9 0 0 Si Li, Bong H. Lian, Shing-Tung Yau 2 t c Abstract O We study the variation of relative cohomology for a pair consisting of a smooth 2 projective hypersurface and an algebraic subvariety in it. We construct an inhomoge- 2 neous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic ] top form on a toric Calabi-Yau hypersurface, and deriving a general formula for the G d-exact form on one side of the equation. We also derive a double residue formula, A giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for Abel-Jacobi map, which has played an important role in recent study of D-branes . h [25]. Usingthevariationformalism,weprovethattherelativeperiodsoftoricB-branes t on a toric Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system a m proposedin physicsliterature[6],anddiscusstherelations betweentheworks[25][21] [6] in recent study of open string mirror symmetry. We also give thegeneral solutions [ to theenhanced hypergeometric system. 1 v 1 Introduction 5 1 2 MirrorsymmetryconnectssymplecticgeometryofCalabi-Yaumanifoldtocomplexgeometry 4 ofitsmirrormanifold. Inclosedstringtheory,this hasledtopredictionsoncountingcurves . 0 onprojectiveCalabi-Yauthreefolds[9][8]. Inopenstringtheory,mirrorsymmetryhasledto 1 predictionsoncountingholomorphicdiscs,firstinthenon-compactcasestudiedin[4][3],and 9 more recently in the compact quintic example, where the instanton sum of disc amplitude 0 with non-trivial boundary on the real locus of the real quintic is shown to be identical to : v the normalized Abel-Jacobi map on the mirror quintic via mirror map [31][25][27]. i X In physics, the Abel-Jacobi map serves as the domain-wall tension of D-branes on the B-model, and is obtained via reduction of the holomorphic Chern-Simons action on curves r a [4]. It is conjectured to have remarkable integrality structure [26]. A key for calculating the Abel-Jacobi map is through inhomogeneous Picard-Fuchs equations [25]. Let X be a z family of Calabi-Yau threefolds parameterizedby variablez, and Ω be a family of nonzero z holomorphic 3-forms on X . Assume that there is a family of pairs of holomorphic curves z C+,C− in X . Let (∂ ) be a Picard-Fuchs operator. Then there exists a 2-form β such z z z D z z that (∂ )Ω = dβ (1.1) z z z D − The exactterm dβ does not contribute when it is integratedover a closed3-cycle Γ in X . z z Theso-calledclosed-stringperiod Ω thensatisfiesahomogeneousPicard-Fuchsequation. Γ z In open string theory, it is necessary to consider the integral of Ω over a 3-chain Γ in X z z R which is not closed, but whose boundary is C+ C−. Because of contributions from the − boundary, this so-called open-string period Ω satisfies an inhomogeneous Picard-Fuchs Γ z equation. Solving the equation gives a precise description of the Abel-Jacobi map up to R closed-stringperiods. To study this map,β plays anessentialrolesince it is this formthat z 1 gives rise to one side of the inhomogeneous Picard-Fuchs equation: (∂ )Ω = β . (1.2) z z z D − ZΓ Z∂Γ The inhomogeneous term on the right side turns out also to encodes important information for predicting the number of holomorphic disks on a mirror Calabi-Yau manifold. There have been several proposals for constructing the inhomogeneous Picard-Fuchs equationanditssolutions. Inthecaseof1-modulifamily[25],itwasdonebyfirstcomputing β usingtheGriffith-Dworkreductionprocedure,andthenbydoinganexplicit(butdelicate) z local analytic calculation of appropriate boundary integrals. Based on the notion of off- shell mirror symmetry, two other proposals [21][6] have been put forth. Roughly speaking, their setup begins with a family of divisors Y which deforms in X under an additional z,u z parameter u. For each relative homology class Γ H (X ,Y ), one considers the integral 3 z z,u ∈ Ω (1.3) z ZΓ which is called a relative period for B-brane. It is proposed that the open-string periods aboveberecoveredasacertaincriticalvalueoftherelativeperiod,regardedasafunctionof u. Tocalculatethe relativeperiods,[21]proposedaproceduresimilarto theGriffith-Dwork reduction. In [6], an enlarged polytope is proposed to encode both the geometry of the Calabi-YauX andthe B-branegeometry. This givesrise to a GKZ hypergeometricsystem z fortherelativeperiods,andaspecialsolutionatacriticalpointinuthenleadstoasolution to the original inhomogeneous Picard-Fuchs equation. Ourgoalinthispaperistofurtherdevelopthemathematicalstructuresunderlyinginho- mogeneousPicard-FuchsequationsandtheAbel-Jacobimap,andtoclarifytherelationships between the three approaches mentioned above. Here is an outline. We begin, in section 2, withadescriptionofaresidueformalismforrelativecohomologyofafamilyofpairs(X ,Y ), z z including anumberofvariationalformulasonthe localsystemHn(X ,Y ). Insection3,we z z derive a general formula for the exact form (the β-term) appearing in the inhomogeneous Picard-Fuchs equation for toric Calabi-Yau hypersurfaces, generalizing GKZ-type differential equation to the level of differential forms instead of cohomology classes. This gives a much more uniform approach to computing the β-term than the Griffith-Dwork reduction. In section 4, we prove a purely algebraic a double residue formula for the inhomogeneous termofthePicard-FuchsequationthatgovernstheAbel-Jacobimap. Thisuniformapproach also allows us to bypass the delicate local analytical calculation of boundary integrals in a previous approach [25][23]. In section 5, using the residue formalism in section 2, we give a simple interpretation of the relative version of the Griffith-Dwork reduction used in [21]. In particular, this gives a mathematical justification for the appearance of log divisor, and elucidates the relationshipbetween relative periods and the Abel-Jacobi map. We also give a uniform descriptionfor the enhancedpolytope method for describing toricB-brane geometry in a general toric Calabi-Yau hypersurface, and show that relative periods satisfy the corresponding enhanced GKZ system. Finally, we give a general formula, modeled on the closed string case [16][17], for solution to the enhanced GKZ system. Acknowledgement. S.L.wouldliketothankJ.Walcherformanystimulatingdiscussions, and thank M.Soroush for answering many questions on his paper. After the completion of a preliminary draft of our paper, three other papers [2][5][15] with some overlap with ours have since been posted on the arXiv. 2 2 Variation of Relative Cohomology Local System of Relative Cohomology and Gauss-Manin Connection Let π : S be a smooth family of n-dimensional projective varieties, and S be a X → Y → family of smooth subvariety . Let s S be a closed point, and denote by X ,Y the s s Y ⊂X ∈ corresponding fiber over s. Consider the family of relative cohomology class Hn(X ,Y ) s s given by the cohomology of the complex of pairs: Γ(Ωn(X )) Γ(Ωn−1(Y )) s s ⊕ with the differential d(α,β)=(dα,α dβ) (2.1) |Ys − Here Ωn(X ) and Ωn−1(Y ) are sheavesofDe Rham differentialn-forms onX and (n 1)- s s s − form on Y , and Γ is the smooth global section. Therefore an element of Hn(X ,Y ) is s s s represented by a differential n-form on X whose restriction to Y is specified by an exact s s form. Lemma 2.1. Hn(X ,Y ) forms a local system on S. s s Proof. Theproofissimilartothecasewithout bychoosingalocaltrivializationof S Y X → which also trivializes S. See e.g.[30]. Y → We denote this local system by n , and let GM be the Gauss-Manin connection. H(X,Y) ∇ There’s a well-defined natural pairing H (X ,Y ) Hn(X ,Y ) C n s s ⊗ s s → (2.2) Γ (α,β) <Γ,(α,β)> α β. ⊗ 7→ ≡ Γ − ∂Γ R R Given a family (α ,β ) Hn(X ,Y ) varying smoothly, which gives a smooth section of s s s s ∈ n denoted by [(α ,β )], and Γ H (X ,Y ) a smooth family of relative cycles, we H(X,Y) s s s ∈ n s s get a function on S given by the pairing <Γ ,(α ,β )>= α β s s s s s − ZΓs Z∂Γs Let v be a vector field on S. We consider the variation <Γ ,(α ,β )> v s s s L where is the Lie derivative with respect to v. Suppose we have a lifting α˜,β˜, which are v L differential forms on , respectively, such that X Y α˜ =α , β˜ =β |Xs s |Ys s and that Γ moves smoothly to form a cycle Γ˜ on : s X Γ =Γ˜ X , ∂Γ . s s ∩ ⊂Y Let v˜ be a lifting of v on , v˜ be a lifting of v on . X Y X Y 3 Proposition 2.2 (Variation Formula). <Γ ,(α ,β )>=<Γ ,(ι ydα˜,ι y(dβ˜ α˜))> (2.3) Lv s s s s v˜X v˜Y − where ι is the contraction with v˜ , and similarly for ι . v˜X X v˜Y Proof. We fix a point s S, andlet σ(t) be alocalintegralcurve ofv suchthat σ(0)=s . 0 0 Let Γ˜ denote the one-∈dimensional family of cycles over σ(s), 0 s t, and we denote σ(t) ≤ ≤ byΓ the cycleoverthe pointσ(t). Alsolet∂Γ be thefamilyofboundarycycle∂Γ over t σ(t) s σ(s),0 s t. Then we have ≤ ≤ f ∂ Γ˜ =Γ Γ ∂Γ (2.4) σ(t) t 0 σ(t) − − (cid:16) (cid:17) therefore f dα˜ = α˜ α˜ α˜ ZΓ˜σ(t) ZΓt −ZΓ0 −Z∂Γσ(t) Similarly f dβ˜= β˜ β˜ − Z∂Γ Z∂Γt Z∂Γ0 Taking the derivative with respefct to t, we get ∂ α˜ β˜ = ι ydα˜+ ι y α˜ dβ˜ ∂t − v˜X v˜Y − (cid:18)ZΓt Z∂Γt (cid:19) ZΓt Z∂Γt (cid:16) (cid:17) The proposition follows. Note that (ι ydα˜,ι y(dβ˜ α˜)) is nothing but the Gauss-Manin connection v˜X v˜Y − GM[(α ,β )]= (ι ydα˜) ,(ι y(dβ˜ α˜)) (2.5) ∇v s s v˜X |Xs v˜Y − |Ys h(cid:16) (cid:17)i and it is straightforward to check using the variation formula that the right side of (2.5) is independent of the choice of α˜,β˜,v˜ ,v˜ , and that the connection is flat. The following X Y corollary also follows from (2.5). Corollary 2.3. <Γ ,(α ,β )>=<Γ , GM[(α ,β )]> (2.6) Lv s s s s ∇v s s Residue Formalism for Relative Cohomology In this section, we assume that X moves as a family of hypersurfaces in a fixed n+1-dim z ambient projective space M with defining equation P = 0. Here P H0(M,[D]) for a z z ∈ fixed divisor class [D], and z is holomorphic coordinate on S parametrizing the family. Let ω H0(M,K (X )) z M z ∈ be a rational (n+1,0)-form on M with pole of order one along X , then we get a famliy of z holomorphic (n,0)-form on X given by z Res ω H0(X ,K ) Xz z ∈ z Xz 4 and also a family of relative cohomology classes (Res ω ,0) Hn(X ,Y ) Xz z ∈ z z Here our convention for Res is that if X is locally given by w = 0, and ω = dw φ, z z w ∧ where φ is locally a smooth form, then Res ω = φ . Note that the map Res : Xz z |Xz Xz H0(M,K (X )) Hn,0(X ) is at the level of forms, not only as cohomology classes since M z z → the order of pole is one. While in the residue formalism of ordinary cohomology, we can ignore exact forms to reduce the order of the pole [14], it is important to keep track of the orderofthepoleinconsideringperiodsofrelativecohomologybecauseoftheboundaryterm. We choose a fixed open cover U of M and a partition of unity ρ subordinate to α α { } { } it. Let P =0 be the defining equation of X on U . Then we can write z,α s α d P M z,α ω = φ (2.7) z z,α P ∧ z,α α X where φ is a smooth (n,0)-formwith supp(φ ) U . On the trivialfamily M S, we z,α z,α α ⊂ × will use d to denote the differential along M only and use d to denote the differential on M the total space. Let φ = φ (2.8) z z,α α X Then φ is a smooth form on M S such that z × φ =Res ω z|Xz Xz z Consider the variation ∂ d P ∂ P M z,α z z,α ω = d ∂ log(P )φ + ∂ φ d φ z M z z,α z,α z z,α M z,α ∂z α ! α (cid:18) Pz,α ∧ − Pz,α (cid:19) X X Let ∂ ∂˜ = +n zX ∂z z,X be a lifting of ∂ to , where n Γ(T ) is along the fiber, which is a normal vector ∂z X z,X ∈ M|Xz field corresponding to the deformation of X in M with respect to z. Then in each U , we z α have ι yd P = ∂ P (2.9) nXz M z,α |Xz − z z,α|Xz It follows easily that (cid:0) (cid:1) ι ydφ =Res (∂ ω d (∂ log(P )φ )) (2.10) ∂˜zX z|Xz Xz z z− M z z z Note that since the transition function of [D] is independent of z, ∂ log(P ) is globally z z well-defined. In general, ∂ ω will have a pole of order two along X , but the substraction z z z ofd (∂ log(P )φ ) makesit logarithmicalongX , hence the residueaboveis well-defined. M z z z z Next we choose arbitrary lifting of ∂ to , and write it as ∂z Y ∂ ∂˜ = +n z,Y z,Y ∂z where n Γ(T ) is along the fiber, which is a normal vector field corresponding to z,X ∈ M|Yz the deformation of Y in M with respect to z. Then the variation formula implies that z 5 Proposition 2.4 (Residue Variation Formula). GM(Res ω ,0)= Res (∂ ω d (∂ log(P )φ )), ι yφ (2.11) ∇∂z Xz z Xz z z− M z z z − nz,Y z (cid:0) (cid:1) To see the effect of the second component on the right side, let us assume that Y = z X H, where H is a fixed hypersurface in M with defining equation Q=0. Then we can z ∩ choose a ǫ-tube T (Γ) [14] of Γ H (X ,Y ) with ∂T (Γ) H. Hence ǫ n z z ǫ ∈ ⊂ 1 1 d (∂ log(P )φ ) = (∂ log(P )φ ) M z z z z z z H 2πi − 2πi | ZTǫ(Γ) ZTǫ(∂Γ) = Res (∂ log(P )φ ) Yz z z z |H Z∂Γ = ι yφ − nz,Y z Z∂Γ which cancels exactly the second component on the right side of (2.11). Therefore, 1 ∂ <Γ,(Res ω ,0)>= ∂ ω z Xz z 2πi z z ZTǫ(Γ) We can localize the above observation and consider the following situation: on each U , suppose we can choose Q independent of z such that P ,Q are transversal and α α z,α α Y U Q = 0,P = 0 . Suppose we have a relative cycle Γ H (X ,Y ) where we z α α z,α n z z ∩ ⊂ { } ∈ can choose a ǫ-tube T (Γ) such that ∂T (Γ) U lies in Q =0 . We have the pairing ǫ ǫ α α ∩ { } 1 <Γ,(Res ω ,0)>= lim ω Xz z ǫ→02πiZTǫ(Γ) z However,the righthandside doesn’t depend onǫ. In fact, let Tǫ(Γ) be a solidannulus over δ Γ. By Stokes’s theorem, we have ω ω = ω z z z − ZTǫ(Γ) ZTσ(Γ) Z∂Tδǫ(Γ) since∂Tǫ(Γ) U Q =0 foreachα,theaboveintegralvanishes. Thereforetheintegral δ ∩ α ⊂{ α } ω doesn’t depend on the position of the ǫ-tube if we impose the boundary condition Tǫ(Γ) z as above. It follows immediately that R 1 (∂ )k <Γ,(Res ω ,0)>= (∂ )kω z Xz z 2πi z z ZTǫ(Γ) where ∂T (Γ) U lies in Q =0 . Applying this to a Picard-Fuchs operator, we get ǫ α α ∩ { } Proposition 2.5. [Inhomogeneous Picard-Fuchs Equation] Let = ( GM) be a Picard- D D ∇∂z Fuchs operator, i.e. (∂ )ω = dβ (2.12) z z z D − for somerational (n 1,0)-formβ withpoles along X . Then undertheabove local choices, z z − we have 1 (∂ )<Γ,(Res ω ,0)>= β . (2.13) D z Xz z 2πi z ZTǫ(∂Γ) 6 In the next section, we will derive a general formula for β using toric method. z More generally, suppose that Q depends on z which is denoted by Q , and we put α z,α { } our ǫ-tube T (Γ ) inside Q =0 on U . Then ǫ z z,α α { } 1 1 Res ω = ω = ρ ω Xz z 2πi z 2πi α z ZΓz ZTǫ(Γz) α ZTǫ(Γz) X andthe integrationdoesn’tdependonǫassumingthe boundaryconditionas above. Apply- ing the variation formula (2.3) we get ∂ 1 1 1 ρ ω = ρ ∂ ω ρ ι yω ∂z 2πi α z 2πi α z z− 2πi α nz,Qα z |Qz,α=0 α ZTǫ(Γz) α ZTǫ(Γz) α ZTǫ(∂Γz) X X X (cid:0) (cid:1) wheren isthenormalvectorfieldcorrespondingtothedeformationof Q =0 inside M. In pza,Qrtαicular, we have ι ydQ = ∂ Q . Hen{cez,α } nz,Qα z,α|{Qz,α=0} − z z,α|{Qz,α=0} ι yω = Res (∂ log(Q )ω ) nz,Qα z |Qz,α=0 − Qz,α=0 z z,α z (cid:0) (cid:1) Putting together the last three equations, we arrive at Proposition 2.6 (cf. [32]). ∂k 1 k ∂k−l 1 ∂l−1 Res ω = ∂kω + ρ Res ∂ log(Q ) ω ∂zk Xz z 2πi z z ∂zk−l 2πi α Qz,α=0 z z,α ∂zl−1 z (cid:18)ZΓz (cid:19) ZTǫ(Γz) l=1 α ZTǫ(∂Γz) (cid:18) (cid:19) X X 3 Exact GKZ Differential Equation and Toric geometry In this section, we study the Picard-Fuchs differential operators arising from a generalized GKZ hypergeometric systems [16][17] for toric Calabi-Yau hypersurfaces and derive a general formula for the β-term of an inhomogeneous Picard-Fuchs equation, from toric data. We first consider the special case of a weighted projective space, where β-term will be much simpler than in the general case, which will be considered at the end of this section. Let P4(w) = P4(w ,w ,w ,w ,w ). We assume that w = 1 and it’s of Fermat-type, i.e., 1 2 3 4 5 5 w dforeachi,whered=w +w +w +w +w . There’sassociated4-dimensionalintegral i 1 2 3 4 5 | convex polyhedron given by the convex hull of the integral vectors 5 ∆= (x , ,x ) R5 w x =0,x 1 1 5 i i i ( ··· ∈ | ≥− ) i=1 X If we choose the basis e =(1,0,0,0, w ),i=1..4 , then the vertices is given by i i { − } d ∆: v =( 1, 1, 1, 1) 1 w − − − − 1 d v =( 1, 1, 1, 1) 2 − w − − − 2 d v =( 1, 1, 1, 1) 3 − − w − − 3 d v =( 1, 1, 1, 1) 4 − − − w − 4 v =( 1, 1, 1, 1) 5 − − − − 7 and the vertices of its dual polytope is given by ∆∗ : v∗ =(1,0,0,0) 1 v∗ =(0,1,0,0) 2 v∗ =(0,0,1,0) 3 v∗ =(0,0,0,1) 4 v∗ =( w , w , w , w ) 5 − 1 − 2 − 3 − 4 Let v∗,k =0,1,2,.., be integral points of ∆∗, where v∗ =(0,0,0,0). We write k 0 f∆∗(x)= akXvk∗ (3.1) v∗∈∆∗ kX whichisthedefiningequationforourCalabi-Yauhypersurfacesintheanti-canonicaldivisor class. Here X = {X1,X2,X3,X4} is the toric coordinate, Xvk∗ = 4 Xjvk∗,j. If we use j=1 homogeneous coordinate [10] z ,1 ρ 5 corresponding to the onQe-dim cone v ,1 ρ ρ { ≤ ≤ } { ≤ ρ 5 , then the toric coordinate can be written by homogeneous coordinate ≤ } zd/wj X = j , j =1,2,3,4 (3.2) j 5 z ρ ρ=1 Q The relevant rational form with pole of order one along the hypersurface is given by 5 w i 4 1 dX Π(a) = i=1 j Qd3 vk∗∈∆∗akXvk∗ jY=1 Xj P Ω0 = a 5 z+1 k ρ v∗∈∆∗ ρ=1 k P Q Ω 0 = (3.3) a 5 z + 5 a zd/wρ+ a 5 z+1 0 ρ ρ ρ k ρ ρ=1 ρ=1 v∗∈∆∗,k>5 ρ=1 k Q P P Q where Ω = 5 ( 1)ρ−1w z dz dˆz dz . Define the relation lattice by 0 ρ=1 − ρ ρ 1∧··· ρ∧··· 5 L=Pl=(l ,l ,...) Z|∆∗|+1 l v¯∗ =0 , where v¯∗ =(1,v∗), v∗ ∆∗ { 0 1 ∈ | i i } i i i ∈ i X The moduli variable associated with the choice of a basis l(k) for L is given by [16] { } xk =( 1)l0(k)al(k) − The key idea here is to consider the following 1-parameter family of automorphisms φt :zρ (a0)wdρtzρ, 1 ρ 5. → a ≤ ≤ ρ Put Π (a)=φ∗Π(a). t t 8 It satisfies the differential equation ∂ (φ∗Π(a))= φ∗Π(a) t t LV t 5 whereV = wρ(log a0)z ∂ isthegeneratingvectorfieldforφ , istheLiederivative. ρ=1 d aρ ρ∂zρ t LV This is solvePd by φ∗Π(a)=etLVΠ(a) (3.4) t Define Π˜(x)=a Π (a) (3.5) 0 1 Π˜(x) is a function of x only. Indeed, k { } Ω Π˜(x)= 0 ρ=51zρ+ ρ=51(aaρ0)wρ/d!ρ=51zρd/wρ +v∗∈∆∗,k>5aak0 ρ=51 aaρ0 wdρρ=51zρ+1 k (cid:16) (cid:17) Q Q P P Q Q Since we have 5 5 w w v =0, v∗ = ρ v∗ ρ ρ k d k ρ ρ ρ=1 ρ=1 X X we see that both ρ=51(aaρ0)wρ/d! and aak0 ρ=51 aaρ0 wdρ can be written in terms of xk’s (cid:16) (cid:17) as an algebraic funcQtion. Q Given an integral point l L, consider the GKZ operator (it differs from the standard GKZ operator by a factor of a∈ ali) 0 li>0 i ∂ li Q ∂ −li D = a ali al a−li l 0(lYi>0 i (cid:18)∂ai(cid:19) − lYi<0 i (cid:18)∂ai(cid:19) ) l0 ∂ li−1 ∂ −l0 ∂ −li−1 ∂ = (a j) (a j) al (a j) (a j) a  0∂a − i∂a − − 0∂a − i∂a −  0 0 i 0 i jY=1 i6=0Y,li>0jY=0 jY=1 i6=0Y,li<0 jY=0  = D˜ a l 0  m where we use the convention that ( )=1 if m 0. From ··· ≤ i=1 Q D Π(a)=0 l We get eLVD˜ e−LVΠ˜(x)=0 (3.6) l Lemma 3.1. ∂ ∂ eLV(ai∂ai)e−LV =ai∂ai +δ1≤i≤5Lwdizi∂∂zi −δi,0Lρ=51wdρzρ∂∂zρ P here δ =1 if 1 i 5 and otherwise 0. 1≤i≤5 ≤ ≤ 9 Proof. Since 5 ∂ w ∂ w ∂ i ρ V,a = δ z δ z i 1≤i≤5 i i,0 ρ a d ∂z − d ∂z (cid:20) i(cid:21) i ρ=1 ρ X ∂ V, V,a = 0 i a (cid:20) (cid:20) i(cid:21)(cid:21) The lemma follows from the formula ∞ ∂ ∂ 1 eLV(a )e−LV =a + i∂ai i∂ai k=1k!L(adV)k ai∂∂ai X “ ” It follows from the lemma that l0 ∂ li−1 ∂ jY=1(a0∂a0 −j) i6=0Q,li>0 jY=0(ai∂ai −j+δ1≤i≤5Lwdizi∂∂zi) −l0 ∂ −li−1 ∂ −aljY=1(a0∂a0 −j) i6=0Q,li<0 jY=0 (ai∂ai −j+δ1≤i≤5Lwdizi∂∂zi)Π˜(x)=0 (3.7) where we have used Π˜(x)=0 to eliminate the terms with  . Lρ=51wdρzρ∂∂zρ Lρ=51wdρzρ∂∂zρ Observe that each LPie derivative Lwdizi∂∂zi commutes with all other opPerators appearing on the left side of (3.7). So we can move every term involving Lwdizi∂∂zi to the rightside, so that (3.7) can now be explicitly written as 5 D˜lΠ˜(x)=− Lwdizi∂∂ziαi i=1 X where the α are (easily computable) d-closed 4-forms depending on l. By the Cartan-Lie i formula =dι +ι d, we obtain the formula X X X L Proposition 3.2. [β-term Formula] D˜ Π˜(x)= dβ l l − where βl = iιwdizi∂∂ziαi. P Next we consider the differential operators of the extended GKZ system induced by the automorphism of the ambient toric variety. The corresponds to the root of ∆∗ [17]. Let v∗ R(∆∗), = 1, 0 for ρ=ρ i ∈ i ρi − i ρ ≥ 6 i then we obtain an equation ∂ 1 1 vk∗X∈∆∗(+1)avk∗∂avk∗+vi∗a0e−LVΠ˜(x)=L(ρ=51zρ)zρi∂z∂ρi a0e−LVΠ˜(x) Q   10

Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces

Similar Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces