Degenerating Hodge structure of one-parameter family of Calabi-Yau threefolds
aa r X i v : . [ m a t h . AG ] S e p DEGENERATING HODGE STRUCTURE OF ONE-PARAMETERFAMILY OF CALABI–YAU THREEFOLDS
TATSUKI HAYAMA ATSUSHI KANAZAWA
Abstract.
To a one-parameter family of Calabi–Yau threefolds, we can associatethe extended period map by the log Hodge theory of Kato and Usui. In the presentpaper, we study the image of a maximally unipotent monodromy point under theextended period map. As an application, we prove the generic Torelli theorem for alarge class of one-parameter families of Calabi–Yau threefolds. Introduction
The present paper is concerned with the limit mixed Hodge structure around amaximally unipotent monodromy (MUM) point of a one-parameter family of Calabi–Yau threefolds whose Kodaira spencer map is generically an isomorphism. For such afamily, the period domain for the Hodge structures and the limit mixed Hodge struc-tures (LMHSs) were previously studied by [KU] and [GGK]. The starting point of thepresent work is the theory of normalization of the LMHS around a MUM point devel-oped in [GGK]. MUM points play a central role in mirror symmetry [CdOGP, Mor].Mirror symmetry is a duality between complex geometry and symplectic geometryamong several Calabi–Yau threefolds. It expects that each MUM point of a family ofCalabi–Yau threefolds corresponds to a mirror
Calabi–Yau threefold of the family. Fora large class of Calabi–Yau threefolds, we observe that the normalization of the LMHSreflects the topological invariants of mirror Calabi–Yau threefold.The idea of this paper is to investigate the degenerating Hodge structures in theframework of the log Hodge theory [KU]. An advantage of our approach is that, byslightly extending the domain and range of the period map, we have a better controlof the period map. As an application, we prove the generic Torelli theorem for a largeclass of one-parameter families of Calabi–Yau threefolds (Theorem 4.3). The genericTorelli theorem was confirmed for the mirror families of Calabi–Yau hypersurfacesin weighted projective spaces by Usui [Usu2] and Shirakawa [Shi]. Our study is aslight refinement of their technique but can be applied to a wider class of Calabi–Yauthreefolds. The result is particularly interesting when a family has multiple MUMpoints and also works for new examples beyond toric geometry such as the mirrorfamily of the Pfaffian–Grassmann Calabi–Yau threefolds (Section 5.2).The layout is this paper is as follows. Section 2 covers some basics of Hodge theoryand the compactification of period domains. This chapter also serves to set notations.Section 3 begins with a review of the normalization of a LMHS obtained in [GGK].We then study the LMHSs using the normalization. Section 4 is devoted to the generic
Mathematics Subject Classification.
Key words and phrases. (log) Hodge theory, Calabi–Yau, Torelli problem, mirror symmetry.
Torelli for a one-parameter family of Calabi–Yau threefolds. Section 5 briefly reviewsmirror symmetry of Calabi–Yau threefolds with a particular emphasis on a monodromytransformation around a MUM point. We also discuss some suggestive examples ofCalabi–Yau threefolds with two MUM points.
Acknowledgement.
It is a pleasure to record our thanks to C. Nakayama for usefulcomments on the preliminary version of the present paper. This research was partiallysupported by Research Fund for International Young Scientists NSFC 11350110209(Hayama).2.
LMHS and partial compactification of period domain
Hodge structure and period domain.
In this section, we recall the definitionof polarized Hodge structures and of period domains. A Hodge structure of weight w with Hodge numbers ( h p,q ) p,q is a pair ( H, F ) consisting of a free Z -module H ofrank P p,q h p,q and a decreasing filtration F on H C := H ⊗ C satisfying the followingconditions:(1) dim C F p = P r ≥ p h r,w − r for all p ;(2) H C = L p + q = w H p,q ( H p,q := F p ∩ F w − p ) . For Hodge structures (
H, F ) and ( H ′ , F ′ ), homomorphism f : H → H ′ is a ( r, r )-morphism of Hodge structures if f ( F p ) ⊂ F ′ p + r and f ( ¯ F p ) ⊂ ¯ F ′ p + r .A polarization h∗ , ∗∗i for a Hodge structure ( H, F ) of weight w is a non-degeneratebilinear form on H , symmetric if w is even and skew-symmetric if w is odd, satisfyingthe following conditions:(3) h F p , F q i = 0 for p + q > w ;(4) i p − q h v, ¯ v i > = v ∈ H p,q .We fix a polarized Hodge structure ( H , F , h∗ , ∗∗i ) of weight w with Hodge numbers( h p,q ) p,q . We define the period domain D which parametrizes all Hodge structures ofthis type by D := (cid:26) F ( H , F, h∗ , ∗∗i ) is a polarized Hodge structureof weight w with Hodge numbers ( h p,q ) p,q (cid:27) . The compact dual ˇ D of D isˇ D := (cid:8) F ( H , F, h∗ , ∗∗i ) satisfies the above (1)–(3) (cid:9) . Let G A := Aut ( H ⊗ A, h∗ , ∗∗i ) for a Z -module A . Then, G R acts transitively on D and G C acts transitively on ˇ D .Let S be a complex manifold. A variation of Hodge structure (VHS) over S is apair ( H , F ) consisting of a Z -local system and a filtration of H ⊗ O S over S satisfyingthe following conditions:(1) The fiber ( H s , F s ) at s ∈ S is a Hodge structure;(2) ∇F p ⊂ F p − ⊗ Ω S for the connection ∇ := id ⊗ d : H ⊗ O S → H ⊗ Ω S .A polarization for a VHS is a bilinear form on the local system which defines a polar-ization on each fiber. In this paper, a VHS is always assumed to be polarized. EGENERATING HODGE STRUCTURE OF CALABI–YAU THREEFOLDS 3
For a VHS over S , we fix a base point s ∈ S . Let D be the period domain for theHodge structure at s . We then have the period map φ : S → Γ \ D via s F s , whereΓ is the monodromy group.2.2. Limit mixed Hodge structure.
Let ¯ S be a smooth compactification of S suchthat ¯ S − S is a normal crossing divisor. For each p ∈ ¯ S − S , there exists a neighbourhood V of around p in ¯ S such that U := V ∩ S ∼ = (∆ ∗ ) m × ∆ n − m where ∆ is the unit disk.We can lift the period map to ˜ φ : ˜ U → D , where ˜ U → U is the universal covering map.Under the identification ˜ U ∼ = H m × ∆ n − m , the covering map ˜ U → U is given by( z , . . . , z n ) (exp (2 πiz ) , . . . , exp (2 πiz m ) , z m +1 , . . . , z n ) . Let T , . . . , T m be a generator of the monodromy around p such that˜ φ ( · · · , z j + 1 , · · · ) = T j ˜ φ ( · · · , z j , · · · ) . Let us assume T j is unipotent. Then N j = log T j is nilpotent in the Lie algebra g Q , and N , . . . , N m are commutating with each other. We define ˜ ψ : ˜ U → ˇ D by z exp ( − P j z j N j ) φ ( z ). Since ˜ ψ ( · · · , z j + 1 , · · · ) = ψ ( · · · , z j , · · · ), ˜ ψ descends to ψ : U → ˇ D , which admits a unique extension to ψ : ∆ n → ˇ D by [Sch]. We call F ∞ := ψ ( ) ∈ ˇ D the limit Hodge filtration (LHF). Remark 2.1.
The LHF is not uniquely determined by a VHS. In fact, for f j ∈ O ∆ ,we obtain new coordinates(exp (2 πif ( z )) z , . . . , exp (2 πif n ( z n )) z n ) , with respect to which, the LHF is given by exp ( − P f j (0) N j ) F ∞ . Moreover, N , . . . , N m depend also on the choice of coordinates. However the nilpotent orbit (to be discussedin the next subsection) is determined by the VHS.Let N := N + · · · + N m . By [Sch], we have an increasing filtration W ( N ) of H R , := H ⊗ R . Denoting by W the shifted filtration of W ( N ) by the weight w , thepair ( W, F ∞ ) has the following properties:(1) the graded quotient (Gr Wk , F ∞ Gr Wk, C ) is a Hodge structure of weight k ;(2) N defines a ( − , − Wk , F ∞ Gr Wk, C ) → (Gr Wk − , F ∞ Gr Wk − , C ) ofHodge structures;(3) N k : (Gr Ww + k , F ∞ Gr Ww + k, C ) → (Gr Ww − k , F ∞ Gr Ww − k, C ) is isomorphism;(4) h∗ , N k ( ∗∗ ) i gives a polarization on (Gr Ww + k , F ∞ Gr Ww + k, C ).The pair ( W, F ∞ ) is called the limit mixed Hodge structure (LMHS).2.3. Partial compactification of period domain.
We call σ ⊂ g R a nilpotent coneif it satisfies the following conditions:(1) σ is a closed cone generated by finitely many elements of g Q ;(2) N ∈ σ is a nilpotent as an endmorphism of H R ;(3) N N ′ = N ′ N for any N, N ′ ∈ σ .For A = R , C , we denote by σ A the A -linear span of σ in g A . TATSUKI HAYAMA ATSUSHI KANAZAWA
Definition 2.2.
Let σ = P nj =1 R ≥ N j be a nilpotent cone and F ∈ ˇ D . Then the pairconsisting of σ and exp ( σ C ) F ⊂ ˇ D is called a nilpotent orbit if it satisfies the followingconditions:(1) exp ( P j iy j N j ) F ∈ D for all y j ≫ N F p ⊂ F p − for all p ∈ Z and for all N ∈ σ .The data ( N , . . . , N m , F ∞ ) given in the previous section generates a nilpotent orbit.Moreover, any nilpotent orbit generates a LMHS. In fact, W ( N ) = W ( N ′ ) for any N and N ′ in the relative interior of σ (see [CK] for example), and the pair ( W ( N )[ w ] , F ′ )is a LMHS for any F ′ ∈ exp ( σ C ) F .Let Σ be a fan consisting of nilpotent cones. We define the set of nilpotent orbits D Σ := { ( σ, Z ) | σ ∈ Σ , ( σ, Z ) is a nilpotent orbit } . For a nilpotent cone σ , the set of faces of σ is a fan, and we abbreviate D { faces of σ } as D σ . Let Γ be a subgroup of G Z and Σ a fan of nilpotent cones. We say Γ iscompatible with Σ if Ad( γ )( σ ) ∈ Σ for all γ ∈ Γ and for all σ ∈ Σ. Then Γ acts on D Σ if Γ is compatible with Σ. Moreover we say Γ is strongly compatible with Σ if it iscompatible with Σ and for all σ ∈ Σ there exists γ , . . . , γ n ∈ Γ( σ ) := Γ ∩ exp ( σ ) suchthat σ = P j R ≥ log ( γ j ).We consider the geometric structure of Γ( σ ) gp \ D σ in the case where σ has rank 1(we will discuss this case in the next section). For a nilpotent cone σ = R ≥ N and the Z -subgroup Γ( σ ) gp = e Z N , we have the partial compactification Γ( σ ) gp \ D σ . We nowshow its geometric structure following the exposition of [KU]. Let us define C × ˇ D ⊃ E σ := (cid:26) ( s, F ) exp ( ℓ ( s ) N ) F ∈ D if s = 0 , ( σ, exp ( C N ) F ) is a nilpotent orbit if s = 0 (cid:27) , where ℓ ( s ) is a branch of log( s ) / πi . Here C is endowed with a log structure as a toricvariety and C × ˇ D is a logarithmic analytic space. By [KU, Theorem A], the subspace E σ is a log manifold with the map E σ → Γ( σ ) gp \ D σ , ( s, F ) ( exp ( ℓ ( s ) N ) F if s = 0 , ( σ, exp ( σ C ) F ) if s = 0 . The geometric structure of Γ( σ ) gp \ D σ is induced by the map above, which is a C -torsor,i.e. Γ( σ ) gp \ D σ ∼ = E σ / C . Theorem 2.3 ([KU, Theorem A]) . Let Σ be a fan of nilpotent cones and let Γ be asubgroup of G Z which is strongly compatible with Σ . Then the following hold:(1) If Γ is neat (i.e., the subgroup of G m ( C ) generated by all the eigenvalues of all γ ∈ Γ is torsion free), then Γ \ D Σ is a logarithmic manifold.(2) The map Γ( σ ) gp \ D σ → Γ \ D Σ is open and locally an isomorphism of logarith-mic manifolds. Logarithmic manifolds are a generalization of analytic spaces introduced in [KU]. Alogarithmic manifold is a subspace of a logarithmic analytic space, whose topology isinduced by the strong topology.For a VHS, locally the period map U → Γ \ D can be extended to the map V → Γ \ D σ .We assume that there exists a fan Σ which includes all nilpotent cones arises from all EGENERATING HODGE STRUCTURE OF CALABI–YAU THREEFOLDS 5 local monodromies arising from ¯ S − S . Note that a construction of fans is still anopen problem in higher dimensional case (cf. [Usu1, § S → Γ \ D Σ . Although the target space is not an analytic space, we havethe following result: Theorem 2.4 ([Usu1, § . The image of ¯ S is a compact analytic space if ¯ S is compact. Moreover, the map is also analytic since the category of logarithmic analytic spacesis a full subcategory of B (log) whose objects are logarithmic manifolds ([KU, § The case where rank H = 4 with h , = h , = 1In this section, we consider Hodge structures with Hodge numbers h p,q = 1 if p + q = 3 , p, q ≥ , and h p,q = 0 otherwise . In this case, the partial compactifications of the period domain D are well-studied in[KU, § H = 4 and G Z = Sp (2 , Z ). The perioddomain D is the flag domain Sp (2 , R ) / ( U (1) × U (1)) of dimension 4. If σ generates anilpotent orbit, then σ = R ≥ N and N is one of the following types:(1) N = 0 and dim Im N = 1;(2) N = 0 and dim Im N = 2;(3) N = 0 and N = 0.The case (3) is called maximally unipotent monodromy (MUM). The goal of this sectionis to analyze MUM and their LMHS in detail.3.1. Normalization of monodromy matrix.
Let T ∈ G Z be a unipotent elementsuch that log T = N is a MUM element. The monodromy weight filtration W = W ( N )[3] is { } = W − ⊂ W = W ⊂ W = W ⊂ W = W ⊂ W = H Q with the graded quotient Gr W p ∼ = Q for 0 ≤ p ≤
3. The pair (Gr W p , F Gr W p, C ) is theTate Hodge structure of weight 2 p if ( N, F ) generates a nilpotent orbit. The LMHScondition induces Gr W N −→ Gr W N −→ Gr W N −→ Gr W , where each N : Gr W p → Gr W p − is an isomorphism of Hodge structures.By [GGK, Lemma (I.B.1) & (I.B.3)], we may choose a symplectic basis e , . . . , e of H Z which satisfies W p = span R { e j | ≤ j ≤ p } (0 ≤ p ≤ , ( h e i , e j i ) i,j = − − . (3.1)By [GGK, (I.B.7)], with respect to this basis, N is of the form N = a e b f e − a . (3.2) TATSUKI HAYAMA ATSUSHI KANAZAWA for some a, b, e, f ∈ Q . The polarization condition of a LMHS yields inequalities: i h e , N e i = a b > , i h e , N e i = b > . Moreover, we have T = e N = a e + ab b f − a b e − ab − a ∈ G Z , which shows that a, b, e ± ab , f − a b ∈ Z . (3.3)The symplectic basis e , . . . , e with the properties (3.1) is not unique; for any A ∈ G Z ( W ) := Aut( H, h∗ , ∗∗i , W ), the new basis Ae , . . . , Ae will do. Any A ∈ G Z ( W ) isrepresented by a lower triangular matrix with 1’s on the diagonal, and thus written as A = e M with M = p r q s r − p where p, q, r, s satisfy the same condition as a, b, e, f in (3.3). Under the transformation N → Ad ( A ) N , the entries a, b, e, f change as follows: a a, b b, e e − bp + aq, (3.4) f f − ep + bp − apq + 2 ar. Proposition 3.1 ([GGK, Proposition I.B.10]) . Under the action of G Z ( W ) , b is in-variant, and a is invariant up to ± . Moreover, for m = gcd( a, b ) , [ e ] ∈ Z /m Z isinvariant if ab is even, and [2 e ] ∈ Z / m Z is invariant if ab is odd. Period map around boundary point.
Let ( H , F ) be a VHS over ∆ ∗ withmonodromy N of the form (3.2). Hereafter, we fix such a presentation with a, b, e, f .For the monodromy group Γ = h T i , we have the period map φ : ∆ ∗ → Γ \ D and itslifting ˜ φ : H → D . Now the new map exp ( − zN ) ˜ φ ( z ) descends to a holomorphic mapover ∆, we denote it by ψ ( s ) where s = exp (2 πiz ). Here F ∞ = ψ (0) is the LHF andthen F ∞ ∩ F ∞ (mod W ) is generated by e . We may choose a generator e + π e + π e + π e of the subspace F ∞ for some π , π , π ∈ C . Then the subspace F ψ ( s ) corresponding to ψ ( s ) ∈ ˇ D is generated by ψ ( s ) e + ψ ( s ) e + ψ ( s ) e + ψ ( s ) e where ψ i for 0 ≤ i ≤ ψ (0) = 1 and ψ i (0) = π i for 0 ≤ i ≤
2. By untwisting ψ , a local frame of the subspace F spanned EGENERATING HODGE STRUCTURE OF CALABI–YAU THREEFOLDS 7 by the period is given by ω ( s ) ω ( s ) ω ( s ) ω ( s ) := exp ( zN ) ψ ( s ) ψ ( s ) ψ ( s ) ψ ( s ) . Here ω ( s ) = ψ ( s ) and ω ( s ) = aω ( s ) log ( s )2 πi + ψ ( s ) . Therefore q ( s ) := exp (cid:18) πi ω ( s ) aω ( s ) (cid:19) = exp (cid:18) πi ψ ( s ) aψ ( s ) (cid:19) s (3.5)defines a new coordinate of ∆, which is known as the mirror map in mirror symmetry.By § φ : ∆ → Γ \ D σ . As we saw in § φ (∆) ⊂ Γ \ D σ is induced by the C -torsor E σ → Γ( σ ) gp \ D σ . Lemma 3.2.
The period map φ : ∆ → φ (∆) is an isomorphism as analytic spaces.Proof. The coordinate q gives a local section of the C -torsor E σ → Γ( σ ) gp \ D σ restrictedon the image φ (∆). In fact, we can define ρ : φ (∆) → E σ ; φ ( s ) ( q ( s ) , exp (cid:18) − ψ ( s ) aψ ( s ) N (cid:19) ψ ( s )) . This induces isomorphsims ∆ ∼ = ρ ( φ (∆)) ∼ = φ (∆) as analytic spaces. (cid:3) Moreover the map ∆ → φ (∆) induces an isomorphism of log structures in a mannersimilar to [Usu2, § Normalization of LHF.
Let ( σ, exp ( σ C ) F ) be a nilpotent orbit, i.e. ( W, F ) isa LMHS. We show that we have a canonical choice of F which has a normalized formwith respect to the symplectic basis e , . . . , e .For the LMHS ( W, F ), we have the Deligne decomposition H C = L ≤ j ≤ I j,j sothat W p = M k ≤ p I k,k , F p = M k ≥ p I k,k for 0 ≤ p ≤
3. We can take a unique generator v p ∈ I p,p such that [ v p ] = [ e p ] in Gr W p, C .By [GGK, Proposition (I.C.2)], with repect to the basis v , . . . , v , the matrix N is isof the form N ω = a b − a . TATSUKI HAYAMA ATSUSHI KANAZAWA
Moreover, by [GGK, Proposition (I.C.4)], the period matrix of F is then written as (cid:2) ω ω ω ω (cid:3) = π π ba π + ea π ea π + fa − π − π . (3.6)By multiplying exp (cid:0) − π a N (cid:1) , we may further choose F so that π = 0. If π = 0, bythe second bilinear relation [GGK, (I.C.10)], the period matrix (3.6) is written as f / a e/a π f / a . (3.7)Here the values f / a , e/a and π correspond to the extension class of the LMHS [GGK, § I.C]. We observe that the boundary component D σ \ D ∼ = C is parametrized by π .Recall that the LHF depends on the choice of coordinates for a VHS (Remark 2.1).If we use the canonical coordinate q of (3.5), the normalized period matrix takes theform of (3.7). In this case, the LHF is given by F ∞ = lim z → exp (cid:18) − log z πi N (cid:19) F z = f / aπ . Generic Torelli theorem
The goal of this section is to show the generic Torelli theorem for one-parameterfamilies of Calabi–Yau threefolds.4.1.
Degree of period map.
Let X → S be a one-parameter family of Calabi–Yau threefolds. Given a smooth compactification ¯ S of S so that ¯ S − S consists offinite points. Let φ : S → Γ \ D be the period map associated to the VHS on H := H ( X, Z ) / Tor for a fixed smooth fiber X . Although the monodromy group Γ is notnecessary a neat subgroup of G Z , there always exists a neat subgroup Γ ′ of Γ of finiteindex. In this situation, we have a lifting ˜ φ of φ ˜ S (cid:15) (cid:15) ˜ φ / / Γ ′ \ D (cid:15) (cid:15) S φ / / Γ \ D where ˜ S is a finite covering of S . To show the generic Torelli theorem for φ , it sufficesto show the theorem for the lifting ˜ φ : ˜ S → Γ ′ \ D . Therefore we henceforth assumethat Γ is neat. We also assume that the Kodaira–Spencer map is an isomorphism onthe base curve S to exclude trivial cases [BG]. To summarize, we assume that:(1) the monodromy group Γ is neat;(2) the Kodaira–Spencer map is an isomorphism on S . EGENERATING HODGE STRUCTURE OF CALABI–YAU THREEFOLDS 9
Let σ , . . . , σ n be the nilpotent cones which arise from the monodromies around¯ S − S . We define a fan Ξ in g R byΞ := Γ · σ ∪ · · · ∪ Γ · σ n ∪ { } . The fan Ξ is strongly compatible with Γ. By [KU], the partial compactification Γ \ D Ξ of Γ \ D is a logarithmic manifold and the period map extends to φ : ¯ S → Γ \ D Ξ . ByTheorem 2.4, the image φ ( ¯ S ) and the map φ is analytic. Moreover φ is proper andthus a finite covering map. Proposition 4.1.
Let p ∈ ¯ S − S be a MUM point. If φ − ( φ ( p )) = { p } , then the map φ : ¯ S → φ ( ¯ S ) is of degree .Proof. By Lemma 3.2, a disk ∆ p around p is isomorphic to the image φ (∆ p ). Since φ ( p ) is not a branch point, the map φ must be of degree 1. (cid:3) For p ∈ ¯ S − S , the image φ ( p ) is the nilpotent orbit determined by the local mon-odromy and the LHF around p . If the family has only one MUM, we clearly have φ − ( φ ( p )) = p , therefore the generic Torelli theorem holds by Proposition 4.1.To show the generic Torelli theorem for a family with multi MUMs, it suffices toshow that there exists a MUM point p such that for any other MUM point p thecondition φ ( p ) = φ ( p ) holds. Let N j be the logarithm of the local monodromy around p j , and let F j be the LHF. Then φ ( p j ) = ( σ j , exp ( σ j, C ) F j ) mod Γwhere σ j = R ≥ N j . As discussed in the previous section, we have the normalizedmatrix (3.2) of N j determined by a j , b j , e j , f j ∈ Q and the canonical choice (3.7) of F j determined by π j ∈ C using a symplectic basis e j , . . . , e j satisfying (3.1). Proposition 4.2. If b = b or π − π Q , then g ( σ , exp ( σ , C ) F ) = ( σ , exp ( σ , C ) F ) for any g ∈ G Z . In other words, we have φ ( p ) = φ ( p ) .Proof. We define g ∈ G Z by e k e k . Then Ad ( g ) N is written as the normal-ized matrices determined by a , b , e , f using the symplectic basis e , . . . , e , andAd ( g ) W ( N ) = W ( N ). We put W = W ( N ). If b = b , there does not exists h ∈ G Z ( W ) such that Ad ( hg ) N ∈ σ since b is invariant for the action of G Z ( W ) byProposition 3.1. Then Ad( γ ) σ = σ mod Γ for any γ ∈ G Z .Now suppose that b = b and that there exists h ∈ G Z ( W ) such that Ad ( hg ) σ = σ . The filtration gF is written as the normalized period matrix determined by π using e , . . . , e . Then the period matrix of the canonical choice in hg exp ( σ , C ) F = exp ( σ , C ) hgF is determined by π + λ with λ ∈ Q since h ∈ G Z and N ∈ g Q . Since π − π Q , weconclude that hg exp ( σ , C ) F = exp ( σ , C ) F . Therefore there does not exist γ ∈ G Z such that Ad( γ ) σ = σ and γ exp ( σ , C ) F = exp ( σ , C ) F . (cid:3) Theorem 4.3 (Generic Torelli Theorem) . Let X → S be a one-parameter family ofCalabi–Yau threefolds with a MUM point. Assume that there exists a MUM point p such that for any other MUM point p ∈ ¯ S − S the condition b = b or π − π Q holds. Then the map φ : ¯ S → φ ( ¯ S ) is the normalization of φ ( ¯ S ) . Proof.
The assertion readily follows from the combination of Proposition 4.2 andProposition 4.1. (cid:3)
Theorem 4.3 in particular applies to the families of Calabi–Yau threefolds with ex-actly one MUM point. Such examples include almost all one-parameter mirror familiesof complete intersection Calabi–Yau threefold in weighted projective spaces and ho-mogeneous spaces (see [vEvS] for more details). We will discuss some Calabi–Yauthreefolds with two MUM points in the next section.5.
Mirror Symmetry
In this section, we see that the Hodge theoretic invariants b and π appear in theframework of mirror symmetry. Mirror symmetry claims, given a family of Calabi–Yau threefolds X → B with a MUM point, there exists another family X ∨ → B ∨ of Calabi–Yau threefolds such that some Hodge theoretic invariants of X around theMUM point and symplectic invariants of X ∨ are equivalent in a certain way. Here X and X ∨ are generic members of X → B and X ∨ → B ∨ respectively. Simply put,mirror symmetry interchanges the complex geometry of one Calabi–Yau threefold X with the symplectic geometry of another, called a mirror threefold X ∨ , and such acorrespondence depends on the choice of a MUM point. We should think that eachMUM point corresponds to a mirror Calabi–Yau threefold. If a family of Calabi–Yauthreefolds X → B has several MUM points, there should be several mirror threefolds.We refer the reader to [CK2] for a detailed treatment of mirror symmetry.In this section, we investigate the interplay between the LMHS at a MUM point andthe corresponding mirror threefold. For the sake of convenience, we restrict ourselvesto one-parameter models, that is, the case when h , ( X ) = h , ( X ∨ ) = 1. Since thecomplex moduli space of X is 1-dimensional, X comes with a family X → S over apunctured curve S . Since mirror symmetry is a statement about a MUM point of S ,we assume that such a point corresponding to X ∨ is chosen.We denote by Ω z a holomorphic 3-form on the mirror Calabi–Yau threefold over apoint z ∈ S of the family X → S . On an open disk ∆ around the MUM point z = 0,there exist solutions ω , . . . , ω of the Picard–Fuchs equation of the following form: ω ( z ) = ψ ( z ) = 1 + O ( z ) , (5.1) 2 πiω ( z ) = ψ ( z ) log( z ) + ψ ( z ) , (2 πi ) ω ( z ) = 2 ψ ( z ) log( z ) + ψ ( z ) log( z ) + ψ ( z ) , (2 πi ) ω ( z ) = 3 ψ ( z ) log( z ) + 3 ψ ( z ) log( z ) + ψ ( z ) log( z ) + ψ ( z ) , where ψ i is a power series in z such that ψ j (0) = 0 for 0 ≤ j ≤
2. An importantobservation is that the local monodromy group at each MUM point is controlled by thetopological invariants of the corresponding mirror threefold as follows. Let z ∈ ∆ ∗ bea reference point. We equip H ( X z , Z ) / Tor with the standard symplectic form (3.1).Then mirror symmetry predicts the existence of a symplectic basis A , A , B , B of EGENERATING HODGE STRUCTURE OF CALABI–YAU THREEFOLDS 11 H ( X z , Z ) / Tor such that R A Ω z R A Ω z R B Ω z R B Ω z = − c ( X ∨ ) · H λ deg X ∨ ζ (3) χ ( X ∨ )(2 πi ) − c ( X ∨ ) · H − deg X ∨ ω ( z ) ω ( z ) ω ( z ) ω ( z ) , (5.2)where λ = 1 if deg X ∨ is even and = − / Proposition 5.1.
Assume the relation (5.2). Then the normalized matrix (3.2) of N = log T and the normalized period matrix (3.7) of the LHF are determined by a = 1 , b = deg X ∨ , e = ( b is even − / b is odd , f = − c ( X ∨ ) · H , π = χ ( X ∨ ) ζ (3)(2 πi ) . Proof.
The monodromy matrix of [ ω , ω , ω , ω ] T is readily available. We rewrite itwith respect to the symplectic basis to obtain N . The LHF is obtained in a similarmanner. (cid:3) Therefore we see that the LMHS reflects the topological invariants of the mirrorthreefold. With this topological interpretation, the integrality condition (3.3) is ex-plained by the Riemann–Roch theorem.
Example 5.1.
For the mirror family of a quintic threefold X ∨ , a, b, e, f and π aredetermined in [GGK, (III.A)]: a = − , b = 5 , e = 11 / , f = − / , π = − ζ (3)(2 πi ) . Here deg X ∨ = 5, c ( X ∨ ) · H = 50 and χ ( X ∨ ) = − a and e into 1 and − / Multiple Mirror Symmetry.
We find Theorem 4.3 and Proposition 5.1 partic-ularly interesting when a family of Calabi–Yau threefolds has two MUM points. Sucha family is of considerable interest because the existence of two MUMs suggests theexistence of two mirror partners. The first concrete example of such a multiple mirrorphenomenon was discovered in [Rod]. Recently, a few more examples were constructed[Kan, HT, Miu]. In this section, we investigate two examples of Calabi–Yau threefoldswith two MUM points.5.2.
Grassmannian and Pfaffian Calabi–Yau threefold.
The Grassmannian Gr(2 , P . The complete in-tersection of 7 hyperplanes sections of this embedding yields a Calabi–Yau threefold X ∨ := Gr(2 , ∩ (1 ) ⊂ P with h , = 1.Let N be a 7 × N = ( n ij ) with [ n ij ] i Conifolds Figure 1. Moduli space P of X = Y deg X ∨ = 42 = deg Y ∨ = 14, Theorem 4.3 applies and the geneic torelli theorem holdsfor the mirror family. An identical argument applies to, for example, the Calabi–Yauthreefolds constructed in [HT, Miu].5.3. Complete Intersection Gr(2 , ∩ Gr(2 , ⊂ P . Let i , i : Gr(2 , ֒ → P be generic Pl¨ucker embeddings. It is shown in [Kan] that the complete intersection X ∨ := i (Gr(2 , ∩ i (Gr(2 , h , = 1. In [Miu2], amirror family X → P of X ∨ was constructed via a toric degeneration of Gr(2 , 5) to aHibi toric variety. An interesting observation is that the mirror family has exactly twoMUM points and both of them correspond to X ∨ . The corresponding Hodge theoreticinvariants around the MUM points are identical and Theorem 4.3 cannot be appliedin this case. For the reader’s convenience, we write down the Picard–Fuchs operatorwith the Euler differential Θ := z∂ z :Θ − z (124Θ + 242Θ + 187Θ + 66Θ + 9)+ z (123Θ − − − − z (123Θ + 738Θ + 689Θ + 210Θ + 12) − z (+124Θ + 254Θ + 205Θ + 78Θ + 12) + z (Θ + 1) References [BCFKS] V.V. Batyrev, I. Ciocan-Fontanine, B. Kim and D. van Straten, Conifold transitions andmirror symmetry for Calabi–Yau complete intersections in Grassmannians , Nuclear Phys. B 514(1998), no.3, 640-666.[BG] R. Bryant and P. Griffiths, Some observations on the infinitesimal period relations for regularthreefolds with trivial canonical bundle , Arithmetic and geometry, Vol. II, 77–102,[CdOGP] P. Candelas, X.C. de la Ossa, P.S. Green, and L. Parkes, A pair of Calabi–Yau manifoldsas an exactly solvable superconformal theory , Nuclear Phys. B 359 (1991), no. 1, 21-74.[CK] E. Cattani and A. Kaplan, Polarized mixed Hodge structures and the local monodromy of avariation of Hodge structure , Invent. Math. (1982), no. 1, 101–115.[CK2] D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys andMonographs, 68. American Mathematical Society, Providence, RI, 1999. EGENERATING HODGE STRUCTURE OF CALABI–YAU THREEFOLDS 13 [vEvS] C. van Enckevort and D. van Straten, Monodromy calculations of fourth order equations ofCalabi–Yau type , Mirror symmetry V, 539-559, AMS/IP Stud. Adv. Math., 38, Amer. Math. Soc.Province, RI, 2006.[GGK] M. Green, P. Griffiths and M. Kerr, Neron models and boundary components for degenerationsof Hodge structures of mirror quintic type , in Curves and Abelian Varieties (V. Alexeev, Ed.),Contemp. Math 465 (2007), AMS, 71-145.[HT] S. Hosono and H. Takagi, Mirror symmetry and projective geometry of Reye congruences I , toappear in J. Alg. Geom.[Kan] A. Kanazawa, Pfaffian Calabi–Yau Threefolds and Mirror Symmetry , Comm. Num. Th. Phys,Vol. 6, Number 3, 661-696, 2012.[KU] K. Kato and S. Usui, Classifying space of degenerating polarized Hodge structures , Annals ofMathematics Studies, . Princeton University Press, 2009.[Miu] M. Miura, Minuscule Schubert varieties and mirror symmetry , arXiv:1301.7632.[Miu2] M. Miura, Hibi toric varieties and mirror symmetry , Ph.D. thesis, University of Tokyo, 2013.[Mor] D. Morrison, Mirror symmetry and rational curves on quintic threefolds: A guide for mathe-maticians, JAMS 6 (1993) 223-247.[Rod] E. Rødland, The Pfaffian Calabi–Yau, its Mirror, and their Link to the Grassmannian Gr (2 , Variation of Hodge structure: the singularities of the period mapping , Invent. Math.22 (1973), 211-319.[Shi] K. Shirakawa, Generic Torelli theorem for one-parameter mirror families to weighted hypersur-faces , Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 10, 167-170.[Tjo] E. Tjøtta, Quantum Cohomology of a Pfaffian Calabi–Yau Variety : Verifying Mirror SymmetryPredictions, Compositio Math. 126 (2001), no. 1, 79-89.[Usu1] S. Usui, Images of extended period maps , J. Alg. Geom. 15 (2006), no. 4, 603-621.[Usu2] S. Usui, Generic Torelli theorem for quintic-mirror family , Proc. Japan Acad. Ser. A Math.Sci. 84 (2008), no.8, 143-146. Mathematical Science Center, Tsinghua University,Haidian District, Beijing 100084, China. [email protected]