# GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui

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1 GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY Sampei Usui Abstract. This article is a geometric application of polarized logarithmic Hodge theory of Kazuya Kato and Sampei Usui. We prove generic Torelli theorem for the well-known quintic-mirror family in two ways by using different logarithmic points at the boundary of the fine moduli of polarized logarithmic Hodge structures. 1. Review of quntic-mirror family We review the construction of the mirror family of the pencil joining quintic hypersurface of Fermat type and the simplex consisting of the coordinate hyperplanes in a 4-dimensional complex projective space after [M1]. Let ψ P 1, and let (1) Q ψ = { 5 j=1 x5 j 5ψ 5 j=1 x j = 0} P 4. Let µ 5 = {α C α 5 = 1}. The singular members of the pencil (1) are listed as follows: (2) Over each point ψ µ 5 C P 1, Q ψ contains 5 3 = 125 ordinary double points (α 1,..., α 5 ) (µ 5 ) 5 /µ 5 P 4 with ψα 1 α 5 = 1. (3) Over P 1, Q = j {x j = 0}. This is a 4-dimensional simplex. Let G = {α (µ 5 ) 5 α 1 α 5 = 1}/µ 5. By multiplication, this becomes a group which acts on P 4 coordinate-wise. This action of G preserves Q ψ. Taking the quotient Q ψ /G, the following canonical singularities appear: (4) For each pair of distinct indices j, k, a compound du Val singularity ca 4 appears as the quotient of the curve Q ψ {x j = x k = 0} \ ( m j,k {x m = 0}). (5) For each triple of distinct indices j, k, l, the point which is the quotient of the five points Q ψ {x j = x k = x l = 0} belongs to the closure of three curves in (4). Key words and phrases. quintic-mirror family, logarithmic Hodge theory, moduli, Torelli theorem Mathematics Subject Classification. Primary 14C30; Secondary 14D07, 32G20. Partly supported by the Grant-in-Aid for Scientific Research (B) No , Japan Society for the Promotion of Science 1 Typeset by AMS-TEX

2 Moreover, we see that holomorphic 3-forms on Q ψ are G-invariant for every ψ C by adjunction formula. For ψ C, it is known that there is a simultaneous minimal desingularization W ψ of these quotient singularities, and that holomorphic 3-forms extend to nowhere vanishing forms on W ψ. We thus have a pencil (W ψ ) ψ P 1 whose singular fibers are listed as follows: (6) Over each point ψ µ 5 C P 1, W ψ has one ordinary double point. (7) Over ψ = P 1, W ψ is a normal crossing divisor in the total space, whose components are all rational. The other members W ψ are smooth with Hodge numbers h p,q = 1 for p + q = 3. By the action of α µ 5, (x 1,..., x 5 ) (α 1 x 1, x 2,..., x 5 ), we have W αψ W ψ. Let λ = ψ 5, and let (W λ ) λ ((W ψ ) ψ )/µ 5 (λ-plan) (ψ-plan)/µ 5. This pencil (W λ ) λ P 1 is the mirror of the original pencil (1). (For more details of the above construction, see e.g. [M1].) 2. Review of fine moduli space Γ\D Ξ We review some facts in [KU] that are necessary in the present article. Let w = 3, and h p,q = 1 (p + q = 3, p, q 0). Let H 0 = 4 j=1 Ze j, and e 3, e 1 0 = e 4, e 2 0 = 1. Let D be the corresponding classifying space of polarized Hodge structures, and Ď the compact dual. Let S = (square free positive integers), and let m S. Define N α, N β, N m End(H 0,, 0 ) as follows. N α (e 3 ) = e 1, N α (e j ) = 0 (j 3); N β (e 4 ) = e 3, N β (e 3 ) = e 1, N β (e 1 ) = e 2, N β (e 2 ) = 0; N m (e 1 ) = e 3, N m (e 4 ) = me 2, N m (e 2 ) = N m (e 3 ) = 0. Then the respective Hodge diamonds are 0 : N α : (3, 0) (2, 1) (1, 2) (0, 3) (2, 2) (3, 0) (0, 3) (1, 1) 2

3 N β : N m : (3, 3) (2, 2) (1, 1) (0, 0) (3, 1) (1, 3) (2, 0) (0, 2) Let σ α = R 0 N α, σ β = R 0 N β, and σ m = R 0 N m (m S). Proposition ([KU, 12.3]). Let Ξ = (rational nilpotent cones in g R of rank 1). Then Ξ = {Ad(g)σ σ = {0}, σ α, σ β, σ m (m S), g G Q }. This is a complete fan, i.e., if there exists Z Ď such that (σ, Z) is a nilpotent orbit, then σ Ξ. Here a pair (σ, Z) is a nilpotent orbit if Z = exp(cn)f, NF p F p 1 ( p) and exp(iyn)f D (y 0) hold for R 0 N = σ and F Z. In [KU], the fine moduli space Γ\D Ξ of polarized logarithmic Hodge structures is constructed. We briefly explain this according to the present case. Let Γ be a neat subgroup of G Z := Aut(H 0,, 0 ) of finite index. As a set, D Ξ = {(σ, Z) : nilpotent orbit σ Ξ, Z Ď}. Let σ Ξ, and let Γ(σ) = Γ exp(σ). If σ = {0}, then D {({0}, F ) F D} D Ξ. If σ {0}, then Γ(σ) N. Let γ be its generator and let N = log γ. Define E σ = and the map { } (q, F ) C Ď exp((2πi) 1 log(q)n)f D if q 0, and, exp(cn)f is a σ-nilpotent orbit if q = 0 { exp((2πi) 1 (1) E σ Γ(σ) gp log(q)n)f mod Γ(σ) gp if q 0, \D σ, (q, F ) (σ, exp(cn)f ) mod Γ(σ) gp if q = 0. Here Γ(σ) gp is the subgroup of Γ generated by Γ(σ). C Ď is obviously an analytic manifold. We endow this with the logarithmic structure M associated to the divisor {0} Ď. The strong topology of E σ in C Ď is defined as follows; a subset U of E σ 3

4 is open if, for any analytic space Y and any analytic morphsm f : Y C Ď such that f(y ) E σ, f 1 (U) is open in Y. By (1), the quotient topology, the sheaf O of local rings over C, and the logarithmic structure M are introduced on Γ(σ) gp \D σ. Introduce the corresponding structures O, M on Γ\D Ξ so that Γ(σ) gp \D σ Γ\D Ξ (σ Ξ) are local isomorphisms and form an open covering. Then, the resulting Γ\D Ξ is a logarithmic manifold, which is nearly a logarithmic analytic space but has slits at the boundaries. In fact, in the present case, we have in [KU, 12.3] that dim C D = 4, but dim C Γ(σ α ) gp \(D σα D) = 2, dim C Γ(σ β ) gp \(D σβ D) = 1 and dim C Γ(σ m ) gp \(D σm D) = Period map Let be a unit disc, = {0}, and h, z q = e 2πiz, the universal covering. Let ϕ : γ \D be a period map, where γ is the monodromy group generated by a unipotent element γ, and ϕ : h D a lifting. Let N = log γ. The map exp( zn) ϕ(z) from h to D drops down to the map ψ : D. Then the nilpotent orbit theorem of Schmid asserts that there exists the limit ψ(0), denoted by F, and that ϕ(z) and exp(zn)f are closing as Im z. Let σ = R 0 N. In our space γ \D σ we have, moreover, exp(zn)f ((σ, exp(σ C )F ) mod γ ) as Im z. Hence ϕ(q) ((σ, exp(σ C )F ) mod γ ) as q 0 in γ \D σ. (For details, see [KU].) Fix a point b P 1 {0, 1, } on the λ-plane, identify H 3 (W b, Z) = H 0, and let (1) Γ = Image(π 1 (P 1 {0, 1, }) G Z ). This Γ is not neat. In fact, the local monodromy around 0 is of order 5. Let (2) P 1 {0, 1, } Γ\D be the period map. Since K Wλ is trivial, the differential of (2) is injective everywhere. Endow P 1 with the logarithmic structure associated to the divisor {1, }. Then, by 1, 2 and [KU], (2) extends to a morphism (3) ϕ : P 1 Γ\D Ξ, 0 (point) mod Γ Γ\D, 1 (Ad(g)(σ α )-nilpotent orbit for some g G Q ) mod Γ, (Ad(g)(σ β )-nilpotent orbit for some g G Q ) mod Γ. of logarithmic ringed spaces (cf. [KU, (i)]). The image of the extended period map ϕ is an analytic curve, which is not affected by the slits of the space Γ\D Ξ. Let X = Γ\D Ξ. Let P 1 = 1, P = P 1, and let Q 1 = ϕ(p 1 ), Q = ϕ(p ) X. Then, by the observation of local monodromy and holomorphic 3-form basing on the descriptions in 1 and 2, we have (4) ϕ 1 (Q λ ) = {P λ } for λ = 1,. 4

5 4. Generic Torelli theorem We use the notation in the previous sections. Theorem. The period map ϕ in 3 (2) is the normalization of analytic spaces over its image. Proof. We use the fs logarithmic point P 1 and its image Q 1 at the boundaries ( 3). Since ϕ 1 (Q 1 ) = {P 1 } ( 3 (4)), it is enough to show that the local ramification index at Q 1 is one, i.e., Claim. (M X /O X ) Q 1 (M P 1/O P 1 ) P1 is surjective. Let N := N α be the nilpotent endomorphism introduced in 2. Let q be a local coordinate on a neighborhood U of P 1 = 1 in P 1, and let z = (2πi) 1 log q be a branch over U {P 1 }. Then exp( zn)e 3 = e 3 ze 1 is single-valued. Let ω( q) be a local frame of the locally free O P 1-module F 3. Write ω( q) = 4 j=1 a j( q)e j, and define t = a 1 ( q)/a 3 ( q). Then t = e 3, ω( q) 0 e 1, ω( q) 0 = exp( zn)e 3, ω( q) 0 + z e 1, ω( q) 0 e 1, ω( q) 0 = z + (single-valued holomorphic function in q). Let q = e 2πit. Then q = u q for some u O P 1,P 1. Let V be a neighborhood of Q 1 in X = Γ\D Ξ. We have a composite morphism of fs logarithmic local ringed spaces U V C, q q = e 2πi( a 1/a 3 ) (= u q). Hence the composite morphism (M P 1/O P 1 ) P1 (M X /O X ) Q 1 (M C /O C ) 0 of reduced logarithmic structures is an isomorphism. The claim follows. In fact, that is an isomorphism since the rank of (M X /O X ) Q 1 is one in the present case. 5. The second proof, and logarithmic generic Torelli theorem In this section, we give another proof of the generic Torelli theorem in 4 by using the fs logarithmic points P and Q at the boundaries. Since ϕ 1 (Q ) = {P } ( 3 (4)), it is enough to show the following: Claim. (M X /O X ) Q (M P 1/O P 1 ) P is surjective. Proof. Let N be the logarithm of the local monodromy at λ =. Let β 1, β 2, α 1, α 2 be the integral symplectic basis of H 0 given in [M1, Appendix C], and let g 0 = α 2, g 1 = 2α 1 + β 1, g 2 = β 2, g 3 = α 1 + β 1. Then g 3, g 2, g 1, g 0 is another integral symplectic basis such that g 0, g 1 is a good integral basis of Image(N 2 ) in the sense of [M1, 2, Appendix C]. Using the result there, we have (N(g 3 ), N(g 2 ), N(g 1 ), N(g 0 )) = (g 3, g 2, g 1, g 0 ). 5 9/ /2 25/

6 Let q be a local coordinate on a neighborhood U of P = in P 1, and let z = (2πi) 1 log q be a branch over U {P }. Then exp( zn)g 1 = g 1 zg 0 is single-valued. Let ω( q) be a local frame of the locally free O P 1-module F 3. Write ω( q) = 3 j=0 b j( q)g j, and define t = b 3 ( q)/b 2 ( q) and q = e 2πit. Then, as in 4, we see q = u q for some u O P 1,P, and the claim follows. Combining the results in 4 and 5, we have the following refinement. Theorem. The period map ϕ in 3 (3) is the normalization of fs logarithmic analytic spaces over its image. Here the normalization of the image of ϕ is endowed with the pull-back logarithmic structure. Note. Canonical coordinates in [M1], [M2], like q in the proofs of Claims in 4 and 5, can be understood more naturally in the context of PLH. Problem 1. Eliminate the possibility that the image ϕ(p 1 ) would have singularities in Theorems in 4 and 5. Problem 2. Describe the global monodromy group Γ explicitly. Generators of Γ are computed ([COGP], [M1, Appendix C]), and Γ is known to be Zariski dense in G Z = Sp(4, Z) ([COGP], [D, 13]). Acknowledgement. The author is grateful to Kazuya Kato and Chikara Nakayama. The author is also grateful to the referee for the useful comments. References [COGP] P. Candelas, C. de la Ossa, P. S. Green, and L. Parks, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Phys. Lett. B 258 (1991), [D] P. Deligne, Local behavior of Hodge structures at infinity, AMS/IP Studies in Advanced Mathematics 1 (1997), [KU] K. Kato and S. Usui, Classifying spaces of degenerating polarized Hodge structures, Ann. Math. Studies, Princeton Univ. Press, Princeton, 2009, pp [M1] D. Morrison, Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians, J. of AMS 6-1 (1993), [M2], Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218 (1993), [S] B. Szendröi, Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem; ibid II, Internat. J. Math. 11; Math. Proc. Cambridge Philos. Soc. 129 (2000), ; [V] C. Voisin, A generic Torelli theorem for the quintic threefold, New Trends in Algebraic Geometry, London Math. Soc., Lecture Note Ser. 264 (1999), Sampei USUI Graduate School of Science Osaka University Toyonaka, Osaka, , Japan 6

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