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Lines on Calabi Yau complete intersections, mirror symmetry,

arXiv:alg-geom/9301001v1 4 Jan 1993Lines on Calabi Yau complete intersections, mirror symmetry,and Picard Fuchs equationsA. Libgober1 and J. Teitelbaum2Department of MathematicsUniversity of Illinois at ChicagoP.O.B.

4348, Chicago, Ill, 60680Internet: u11377@uicvm.uic.edujeremy@math.uic.eduIntroduction and statement of the result. It was suggested (cf.

[COGP], [GP])that (in some circumstances) if V is a Calabi–Yau threefold then one can relate to Va family W(V )t, t ∈C of Calabi–Yau manifolds which are ”mirrors” of V , such thatone has the following relation between their Euler characteristics: χ(V ) = −χ(W(V )t).One of the properties of this correspondence should be the following: the coefficients of theexpansion of certain integrals attached to W(V )t (so called Yukawa couplings) relative to anappropriately chosen parameter are integers from which one may calculate the numbers rdof rational curves of degree d on a generic Calabi–Yau manifold which is a deformation of V .This was verified in [COGP] and [M1],[M2] in the case when V is the quintic hypersurfacein CP4 for rational curves of a small degree. Other authors ([CL],[S]) have suggested alarge list of mirrors of hypersurfaces in weighted projective spaces.

The purpose of thisnote is to verify the above predictions for the remaining types of Calabi–Yau completeintersections in complex projective space when d = 1 i.e. the case of lines.A Calabi–Yau threefold W is a Kahler manifold such that dim W = 3, the canonicalbundle of W is trivial, and the Hodge numbers satisfy h1,0 = h2,0 = 0.

Let Wt be afamily of such manifolds and let ωt be a family of holomorphic 3-forms on Wt (unique upto constant for each t because h3,0(Wt) = 1). According to Griffiths transversality ([G]):κt(k) =ZWω ∧dkωtdtk(1)is equal to zero for k ≤2.

Let κttt = κt(3). We assume that the monodromy T about t = ∞acting on H3(Wt, Z) is maximally unipotent i.e.

that (T −I)3 ̸= 0 and (T −I)4 = 0. If thisis the case then ([M1],[M2]) for N = logT one has dim(ImN 3)⊗C = 1 ( as a consequenceof h3,0 = 1).

Let γ1, γ0 ∈H3(Wt, Z) be a basis of (ImN 2) ⊗C such that γ0 ∈(ImN 3) isan indivisible element and γ1 = 1/λN 2˜γ1 where γ1 is indivisible and the intersection indexof ˜γ1 and γ0 is 1. Let m be defined from the relation Nγ1 = m · γ0 and lets =1mRγ1 ωRγ0 ω , q = e2πis.

(2)Then q is indepenedent of a choice of the basis γ0, γ1 and the form ω up to root of unityof degree |m| (cf. [M1]).1Supported by NSF grant DMS–910798.2Supported by NSF grant DMS–9204265 and by a Sloan Research Fellowship.1

2In this paper, we use an a priori different normalization for the parameter s determinedby specifying the asymptotic behavior of s as t →∞. This normalization is described in(18); it is analogous to that exploited in [COGP] and [M2].Let Vλ be given in CP5 byQ1 = x31 + x32 + x33 −3λx4x5x6 = 0Q2 = x34 + x35 + x36 −3λx1x2x3 = 0(3)This is a complete intersection which is a Calabi–Yau threefold for generic λ .

Let G81 ⊂PGL(5, C) be the subgroup (of order 81) of transformations gα,β,δ,ǫ,µ where α, β, δ, ǫ ∈Z mod 3 µ ∈Z9 and 3 · µ = α + β = δ + ǫ mod 3. These transformations act as:gα,β,δ,ǫ,µ : (x1, x2, x3, x4, x5, x6) →→(ζα3 · ζµ9 · x1, ζβ3 · ζµ9 · x2, ζµ9 · x3, ζ−δ3· ζ−µ9· x4, ζ−ǫ3· ζ−µ9· x5, ζ−µ9· x6)(4)and preserve both hypersurfaces ˜Qi given by the equations Qi = 0 (i = 1, 2).Theorem.

The resolution of singularities W(Vλ) of the quotient of Vλ by the action ofG81 which is a Calabi–Yau manifold satisfies: χ(Vλ) = −χ(W(Vλ)). The monodromy ofW(Vλ) about infinity is maximally unipotent.

For q defined for the family W(Vλ) by theasymptotic normalization (18), the coefficient of q in the q-expansion of κsss is equal to thenumber of lines on a generic non singular complete intersection of two cubic hypersurfacesin CP5.A calculation of the Euler characteristic. The statement on Euler characteristic(as well as the statement on the number of lines) are verified by direct calculation of thequantities involved.

The total Chern class c = 1 + c1 + c2 + c3 ∈H∗(Vλ, Z), of the tangentbundle of Vλ satisfies c · (1 + 3 · h)2 = (1 + h)6 where h is the generator H2(Vλ, Z) (here(1 + 3 · h)2 and (1 + h)6 respectively the total Chern class of the normal bundle to Vλ inCP5 and the pulback on Vλ of the total Chern class of CP5) . The Euler charcateristic ofVλ is c3 evaluated on its fundamental class which (using the fact that h3 evaluated on thefundamental class is 9) gives χ(Vλ) = −144.On the other hand according to the “physicist’s formula” (cf.

[DHVW]) or rather toits reformulation due to Hirzebruch and Hofer (cf. [HH]) the Euler characteristic of aCalabi–Yau resolution of the quotient Vλ/G81 can be found asΣ[g]χ(V gλ /C(g))(5)where the summation is over all conjugacy classes [g] of elements of G, C(g) denotes thecentralizer of g and Xg is the fixed point set of an element g. Because G81 is abelian theformula reduces to Σgχ(V gλ /G81) where the summation is over all elements of the group.There are 6 curves Ci,j having non–trivial stabilizer corresponding to the vanishing oftwo variables in either of the two sets (x1, x2, x3) or (x4, x5, x6).

The Euler characteristic

3of such a curve, which is a complete intersection of two cubic surfaces in P3, is −18.The stabilizer of each curve contains 3 elements since for each curve there are 2 elementswhich have this curve as the fixed point set. Hence the number of elements which haveone dimensional fixed point set is 12.

The Euler characteristic of the quotient of the onedimensional fixed point set is 2. The zero dimensional fixed point sets Di,j,k on a curveCi,j (i, j are in the same group of variables, and k in another) are obtained by equatingto zero a variable in another group.

The stabilizer of such zero dimensional fixed pointset has order 27. Each zero dimensional fixed point set Di,j,k belongs to 3 curves Ci,j.Hence the number of elements stabilizing Di,j,k is 27 −3 × 2 −1 = 20.

The number ofzero dimensional fixed point sets Di,j,k is 6 and each element with zero dimensional fixedpoint set stabilizes 2 sets Di,j,k. Hence the number of elements with zero dimensionalstabilizer is 20×6/2 = 60 and each such element stablizes 6 points.

The quotient of a zerodimensional fixed point set of an element by the group has the Euler characteristic equalto 2. The contribution in (5) from the identity element isχ(Vλ/G81) =1|G81|Σgχ(V gλ )which is equal to 1/81(−144 + 60 × 6 + 12 × (−18)) = 0 .

Hence using (5) the Eulercharacteristic of a Calabi–Yau resolution is 0 + 2 × 60 + 2 × 12 = +144.A method for constructing the Picard Fuchs equations. To find the PicardFuchs equations for the periods of W(Vλ) we shall extend to complete intersections theGriffiths description of cohomology classes of hypersurfaces using meromorphic forms onthe ambient space.

Let T( ˜Q1 ∩˜Q2) be a small tubular neighbourhood of ˜Q1 ∩˜Q2 in CP5and ∂(T( ˜Q1 ∩˜Q2)) be the boundary of T( ˜Q1 ∩˜Q2). ThenH3( ˜Q1∩˜Q2) = H3( ˜Q1∩˜Q2)∗= H7(T( ˜Q1∩˜Q2, ∂T( ˜Q1 ˜Q2)) = H7(CP5, CP5−T( ˜Q1∩˜Q2))(use Poincare duality, retraction combined with Lefschetz duality, and excision).

The lattergroup is isomorphic to H6(CP5 −˜Q1 ∩˜Q2) as follows from the exact sequence of the pair.The Mayer Vietoris sequence combined with these isomorphisms gives the identification:H5(CP5 −( ˜Q1 ∪˜Q2)/Im (H5(CP5 −˜Q1) ⊕H5(CP5 −˜Q2)) = H3( ˜Q1 ∩˜Q2)(7)An alternative description of this isomorphism can be obtained by interpreting a mero-morphic 5-form on CP5 having poles along ˜Q1 ∪˜Q2 as a functional on H3( ˜Q1 ∩˜Q2) whichis given by assigning to a 3-cycle γ representing a homology class in the latter group theintegral over a 5-cycle in CP5 −( ˜Q1 ∪˜Q2); This 5-cycle is the restriction to γ of a torusfibration on which T( ˜Q1 ∩˜Q2) −( ˜Q1 ∪˜Q2) ∩T( ˜Q1 ∩˜Q2) retracts as a consequence of thenon–singularity of ˜Q1 ∩˜Q2. Moreover in the isomorphism (7) the filtration by the totalorder of the pole corresponds to the Hodge filtration on H3( ˜Q1 ∩˜Q2) (details of this willappear elsewhere).

The residues of the meromorphic 5-forms which are G81-invariant give

4the forms on Vλ which descend to Vλ/G81; The pull–back of these forms, which give abasis of H3(W(Vλ), are(x1x2x3)i−1(x4x5x6)n−i−1ΩQi1Qn−i2(8)where n = 2, 3, 4, 5 and Ωis the Euler form:Ω=X(−1)ixidx1 ∧· · · ∧ˆdxi ∧· · · ∧dx6.Calculating the Picard–Fuchs Equation. A cohomology class in H3(Vλ) by (7) isrepresented by a differential formη =nXi=1PiQi1Qn−i2Ωwhere deg(Pi) = 3(n −2) and n ≥2.Relations among forms of this type arise fromconsideration of forms dφ, whereφ =P(xiAj −Aixj)dx1 · · · ˆdxi · · · ˆdxj · · · dx6Qi1Qj2.The relations take the formi P Ai∂Q1∂xiQi+11Qj2+j P Ai∂Q2∂xjQi1Qj+12≡P ∂Ai∂xiQi1Qj2(mod exact)(9)In addition, a form with poles along only one of the forms Qi is equivalent to zero:PQjiΩ≡0(mod exact)(10)We will now describe a procedure for finding canonical representations for meromorphicforms modulo the relations (1) and (2), by constructing an explicit representation of theserelations.Let J1 and J2 represent the rows of the jacobian matrix of (Q1, Q2):Ji = ∂Qi∂x1.

. .∂Qi∂x6If n > 2 is an integer, we construct an (n −1) × 6(n −2) matrix Bn as follows:Bn =(n −2)J100.

. .00J2(n −3)J10.

. .0002J2(n −4)J1.

. .00003J2.

. .00............2J100000(n −3)J2J100000(n −2)J2

5Let In−1 denote the (n −1) × (n −1) identity matrix. We consider the module presentedby the (n −1) × (8n −14) matrix Kn = (Bn Q1In−1 Q2In−1):S8n−14 →Sn−1 →M ∗n →0,where S is the graded polynomial ring in the variables x1, .

. .

, x6.When n > 2 Let Mn denote the part of M ∗n which is homogeneous of degree 3(n −2),and let M2 = C.To see the relationship between Mn and H3(Vλ), suppose that ω belongs to FiliH3(Vλ)i.e. the Hodge filtration of ω is i.

We may represent ω in the formω = (i+1Xk=1pkQk1Qi+1−k2)Ω.Define a homogeneous map φi from FiliH3(Vλ) to Si+1 by setting φi(ω) = (p1, . .

. , pi+1),and we let φi denote the composition of φi with the projection map to M ∗i+2.

It followsfrom our description of the relations (9) and (10) that0 →Fili−1H3(Vλ) →FiliH3(Vλ) →M ∗i+2 →0(11)is exact. We may now briefly describe an algorithm for putting a cohomology class ω,presented as above, into a standard form.

This standard form will consist of elementsmk(ω) ∈Mk for k = 2, . .

. , n + 2 with the property that two forms ω and ω′ represent thesame cohomology class if and only if mk(ω) = mk(ω′) for all k in this range.Step 1.

Compute Grobner bases for the modules Mn for n = 0, . .

. , i + 1.

Such acalculation provides a canonical form for elements of Mi+2 represented as vectors in Si+1.Step 2. Reduce the vector (p1, .

. .

, pi+1) to canonical form modulo the image of Ii+2using Step 1. Suppose that mi+2(ω) is this canonical form.In the reduction process,compute a vector A so thatp1...pi+1= mi+2(ω) + Ki+2A.Step 3.Let Ai,j denote the subvector of A consisting of the entries Ai, .

. .

, Aj.Wedenote by ∇· Ai,i+5 the usual “divergence” of the 6–vector Ai,i+5 relative to the xi:∇· Ai,i+5 = Σk∂Ak∂xk . Construct a new vector p′ in Si representing ω −mi+2(ω) (which, bythe lemma, belongs to Fili−1) by defining:p1 = ∇· A1,6 + A6i+1p2 = ∇· A7,12 + A6i+2 + A7i+3...pi−1 = ∇· A6i−11,6i−6 + A7i−1 + A8i+1pi = ∇· A6i−5,6i + A8i+2

6Repeat Steps 2 and 3 for p′, and continue decreasing i by one each time, until i = 2.We must apply this algorithm in one concrete situation, which we now describe. Define3-forms ωi, for i = 2, 3, .

. .

by the formulaωn = (−1)n(n −2)!n−1Xi=1λn(x1x2x3)i−1(x4x5x6)n−i−1Qi1Qn−i2Ω.These define forms on the complement of ˜Q1 ∪˜Q2 which, by the residue construction,define cohomology classes on Vλ invariant under the automorphism group G81. In fact,these forms span the space of G81–invariant three forms on Vλ, and therefore span H3(Wλ)for the mirror manifold.Let z = λ−6, so that z is a uniformizing parameter at ∞for the parameter space of Vλ.In terms of the derivationΘ = z ddz = −16λ ddλ(12)we have the following fundamental relation:Θωi = −i6ωi + ωi+1.

(13)It follows from this relation, rkH3(W(Vλ) = 4, and the G81 invariance of the formsωi that ω6 is dependent on the forms ω2, . .

. , ω5.

By analogy with Morrison ([M2]), wepostulate a relationship of the following form:ω6(z) =5Xi=2aiz + biz −1 ωi(z)(14)where the ai and bi are small rational numbers.Once the ai and bi are known, it isstraightforward to compute the Picard–Fuchs equation as in [M2]. The most powerfultool available for carrying out the calculations described in the reduction algorithm andcomputing the relation (14) is the Macaulay program of Bayer and Stillman ([Mac]).

Ithas one sizeable limitation which limits its direct application to our problem – it computesGrobner bases over a finite field, whereas at first glance our problem requires computingover the rational function field C(λ).However, if we assume the form of the relationwe seek is as in (14), we may avoid this problem by exploiting the Chinese RemainderTheorem:Step 1. Set the parameter value λ to various constant values λ0 in the finite fieldFp.

Now use Macaulay to apply the reduction algorithm in the corresponding fiber of thefamily and find the relations:ω6(λ−60 ) =Xhi(λ−60 )ωi(λ−60 ).Here the hi are constants in Fp, and these relations are the specializations of the relation(14).

7Step 2. Knowledge of the values of the hi for, say, three distinct λ0 determines the aiand bi mod p. Now repeat the calculation in Step 1 for various different choices of p (againusing Macaulay), then apply the Chinese remainder theorem.

(This is not totally straight-forward, since the ai and bi are rational numbers, not integers, and we have no proved apriori estimate on their denominators; we guessed that the denominators involved powersof two and three, found some reasonable ai and bi, then verified that those coefficientsworked for many choices of prime p.)Using this method, we found the following relation:ω6 =z −73(z −1)ω5 +z + 5536(z −1)ω4 +z −65216(z −1)ω3 +181(z −1)ω2. (15)The associated Picard–Fuchs equation, calculated using this relation and (13), is the gen-eralized hypergeometric equation:(Θ4 −z(Θ + 1/3)2(Θ + 2/3)2)F = 0(16)In particular, this implies that the monodromy at λ = ∞is maximally unipotent.Computing the Yukawa Coupling.To determine the expansion of the Yukawacoupling from the equation, we again follow [M2].

The holomorphic solution F0 to (5) isF0(z) =∞Xn=0(3n)!(n! )32 z36n.We let F1 denote the unique solution to (16) which involves log(z) (but no higher powersof log(z)) and such thats(z) = F1(z)/F0(z)has the propertys(z) ∼log(3−6z) = −6 log(3λ)as z →0.

(17)(This is the asymptotic normalization mentioned at the beginning of the paper, in thisspecial case.) If we letW = F0ΘF1 −F1ΘF0then the Yukawa potential κsss, expressed in the canonical parameter q(z) = exp(s(z)),and normalized so that its leading term is 9 (=the degree of our Calabi–Yau family Vλ) isκsss = −9F 40W 3(s(q) −1).To determine the predicted number of rational curves of given degree, we write κsss inthe formκsss = 9 +X ndd3qd1 −qd .

8With our choices of normalization, we obtain integral values for the nd, and record themin Table 1.Extrapolations.We know that the Picard–Fuchs equation associated to the quintichypersurface is the generalized hypergeometric equation with parameters {1/5, . .

. , 4/5},while that for the complete intersection of two cubics is the hypergeometric equation withparameters {1/3, 1/3, 2/3, 2/3}.

It seems reasonable us to extrapolate from this that theequations for the remaining types of Calabi–Yau complete intersections are hypergeometricas well; with parameters as given in the following table:DescriptionParameters4 quadrics in P7{ 12, 12, 12, 12}2 quadrics and cubic in P6{ 12, 12, 13, 23}2 cubics in P5{ 13, 23, 13, 23}Quartic and quadric in P5{ 14, 12, 34, 12}Quintic in P4{ 15, 25, 35, 45}Based on this hypothesis we calculated the Yukawa potential in each of these cases.There are two constants which must be chosen for each such calculation; one of theseforces κsss to have initial term the degree of the variety, while the other determines theasymptotic behavior of the coordinate s in terms of the “hypergeometric” variable z as inequation (16). In each case, we made the choices(z) ∼log(z) −Xdi log(di)(18)where the di are the degrees of the hypersurfaces defining the complete intersection.

Withthese choices, we obtained the correct values for the number of straight lines in eachcase, and integral values for the predicted number of rational curves. The results of ourcalculations are summarized in the Table 1.

9Table 1.Numerical ResultsPredicted Number of Rational Curves of Given DegreeFor Various Types of Complete Intersection Calabi–Yau ManifoldsDegreeV3,3 ⊂P5V2,4 ⊂P511053*1280*25281292288364243261565516841139448384388390252852497878925831190923282176662660964509532417874605342336717256453900822009160964588281789696850888425684261629606639289562562563948891581250717976557887945288550603166164883599361051204524190720910682860813069047760169269024822656DegreeV2,2,2,2 ⊂P7V2,2,3 ⊂P61512*720*297282242834162561611504425703936168199200519579837442167693171261705359232003195557904564716300354777600517064870788848816680630963870728958096559960675291798457560643297281635230376937591084810202064979838915548163110686153486233022944(*)These numbers coincide with those given in [L] p. 52. The number of lines onV2,2,2,2 (resp.

V2,2,3) is not given explicitly there (only as part of theorem 3). It is easy to

10check that the lines belonging to a quadric in P7 form a cycle on the Grassmanian Gr(1, 7)of lines in P7 which is homologous to 4Ω4,6 (Ωp,q denotes the Schubert cycle consisting oflines in a generic Pq intersecting generic Pp ⊂Pq ). Its 4-fold self–intersection equals 512,which gives the number of lines on V2,2,2,2.

On the other hand, the lines in P6 which belongto a generic hypersurface of degree 3 (resp. 2) form the cycle in Gr(1, 6) homologous to18Ω2,5 + 27Ω3,4 (resp.

4Ω3,5). The intersection index: (18Ω2,5 + 27Ω3,4)(4Ω3,5)2 equals720 which gives the number of lines on V3,2,2.References[COGP] P.Candelas, X. de la Ossa, P.Green, L.Parkes, Nuclear Physics.

B359 (1990).21. [CL] P.Candelas, M.Lynker, R.Schimmrigk, Calabi–Yau manifolds in weighted P4.

Nu-clear Physics, B341 (1990), 383-402. [DHVW] L.Dixon, J.Harvey, C.Vafa, E.Witten, Strings on orbifolds, I, Nuclear Phys.B261, 678-686 (1985) and II (ibid) B 274, 285-314 (1986).

[G] P.Griffiths, Periods of integrals on algebraic manifolds: summary of results anddiscussions of open problems, Bull. AMS.

1970, 76, 228-296. [GP] B.Greene and M.Plesser, Duality in Calabi Yau moduli space , Nuclear PhysicsB338 (1990).

[HH] F.Hirzebruch and T.Hofer, On Euler number of an orbifold, Math. Ann.

286,255-260, (1990). [L] A.Libgober, Numerical characteristics of systems of straight lines on complete in-tersections, Math.

Notes, 13(1973) p. 51–56, Plenum Publishing, translated from Math.Zametki 13(1973) p. 87–96.

[Mac] D. Bayer and M. Stillman, Macaulay: A system for computation in algebraic ge-ometry and commutative algebra. Source and object code available for Unix and Macintoshcomputers.

Contact the authors, or download from zariski.harvard.edu via anonymousftp. [M1] D.Morrison, Mirror symmetry and rational curves on quintic threefolds: a guidefor mathematicians, Preprint, Duke University, 1991.

[M2] D.Morrison, Picard Fuchs equations and the mirror map for hypersurfaces, Preprint,Duke University, 1991. [S] R.Schimmrigk, The construction of mirror symmetry.

Preprint, Sept. 1992, Heidel-berg and CERN.P.S. Besides calculation in [L] of the number of lines on generic complete intersectionof arbitrary dimension on the case when it is finite the case of lines on three dimensionalcomplete intersections with K = 0 also treated in S.Katz paper in Math.

Zeit. 191 (1986).293-296.

We are thanking S.Katz for pointing this out.

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