MIRROR SYMMETRY and RIGID STRUCTURES of GENERALIZED K3 SURFACES3 with the Classical Formulation for K3 Surfaces

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po and Gr4 (H∗(M, R)) ∼= O(4, 20)/(SO(4) O(20)) respectively. Then mirror symmetry is realized as a certain discrete× group action on these moduli spaces. On the other hand, the moduli space MHK of B-field shifts of hy- perKähler metrics can be identified with an open dense subset of M(4,4) by the period map perHK. Inspired by this fact, in the beautiful article [5], Huy- brechts showed that the counterpart of M(2,2) is given by the moduli space MgK3 of generalized K3 surfaces, which are the K3 version of the generalized Calabi–Yau (CY) structures introduced by Hitchin [4]. The moduli space M H2(M, R) of B-field shifts of K3 surfaces endowed with a Ricci-flat K3 × metric are naturally contained in MgK3 of real codimension 2.

ι pergK3 M H2(M, R) / M / Grpo (H (M, R)) = M K3 × gK3 2,2 ∗ (2,2)

perHK / po MHK Gr4 (H∗(M, R)) = M(4,4)

Then, as Huybrechts pointed out, it is natural to expect that points in MK3 might be mirror symmetric to points that are no longer in MK3. We will show that this is precisely the source of the problem of mirror symmetry for the Shioda–Inose K3 surfaces. There is a good chemistry between Hitchin’s generalized CY structures and mirror symmetry as they both embrace A-structures (symplectic) and B-structures (complex) on an equal footing. Inspired by the works of Dol- gachev, Aspinwall–Morrison and Huybrechts, we introduce a formulation of mirror symmetry for generalized K3 surfaces (Section 4). The key features of our formulation is twofold, and both are guided by Hitchin’s theory. The first is to extend the scope of lattice polarizations to the Mukai lattice. Mix- ture of degrees of cycles is indispensable for generalized K3 surfaces. The second is to treat the A- and B-structures on a completely equal footing. The Néron–Severi lattice and transcendental lattice are defined in the same fashion and moreover the lattice polarizations are imposed on both. Along the way, we investigate complex and Kähler rigid structures of generalized K3 surfaces (Section 3). The notion of a complex rigid structure is enhanced to the level of generalized K3 surfaces, and we give a classification theorem which generalizes the famous Shioda–Inose theorem. A rigid Kähler structure is defined by using a generalized CY structure, which captures a hidden integral structure of the Kähler moduli space. Such a structure has been anticipated for a long time also from the viewpoint of mirror symmetry. We give a conjectural explicit mirror correspondence between the complex rigid K3 surfaces and Kähler rigid K3 surfaces.

Structure of article. Section 2 provides a brief summary of generalized K3 structures based on Huybrechts’ work [5]. Section 3 investigates the complex and K¨ahler rigid structures on generalized K3 surfaces. Section 4 is devoted to mirror symmetry for generalized K3 surfaces in comparison MIRROR SYMMETRY AND RIGID STRUCTURES OF GENERALIZED K3 SURFACES3 with the classical formulation for K3 surfaces. Section 5 settles the long- standing problem of mirror symmetry for Shioda–Inose K3 surfaces. Acknowledgement. The author thanks Yu-Wei Fan and Shinobu Hosono for many valuable discussions and helpful comments. Some parts of this work was done while the author was aﬃliated by Kyoto University with Leading Initiative for Excellent Young Researchers Grant. This work is partially supported by the JSPS Grant-in-Aid Wakate(B)17K17817.

2. Generalized K3 surfaces Hitchin’s invention of generalized CY structures is the key to unify the symplectic and complex structure [4]. Such structures have been extensively studied in 2-dimensions by Huybrechts [5]. In this section, for the sake of completeness, we provide a brief review of Huybrechts’ work [5]. 2.1. Generalized CY structures. Let M be a differentiable manifold un- 2 2 2i derlying a K3 surface and AC∗(M)= i=0AC (M) the space of even differen- ⊕ 2 tial forms with C-coefficients. Let ϕi denote the degree i part of ϕ A (M). ∈ C∗ The Mukai pairing for ϕ, ϕ A2 (M) is defined as ′ ∈ C∗ 4 ϕ, ϕ′ = ϕ ϕ′ ϕ ϕ′ ϕ ϕ′ A (M) h i 2 ∧ 2 − 0 ∧ 4 − 4 ∧ 0 ∈ C Definition 2.1 (generalized CY structure). A generalized CY structure on M is a closed form ϕ A2 (M) such that ∈ C∗ ϕ, ϕ = 0, ϕ, ϕ > 0. h i h i The special appeal of generalized CY manifolds resides in the fact that they embrace complex and symplectic structures on an equal footing. There are two fundamental CY structures: (1) A symplectic structure ω on M induces a generalized CY structure

√ 1ω 1 2 ϕ = e − =1+ √ 1ω ω . − − 2 (2) A complex structure J of M makes M a K3 surface MJ . A holomorphic 2-form σ of MJ , which is unique up to scaling, defines a generalized CY structure ϕ = σ. We also write Mσ = MJ . For B A2 (M), eB acts on A2 (M) by exterior product ∈ C C∗ 1 eBϕ = (1+ B + B B) ϕ. 2 ∧ ∧ This action is orthogonal with respect to the Mukai pairing B B e ϕ, e ϕ′ = ϕ, ϕ′ . h i h i A real closed 2-form is called a B-field. For a B-field B and a generalized CY structure ϕ, the B-field transform eBϕ is a generalized CY structure. We will later show that a B-field is indispensable when we view complex and symplectic structures as special instance of a more general notion. The 4 ATSUSHI KANAZAWA following shows that a generalized CY structure is a B-field transform of either one of the fundamental CY structures. Proposition 2.2 ([4]). Let ϕ be a generalized CY structure.

(1) If ϕ0 = 0, then 6 B+√ 1ω ϕ = ϕ0e − with a symplectic ω and a B-field B. (2) If ϕ0 = 0, then ϕ = eBσ = σ + B0,2 σ ∧ with is a holomorphic 2-form σ with respect to a complex structure J and a B-field B. Definition 2.3. Generalized CY structure of (1) and (2) in Proposition 2.2 are called type A and type B respectively. We consider the group Diff (M) of all diffeomorphisms f : M M such that the induced action f : H∗ 2(M, Z) H2(M, Z) is trivial. Generalized→ ∗ → CY structures ϕ and ϕ′ are called isomorphic if there exists an exact B-field B B and a diffeomoprhism f Diff (M) such that ϕ = e f ∗ϕ′. The most fascinating aspects∈ of∗ generalized CY structures is the occur- rence of the classical CY structure σ (type B) as well as of symplectic √ 1ω generalized CY structure e − (type A) in the same moduli space. This allows us to pass from the symplectic to the complex world in a continuous fashion. Example 2.4 ([4]). For a complex structure σ, the real and imaginary parts Re(σ), Im(σ) are themselves real symplectic forms. A family of generalized CY structures of type A 1 (Re(σ)+√ 1Im(σ)) ϕt = te t − converges, as t goes to 0, to the generalized CY structure σ of type B. In this way, the B-fields interpolate between generalized Calabi–Yau structures of type A and B. 2.2. (hyper)Kähler structures.

Deﬁnition 2.5. Let ϕ be a generalized CY structure. Then Pϕ A∗(M) and P H (M, R) denote the real 2-planes ⊂ [ϕ] ⊂ ∗ Pϕ = RReϕ RImϕ A∗(M), ⊕ ⊂ P = R[Reϕ] R[Imϕ] H∗(M, R) [ϕ] ⊕ ⊂ spanned by the real and imaginary parts of ϕ and [ϕ] respectively. They are positive plane with respect to the Makai pairings.

We say that two generalized CY structures ϕ and ϕ′ are orthogonal if Pϕ and Pϕ′ are pointwise orthogonal. The orthogonality of 2-planes Pϕ and Pϕ′ is in general a stronger condition than just ϕ, ϕ = 0. h ′i MIRROR SYMMETRY AND RIGID STRUCTURES OF GENERALIZED K3 SURFACES5

Definition 2.6 (Kähler). A CY structure ϕ is called Kähler if there exists another generalized CY structure ϕ′ orthogonal to ϕ. In this case, ϕ′ is called a Kähler structure for φ.

Let ϕ = σ a holomorphic 2-form for a complex structure J on M. If ϕ′ B+√ 1ω is a Kähler structure for ϕ, then it is of the form ϕ′ = ϕ0′ e − (type A) for some B-field B and symplectic form ω. The orthogonality is equivalent to σ B = σ ω = 0. ∧ ∧ Therefore B is a closed real (1, 1)-form and ω is a Kähler form with respect to J. ± A hyperKähler structure is then defined as a special instance of a Kähler structure. Recall first that a Kähler form ω on a K3 surface is a hyperKähler form if 2ω2 = Cσ σ ∧ for some C R. We may assume C = 1 by rescaling σ. ∈ Definition 2.7 (hyperKähler). A generalized CY structure ϕ is hyperKähler if there exists another generalized CY structure ϕ′ such that (1) ϕ and ϕ′ are orthogonal (2) ϕ, ϕ = ϕ , ϕ . h i h ′ ′i Such ϕ′ is called a hyperKähler structure for ϕ. Remark 2.8. (1) The definition of a (hyper)Kähler structure is symmetric for ϕ and ϕ′. (2) Let ϕ be a generalized CY structure and ϕ′ a (hyper)Kähler for ϕ. B B Then e ϕ′ is a (hyper)Kähler structure for e ϕ. We give a classification of the hyperKähler structures for a generalized CY structure ϕ. By remark 2.8, we may assume that ϕ is either of the form √ 1ω ϕ = λe − (type A) or ϕ = σ (type B). (1) If ϕ = σ, then a hyperKähler structure is a generalized CY structure B+√ 1ω ϕ′ = λe − of type A, where B is a closed (1, 1)-form and ω is a hyperKähler form such that 2 λ 2ω2 = σ σ. ± √ 1ω | | ∧ (2) If ϕ = λe − , then a hyperKähler structure is either (a) a generalized CY structure ϕ′ = σ of type B, where ω is a hyperKähler form, ± B +√ 1ω (b) a generalized CY structure ϕ′ = λ′e ′ − ′ of type A such that 2 2 2 (i) ω ω′ = ω B′ = ω′ B = 0, B′ = ω + ω′ , (ii) λ∧2ω2 = λ∧2ω 2. ∧ | | | ′| ′ 2.3. Generalized K3 surfaces. Definition 2.9 (K3 surface). A K3 surface is a pair (σ, ω) of a closed 2- 2 2 form σ AC(M) with σ σ = 0 and a symplectic form ω A (M) such that ∈ ∧ ∈ 6 ATSUSHI KANAZAWA

(1) ω σ = 0 (2) 2ω∧2 = σ σ> 0. ∧ A K3 surface (σ, ω) is simply a classical K3 surface Mσ with a chosen hyperKähler structure ω. To avoid confusion, we usually denote by Mσ or S a classical K3 surface. Definition 2.10 (generalized K3 surface). A generalized K3 surface is a pair (ϕA, ϕB) of generalized CY structures such that ϕA is a hyperKähler structure for ϕB. A K3 surface (σ, ω) is often identified with a generalized K3 surface √ 1ω B B B (e − ,σ). A B-field transform e (ϕA, ϕB) = (e ϕA, e ϕB ) of a generalized K3 surface (ϕA, ϕB) is a generalized K3 surface. Generalized K3 surfaces (ϕA, ϕB) and (ψA, ψB) are called isomorphic if there exists a dif- feomorphism f Diff (M) and an exact real 2-form B A2(M) such that B ∈ ∗ B B ∈ (ϕA, ϕB )= e f ∗(ψA, ψB) = (e f ∗ψA, e f ∗ψB).

Definition 2.11. We associate to a generalized K3 surface (ϕA, ϕB) the oriented positive 4-spaces 2 Π = PϕA PϕB A ∗(M), (ϕA,ϕB) ⊕ ⊂ Π = P P H∗(M, R). ([ϕA],[ϕB]) [ϕA] ⊕ [ϕB] ⊂ The set TΠ of the generalized K3 surface (ϕA, ϕB) with fixed positive 4-space Π is naturally isomorphic to the Grassmannian of oriented planes Gro(Π) = S2 S2 = P1 P1, 2 × × which is identified with a quadric QΠ P(ΠC). √ 1ω ⊂ For a K3 surface (e − ,σ), the 4-space Π √ 1ω is spanned by the (e − ,σ) oriented basis 1 1 ω2, ω, Re(σ), Im(σ). − 2 We may write ωI = ω, ωJ = Re(σ), ωK = Im(σ), where I, J, K are the complex structures induced by the hyperKähler form. Then there is a 2- sphere S2 = aI + bJ + cK a2 + b2 + c2 = 1 { | } of complex structures with respect to which the metric is Kähler. This 2 1 S ∼= P is identified with the hyperplane section QΠ P(C ωI,ωJ ,ωK ). The other generalized K3 surfaces parametrized by P1∩ P1h S2 are noti realized as the B-field transforms of points in S2. × \

Proposition 2.12. Let (ϕ, ϕ′) be a generalized K3 surface. Then there exists a K3 surface (σ, ω) and a closed B-ﬁeld B such that

B Π ϕ,ϕ = e Π √ 1ω . ( ′) (e − ,σ) MIRROR SYMMETRY AND RIGID STRUCTURES OF GENERALIZED K3 SURFACES7

2.4. Moduli spaces and Torelli theorems. Let MK3 and MgK3 be the moduli spaces of K3 surfaces and generalized K3 structures respectively. We also define the moduli space of the B-field shifts of the hyperKähler metrics on M HK 2 MHK = Met (M)/Diff (M) H (M, R). ∗ × Proposition 2.12 implies that there exists a natural S2 S2-fibration × M M , (ϕ, ϕ′) (g,B), gK3 −→ HK 7→ B where B is given by Π ϕ,ϕ = e Π √ 1ω and g is the hyperKähler metric ( ′) (e − ,σ) associated with σ and ω. We have the canonical inclusion

2 B √ 1ω ι : M H (M, R) M , (σ,ω,B) e (σ, e − ). K3 × −→ gK3 7→ Definition 2.13. The period of a generalized K3 surface (ϕ, ϕ′) is the orthogonal pair of positive oriented planes po (P , P ) Gr (H∗(M, R)). [ϕ] [ϕ′] ∈ 2,2 Then the period map is defined by po per : M Gr (H∗(M, R)), (ϕ, ϕ′) (P , P ). gK3 gK3 −→ 2,2 7→ [ϕ] [ϕ′] Definition 2.14. The period of a B-field shift of hyperKähler metric (g,B) is the positive 4-space po Π Gr (H∗(M, R)) ([ϕ],[ϕ′]) ∈ 4 where g is the hyperKähler metric associated with σ and ω and Π(ϕ,ϕ′) = B e Π √ 1ω . Then the period map is defined by (e − ,σ) po per : M Gr (H∗(M, R)), (g,B) Π . HK HK −→ 4 7→ ([ϕ],[ϕ′]) To summarize, we obtain the following commutative diagram:

per 2 ι / gK3 / po MK3 H (M, R) MgK3 Gr2,2(H∗(M, R)) × ❖❖❖ ❖❖❖ ❖❖ S2 S2 S2 S2 2 ❖❖ S ❖❖' × × perHK / po MHK Gr4 (H∗(M, R)) where S2 and S2 S2 denote the fibers of corresponding the vertical arrows. × An important consequence of the diagram is that pergK3 is an immersion with dense image. Finally we compare the moduli space N = Cϕ / = of the gener- gCY { } ∼ alized CY structures of hyperKähler type, and the moduli space NK3 = Cσ /Diff (M) of the classical marked K3 surfaces. { } ∗ 8 ATSUSHI KANAZAWA

Theorem 2.15 ([5]). The classical period map perK3 naturally extends to the period map pergCY for the generalized CY structures of hyperK¨ahler type. per gCY / NgCY D = [ϕ] P(H∗(M, C)) ϕ, ϕ = 0, ϕ, ϕ > 0 Cϕ [φ] → { ∈ | h i h i } S e S per K3 / 2 NK3 D = [σ] P(H (M, C)) σ, σ = 0, σ, σ > 0 Cσ [σ] → { ∈ | h i h i } pergCY is ´etale surjective, and bijective over the complement of the hyperplane section P(H2(M, C) H4(M, C)) D. ⊕ ∩ Therefore the generalized K3 surfaces coulde be considered as geometric realizations of points in the extended period domain D. This Torelli theorem will be a foundation of our formulation of mirror symmetry in Section 4. e

Figure 1. Moduli space NgCY

3. Rigid structures 3.1. Néron–Severi lattice and transcendental lattice. We begin by introducing the Néron–Severi lattice and the transcendental lattice of a generalized K3 surface. Definition 3.1. The Néron–Severi lattice and transcendental lattices of X = (ϕA, ϕB ) are defined respectively by

NS(X)= δ H∗(M, Z) δ, ϕB = 0 , { ∈ | h i } gT (X)= δ H∗(M, Z) δ, ϕA = 0 . { ∈ | h i } It is worth mentioninge that NS(X) T (X) can be non-trivial. Moreover, ∩ neither ϕA NS(X)C nor ϕB T (X)C holds in general. Our definition of ∈ g∈ e the transcendental lattice T (X) differs from Huybrechts’ [5] (the orthogonal g e complement of NS(X)). We define NS(X) and T (X) on a completely equal e footing. g g e MIRROR SYMMETRY AND RIGID STRUCTURES OF GENERALIZED K3 SURFACES9

For later use, we introduce one more notation. For ψ H (M, C), we ∈ ∗ denote by Lψ the smallest sublattice L H∗(M, Z) such that ψ LC. Then we may write ⊂ ∈

NS(X)= Lϕ⊥B , T (X)= Lϕ⊥A . This in particular showsg that the signaturese of NS(X) and T (X) are of the form (t, ) with 0 t 2. ∗ ≤ ≤ g e It is instructive to compute basic examples. Let δi denote the degree i part of δ H∗(M, Z). ∈ 0,2 2 (1) If ϕB = σ + B σ, a twist of σ by a B-ﬁeld B H (M, R), then ∧ ∈ 0,2 4 NS(X)= δ0 + δ2 δ2 σ = δ0 B σ H (M, Z). { | ZM ∧ ZM ∧ }⊕ g 4 There is a natural inclusion H (M, Z) NS(Mσ) NS(S) (the inclusion is strict if B H2(M, Q)). In⊕ particular, for⊂ the classical ∈ g case ϕB = σ, we recover 0 4 NS(X)= H (M, Z) NS(Mσ) H (M, Z) ⊕ ⊕ g ∼= NS′(Mσ) B+√ 1ω 2 (2) If ϕA = e for B H (M, R) and a symplectic form ω, the − ∈ orthogonality condition δ, ϕA = 0 is equivalent to h i δ0 2 2 δ2 B (B ω )+ δ4 = 0, ZM ∧ − 2 ZM − ZM

δ2 ω δ0 B ω = 0. ZM ∧ − ZM ∧ √ 1ω In particular, if a B-ﬁeld is absent, i.e. ϕA = e − , then

2 2 T (X)= H (M, Z)ω δ0 + δ4 δ0 ω = 2 δ4 , ⊕ { | ZM ZM } e 2 where H (M, Z)ω is the space of ω-primitive classes. 3.2. Complex rigidity. The Shioda–Inose K3 surfaces are known as complex rigid K3 surfaces in the sense that they do not admit any complex deformation keeping the maximal Picard number 20. We ﬁrst give a generalization of complex rigid K3 surfaces.

Deﬁnition 3.2. A generalized K3 surface X = (ϕA, ϕB ) is called complex rigid if

(1) ϕB is of type B, (2) rank(NS(X)) = 22. Theorem 3.3.gA complex rigid generalized K3 surface is of the form X = (λeB+B′+√ 1ω,σ + B σ), where − ′ ∧ (1) Mσ is a complex rigid K3 surface, 1,1 (2) B H (Mσ, R), ∈ 10 ATSUSHI KANAZAWA

2 (3) B′ H (M, Q), (4) ω∈is a hyperKähler form for σ. ± In other words, it is a rational B-field B′ shift of a rigid K3 surface B √ 1ω equipped with a complexified Kähler structure (λe ± − ,σ).

2 Proof. Let X = (ϕA, ϕB) be a complex rigid K3 surface. Since H (M, Z) is of signature (3, 19), ϕA must be of type A. Hence we can write X = B B+√ 1ω B+√ 1ω e ′ (λe − ,σ). Then Y = (λe − ,σ) is a generalized K3 surface, i.e. B is a closed real (1, 1)-form and ω is a hyperK¨ahler form for σ. On the other hand, for ±

ϕB = σ + B′ σ, ∧ rank(LϕB ) = 2 if and only if rank(T (Mσ))= 2 and B′ is rational.

The following is a generalization of the celebrated Shioda–Inose theorem concerning the classiﬁcation of rigid K3 surfaces [8].

ev Theorem 3.4. Let L(2,0) be the set of isomorphism classes of positive deﬁ- rigid nite even lattices of rank 2. Let MCpx be the set of isomorphism classes of complex rigid generalized K3 surfaces. Then

rigid ev 2 M L H (M, Q), X (T (Mσ),B′) Cpx −→ (2,0) × 7→ B B+√ 1ω where X = e ′ (λe − ,σ) as in Theorem 3.3, is surjective. The fiber over 1,1 (T (Mσ),B′) is given by the 2 copies of complexified Kähler cone H (Mσ)+ √ 1 (Mσ) − K Proof. The Shioda–Inose theorem [8] asserts that there is a bijective correspondence between the isomorphism classes of rigid K3 surfaces and the ev set L(2,0) of isomorphism classes of positive definite even lattices of rank 2. The correspondence is given by S T (S). Then the assertion follows from Theorem 3.3. →

3.3. Kähler rigidity. In light of mirror symmetry, a rigid Kähler structure has been anticipated for a long time. A Kähler rigidity comparable with a complex rigidity naturally appears in the framework of generalized CY structures. We begin with a simple observation. Let S be a K3 surface with Picard number 1. We write NS(S)= ZH with H2 = 2n> 0, and consider

v = (1, 0, n), v = (0, H, 0) NS′(S). 1 − 2 ∈ Then

√ 1H e − = (1, √ 1H, n) − − = v + √ 1v (Zv + Zv )C ( NS′(S)C 1 − 2 ∈ 1 2 MIRROR SYMMETRY AND RIGID STRUCTURES OF GENERALIZED K3 SURFACES11

On the other hand, for ǫ2 / Q, ∈ √ 1ǫH 2 e − = (1, √ 1ǫH, ǫ n) − − = (1, 0, ǫ2n)+ √ 1ǫ(0, H, 0) − − 2 = (1, 0, 0) ǫ (0, 0,n)+ √ 1ǫ(0, H, 0) NS′(S)C. − − ∈ √ 1ǫH Hence there is no proper sublattice L ( NS′(S) such that e − LC. Therefore the Kähler structure H is not deformable in such a way∈ that rank(L √ 1ǫH ) = 2. This calculation illustrates that a generalized CY struc- e − B+√ 1ω ture e − is able to detect a hidden integral structure of the Kähler moduli space. Definition 3.5. A symplectic manifold (M,ω) is called Kähler rigid if ω2 H4(M, Q). ∈ As the previous calculation shows, a symplectic manifold (M,ω) is Kähler rigid if and only if rank(L √ 1ω ) = 2. e − Definition 3.6. A generalized K3 surface X = (ϕA, ϕB) is called Kähler rigid if (1) ϕA is of type A, (2) rank(T (X)) = 22. We providee a characterization of the Kähler rigid generalized K3 surfaces (ϕA, ϕB ) as follows. In contrast to the complex rigid case, the partner ϕB can be either of type A or B. Theorem 3.7. A Kähler rigid generalized K3 surface is of the form X = B+√ 1ω (λe − , ϕB ), where (1) B H2(M, Q), (2) ω2∈ H4(M, Q). ∈ In other words, it is a rational B-field B shift of a Kähler rigid symplectic √ 1ω manifold equipped with a hyperKähler structure (λe − , ϕB ). B+√ 1ω Proof. Let X = (e − , ϕB ) be a Kähler rigid K3 surface. We consider an existence condition of a rank 2 sublattice L H (M, Z) such that ⊂ ∗ B+√ 1ω 1 2 2 e − =1+ B + (B ω )+ √ 1(ω + B ω) LC. 2 − − ∧ ∈ First, B needs to be rational, and hence so is ω2. Then we may write the symplectic class ω = κH for κ2 Q and H H2(M, Z) with H2 > 0. Indeed, in this case, there exist m,n∈ N such that∈ ∈ B+√ 1κH B+√ 1κH mRe(e − ), nIm(e − ) H∗(M, Z). ∈ Then the complexification LC of the lattice B+√ 1κH B+√ 1κH L = ZmRe(e − )+ ZnIm(e − ) H∗(M, Z) ⊂ B+√ 1κH contains e − . 12 ATSUSHI KANAZAWA

By analogy with the complex rigid case, we conjecture the following mirror assertion of the Shioda–Inose theorem. Conjecture 3.8. There is a bijective correspondence between the isomor- ev phism classes of Kähler rigid structure on M and the set L(2,0) of isomorphism classes of positive definite even lattices of rank 2. The correspondence is given by (M,ω) L √ 1ω . → e − The conjecture in particular implies a natural bijective correspondence between the complex rigid structures and the Kahler rigid symplectic structures on M (and between their generalized versions by the rational B-field shifts). This mirror correspondence is essentially given by comparison of the Néron–Severi lattices and transcendental lattices. Complex rigid o mirror / Kähler rigid { ◆◆◆ } { qq } ◆◆◆ qq ◆◆ qqq ◆◆ q σ ω Le√ 1ω ◆◆ qq σ L 7→ − ' qx 7→ ev L(2,0)

Remark 3.9. Clearly, there exist parallel theories for T 4, the 4-manifold underlying a 2-dimensional complex torus.

4. Mirror symmetry for generalized K3 surfaces Our formulation of mirror symmetry for generalized K3 surfaces builds upon the combination of two novel ideas: (1) lattice polarizations of cycles (Dolgachev [2]) and (2) generalized K3 structures (Hitchin [4] and Huy- brechts [5]). As is discussed Section 3, lattices are sensitive to the integral structures associated with generalized K3 structures.

4.1. Lattices. The hyperbolic lattice U is the rank 2 even unimodular lat- 0 1 tice deﬁned by the Gram matrix . E is the rank 8 even unimodular 1 0 8 3 2 lattices deﬁned by the corresponding Cartan matrix. Let ΛK3 = U ⊕ E8⊕ be the K3 lattice. ⊕

4.2. Mirror symmetry for K3 surfaces ala Dolgachev. Let S be a K3 surface. The Néron-Severi lattice and transcendental lattice are defined respectively by 1,1 2 NS(S)= H (S, R) H (S, Z), T (S)= NS(S)⊥, ∩ 2 where the orthogonal complement is taken in H (S, Z) ∼= ΛK3. The Picard number is ρ(S) = rank(NS(S)). The extended Néron-Severi lattice NS′(S) by the sublattice 0 4 NS′(S)= H (S, Z) NS(S) H (S, Z). ⊕ ⊕ 4 2 of the Mukai lattice H (S, Z) = U E⊕ . ∗ ∼ ⊕ ⊕ 8 MIRROR SYMMETRY AND RIGID STRUCTURES OF GENERALIZED K3 SURFACES13

Deﬁnition 4.1. Let L be an even non-degenerate lattice of signature (1, ). An L-polarized K3 surface is a pair (S, i) of a K3 surface S and a primitive∗ lattice embedding i : L ֒ NS(S). It is called ample if i(L) contains a K¨ahler class. →

There may be several families of ample L-polarized K3 surfaces, and we usually choose one for the purpose of mirror symmetry. In the foundational article [2], Dolgachev formulated mirror symmetry for lattice polarized K3 surfaces as follows.

Deﬁnition 4.2 ([2]). For a sublattice L ΛK3 of signature (1,t), assume there exists a lattice N such that L = N⊂ U. Then ⊥ ⊕ (1) a family of ample L-polarized K3 surfaces S (2) a family of ample N-polarized K3 surfaces { S} { ∨} are mirror symmetric.

In fact, when the assumption holds, for generic L-polarized K3 surface S and N-polarized K3 surface S∨, we have

NS′(S) = L U = T (S∨), T (S) = N U = NS′(S∨). ∼ ⊕ ∼ ∼ ⊕ ∼ These are considered as a duality between algebraic cycles of S and transcendental cycles of S∨, and vice versa. Mirror duality can also be investigated at the level of the moduli spaces. We refer the reader to the original article [2] for more details. The crucial assumption in Dolgachev’s formulation is the existence of a decomposition of the form L = N U for L ΛK (or more generally ⊥ ⊕ ⊂ 3 L⊥ = N U(k) for some k N). There are geometric explanations for such a decomposition;⊕ the hyperbolic∈ lattice U corresponds to (1) the standard cusps of the Baily–Borel compactification of the period domains [7], (2) the lattice spanned by the fiber and section classes of the SYZ fibra- tions on a mirror K3 surface. Although this formulation works beautifully in many case, it cannot be a definitive one as this artificial assumption does not hold in general. A prototypical example of such K3 surfaces is a Shioda–Inose K3 surface, which is a K3 surface S with the maximal Picard number 20. Hence, by the Hodge index theorem, T (S) is of signature (2, 0) and the assumption is never satisfied. A natural question ”How can we understand mirror symmetry for Shioda–Inose K3 surfaces?” is a long-standing puzzle in the mirror symmetry community. Indeed, its Kähler moduli space is of dimension 20 and complex moduli space is of dimension 0, and there seems no mirror partner for a Shioda–Inose K3 surface. We will provide an answer to this puzzling problem in the framework of mirror symmetry for generalized K3 surfaces. 14 ATSUSHI KANAZAWA

Shioda–Inose K3 surface mirror ?? K¨ahler 20-dim 0-dim complex 0-dim 20-dim

4.3. Lattice polarization via Mukai lattice. For integers κ, λ 2 such that κ + λ = 24, let K and L be even lattice of signature (2, κ ≥2) and (2, λ 2) respectively. − − Definition 4.3. A (K,L)-polarized generalized K3 surface is a pair (X, i) of a generalized K3 surface X = (ϕA, ϕB) and a primitive embedding i : K L ֒ H (M, Z) such that ⊕ → ∗ (1) K NS(X) and KC contains a generalized CY structure of type A, ⊂ (2) L T (X) and LC contains a generalized CY structure of type B. ⊂ g The containmente of a generalized CY structure of type A or type B is comparable with the ampleness condition for the conventional lattice polarizations. Without these, swapping ϕA and ϕB produces a trivial mirror families. Remark 4.4. In the conventional L-polarization, there is asymmetry between algebraic and transcendental cycles; the inclusions L NS(S) and L T (S) are not symmetric. On the other hand, a (K,L⊂)-polarization ⊥ ⊃ imposes the inclusions K NS(X) and L T (X) evenly. ⊂ ⊂ 4.4. Mirror symmetry forg generalized K3e surfaces. As in the case of K3 surfaces, there may be several families of (K,L)-polarized generalized K3 surfaces, and we choose one for the purpose of mirror symmetry. Definition 4.5. For integers κ, λ 2 such that κ+λ = 24, let K,L be even lattice of signature (2, κ 2) and (2≥, λ 2) respectively. Then − − (1) a family of (K,L)-polarized generalized K3 surfaces X (2) a family of (L, K)-polarized generalized K3 surfaces {X} { ∨} are mirror symmetric. The fundamental idea is comparable with Dolgachev’s formulation. A major difference is the mixture of the degrees of cycles. For example, NS(X) may not contain a pure zero cycle (a point is no longer algebraic in this g sense). Another notable difference is that there is no artificial assumption on the lattices. Let X = (ϕA, ϕB) bea(K,L)-polarized generalized K3 surface. Note that the inclusions K NS(X) and L T (X) are equivalent to ϕB LC ⊂ ⊂ ∈ and ϕA KC respectively. ∈ g e Definition 4.6. A deformation of CϕA keeping X as a (K,L)-polarized X generalized K3 surface is called an A-deformation. The A-moduli space MA is defined as the space of A-deformations. A B-deformation and the B- X moduli space MB are defined similarly. MIRROR SYMMETRY AND RIGID STRUCTURES OF GENERALIZED K3 SURFACES15

X X By the Torelli theorem (Theorem 2.15), dim MA = κ 2 and dim MB = λ 2. The construction of moduli spaces works in largely− the same way as that− for lattice polarized K3 surfaces (see Dolgachev’s work [2] for details). We ﬁrst deﬁne

DK = [ϕ] P(KC) σ, σ = 0, ϕ, ϕ > 0 , { ∈ | h i h i } po which can be identified with Gr2 (KR) ∼= O(2, κ 2)/(SO(2) O(κ 2)). It is a complex manifold of dimension κ 2 with− two connected× components− + − DL and DL−, each of which is a symmetric domain of type IV. Independent of the choice of lattices, the (K,L)-polarized generalized K3 surface X form a 20-dimensional family. By the Torelli theorem, the moduli space MX MX A × B is identified with an open subset of D = D+ D+. (K,L) K × L 4.5. Comparison: classical and new. We will next confirm that our formulation of mirror symmetry naturally includes Dolgachev’s. Let K′ ΛK3 be a sublattice of signature (1, ρ 1). Assume that there exists a decomposition⊂ (K ) = L U for a lattice−L of signature (1, 19 ′ ⊥ ′ ⊕ ′ − ρ). Mirror symmetry for lattice polarized K3 surfaces asserts that the K′- polarized K3 surfaces and L′-polarized K3 surfaces are mirror symmetric to each other. Let Mσ be a generic K′-polarized K3 surface equipped with a B-field B shift of a hyperKähler form ω (B,ω KR′ are also taken generically). It defines a generalized K3 surface ∈

B+√ 1ω X = (ϕA, ϕB) = (e − ,σ). and we have

NS(X)= NS′(Mσ) = K′ U, ∼ ⊕ gT (X)= T (Mσ) = L′ U. ∼ ⊕ e Similarly, let Mσ∨ be a generic L′-polarized K3 surface equipped with a B-field B∨ shift of a hyperKähler form ω∨. It defines a generalized K3 surface

B∨+√ 1ω∨ X∨ = (ϕA, ϕB ) = (e − ,σ∨). and we have

NS(X∨)= NS′(Mσ ) = L′ U, ∨ ∼ ⊕ gT (X∨)= T (Mσ ) = K U. ∨ ∼ ⊕ By setting K = K U eand L = L U, X is a (K,L)-polarized generalized ′ ⊕ ′ ⊕ K3 surface and X∨ is a (L, K)-polarized generalized K3 surface. Therefore the classical formulation of mirror symmetry is naturally considered as a special case of ours. 16 ATSUSHI KANAZAWA

5. Mirror symmetry for Shioda–Inose K3 surfaces 5.1. Mirror symmetry for Shioda–Inose K3 surfaces. Our formulation of mirror symmetry will really comes into its own when classical formulation fails. We will ﬁnally discuss mirror symmetry for Shioda–Inose (complex rigid) K3 surfaces in the framework of mirror symmetry for generalized K3 surfaces. For an integer n> 0, we deﬁne 2 2 2 K = 2n ⊕ U E⊕ , L = 2n ⊕ , h− i ⊕ ⊕ 8 h i where k denotes the lattice of rank 1 generated by v with v2 = k. h i (1) A family of the (K,L)-polarized generalized K3 surfaces is given by the family of the Shioda–Inose K3 surfaces X = (eB+√ 1ω,σ) , { − } where σ is the complex structure of M such that T (Mσ)= L (such a complex structure is unique up to isomorphism) and B,ω NS(Mσ)R. For a generic X, we have ∈

NS(X)= NS′(Mσ), T (X)= T (Mσ), which are isomorphicg to K and L erespectively. (2) A family of the (L, K)-polarized generalized K3 surfaces contains the family of the classical K3 surfaces of the form X = (e√ 1H ,σ ) { ∨ − ∨ } as a subfamily. Here σ∨ is a complex structure of M such that 2 NS(Mσ∨ )= ZH,H = 2n> 0. For a generic X∨, we have

NS(X∨) NS′(Mσ ), T (X∨) T (Mσ ) ⊂ ∨ ⊃ ∨ and NS(gX∨)= L, T (X∨)= K. e We observeg that the dimensionse of the A- and B-moduli spaces are inter- changed between the mirror families. (cf. table in Section 4.2)

(K,L)-polarization (L, K)-polarization MA 20-dim 0-dim MB 0-dim 20-dim

To summarize, the mirror partner of a Shioda–Inose K3 surfaces is in general given by a generalized K3 surface. The family of the K3 surfaces of the √ 1H form X∨ = (e − ,σ∨) is a 19-dimensional family contained in the gen- uine 20-dimensional{ mirror} family. In order to understand mirror symmetry for K3 surfaces, we need to incorporate the deformations of generalized K3 surfaces in general. Remark 5.1. The combination of generalized CY geometry and lattice po- rarlization via the Muaki lattice is also the correct framework for mirror symmetry for 4-tori T 4. MIRROR SYMMETRY AND RIGID STRUCTURES OF GENERALIZED K3 SURFACES17

5.2. Attractor mechanisms. The initial motivation of the present article comes from the attractor mechanisms on moduli spaces of CY 3-folds [6, 3]. Let X be a projective CY 3-fold. Let π : MCpx MCpx be the universal covering of the complex moduli space M of X. The→ normal central charge Cpx f of a 3-cycle γ H (X, Z) is deﬁned by ∈ 3 B K (z) Z(ΩXz , γ)= e 2 ΩXz Zγ B where ΩXz is a holomorphic volume form of Xz and K (z) is the Weil– Petersson potential

B K (z)= log(√ 1 ΩXz ΩXz ) − − ZXz ∧ on MCpx. It induces a function

f Z( , γ) : MCpx R 0, z Z(Ωz, γ) , | − | −→ ≥ 7→ | | called the mass function offγ. The stationary points of the mass functions are called the complex attractors and corresponding CY 3-folds are called the complex attractor varieties. This new class of CY 3-folds are a vast generalization of rigid CY 3-folds and expected to posses rich structures. In light of mirror symmetry, the Kähler attractor mechanisms are devel- oped in our recent article [3]. The Kähler attractor varieties are defined in the same fashion (they correspond to the stationary points of the Kähler mass functions). If X and Y are mirror CY 3-folds, then the mirror map X should induce a bijective correspondence between the set AttrCpx of complex Y attractors of X and the set AttrKah of Kähler attractors of Y : MX MY Cpx ∼= Kah S S X Y AttrCpx ∼= AttrKah An interesting observation is obtained when X and Y are the product of a K3 surface and an elliptic curve. On the complex side, a complex attractor variety is the product of a Shioda–Inose K3 surface Mσ and an elliptic curve. On the Kähler side, a Kähler attractor variety is the product of a Kähler rigid symplectic manifold (M,ω) and an elliptic curve. This observation led us to the hidden integral structure on the Kähler moduli space and the idea of mirror symmetry in the framework of generalized K3 surfaces.

References [1] P. Aspinwal and D. Morriron, String theory on K3 surfaces, Mirror symmetry, II, AMS/IP Stud. Adv. Math., vol. 1, Amer. Math. Soc., Providence, RI, 1997, 703-716. [2] I. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, J. Math. Sci. 81 (1996), no. 3, 2599-2630, Algebraic geometry, 4. 18 ATSUSHI KANAZAWA

[3] Y.-W. Fan and A. Kanazawa, Attractor mechanisms on moduli spaces of Calabi–Yau 3-folds, preprint. [4] N. Hitchin, Generalized Calabi–Yau manifolds, Quart. J. Math. Oxford Ser. 54: 281- 308, 2003. [5] D. Huybrechts, Generalized Calabi–Yau structures, K3 surfaces, and B-ﬁelds, Int. J. Math. 16 (2005), 13-36. [6] G. Moore, Arithmetic and Attractors, hep-th/9807087. [7] F. Scattone, On the compactiﬁcation of moduli spaces for algebraic K3 surfaces, Mem. AMS 70 (1987) No.374. [8] T. Shioda and H. Inose, On singular K3 surfaces, Complex analysis and algebraic geometry, Cambridge University Press, Cambridge, 1977. Faculty of Policy Management, Keio University Endo 5322, Fujisawa, Kanagawa, 252-0882, Japan [email protected]