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. 2000 Mathematics Subject Classification. Primary 14C30; Secondary 14D07, 32G20. Partly supported by the Grant-in-Aid for Scientific Research (B) No.19340008, Japan Society for the Promotion of Science 1 Typeset by AMS-TEX
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
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
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) = 1. 3. 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, 4.3.1 (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
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 0 1 0 0 0 0 0 0 (N(g 3 ), N(g 2 ), N(g 1 ), N(g 0 )) = (g 3, g 2, g 1, g 0 ). 5 9/2 0 0 9/2 25/6 1 0 5
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), 21-74. [D] P. Deligne, Local behavior of Hodge structures at infinity, AMS/IP Studies in Advanced Mathematics 1 (1997), 683 699. [KU] K. Kato and S. Usui, Classifying spaces of degenerating polarized Hodge structures, Ann. Math. Studies, Princeton Univ. Press, Princeton, 2009, pp. 288. [M1] D. Morrison, Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians, J. of AMS 6-1 (1993), 223 247. [M2], Compactifications of moduli spaces inspired by mirror symmetry, Astérisque 218 (1993), 243 271. [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), 449 459; 193 204. [V] C. Voisin, A generic Torelli theorem for the quintic threefold, New Trends in Algebraic Geometry, London Math. Soc., Lecture Note Ser. 264 (1999), 425 463. Sampei USUI Graduate School of Science Osaka University Toyonaka, Osaka, 560-0043, Japan usui@math.sci.osaka-u.ac.jp 6