Mirror Symmetry and Projective Geometry of Fourier-Mukai Partners
aa r X i v : . [ m a t h . AG ] D ec Mirror Symmetry and Projective Geometry ofFourier-Mukai Partners
Shinobu Hosono and Hiromichi Takagi
Abstract. This is a survey article on mirror symmetry and Fourier-Mukai partners of Calabi-Yau threefolds with Picard number one basedon recent works [HoTa1,2,3,4]. For completeness, mirror symmetry andFourier-Mukai partners of K3 surfaces are also discussed.
Contents Introduction
Fourier-Mukai partners of K3 surfaces M -polarized K3 surfaces 42.3. M -polarizable K3 surfaces 52.4. Mirror symmetry of K3 surfaces 52.5. Homological mirror symmetry 62.6. FM(X) and mirror symmetry 62.7. An example due to Mukai 72.8. Some other aspects 113. Fourier-Mukai partners of Calabi-Yau threefolds I D b ( X ) ≃ D b ( Y ) Fourier-Mukai partners of Calabi-Yau threefolds II D b ( X ) ≃ D b ( Y ) Birational Geometry of the Double Symmetroid Y Z → Y Y of Y Y and resolutions of Y ρ e Y : f Y → Y G(3 , Introduction
Derived categories of coherent sheaves on projective varieties are attracting at-tentions from many aspects of mathematics for the last decades. Among them, thederived categories of coherent sheaves on Calabi-Yau manifolds have been attract-ing special attentions since they are conjecturally related to symplectic geometryby the homological mirror symmetry due to Kontsevich [Ko] and also to the geo-metric mirror symmetry due to Strominger-Yau-Zaslow [SYZ]. In this article, wewill survey on the derived categories of Calabi-Yau manifolds of dimension two andthree focusing on the so-called Fourier-Mukai partners and their mirror symmetry.As defined in the text, smooth projective projective varieties X and Y are calledFourier-Mukai partners to each other if their derived categories of bounded com-plexes of coherent sheaves are equivalent, D b ( X ) ≃ D b ( Y ) . When X and Y areK3 surfaces, the study of the derived equivalence goes back to the works by Mukaiin ’80s [Mu1] and Orlov in ’90s [Or]. For completeness, we start our survey witha brief summary of their results, and also the mirror symmetry interpretationsmade in [HLOY1]. About the Fourier-Mukai partners of Calabi-Yau threefolds,little is known except a general result that two Calabi-Yau threefolds are derivedequivalent if they are birational [Br2]. In [BC][Ku2], it has been shown that aninteresting example of a pair of Calabi-Yau threefolds X , Y of Picard number one(Grassmannian-Pfaffian Calabi-Yau threefolds) due to Rødland [Ro] is the case ofnon-trivial Fourier-Mukai partners which are not birational. In particular, it hasbeen recognized in [Ku2, Ku1] that the classical projective duality between theGrassmannian G(2 , and the Pfaffian variety Pf(4 , in the construction of X and Y plays a prominent role, and a notion called homological projective dualityhas been introduced in [Ku1]. Recently, it has been found by the present authors[HoTa1,2,3,4] that the projective duality of G(2 , and Pf(4 , has a natural coun-terpart in the projective duality between the secant varieties of symmetric formsand these of the dual forms. In this setting, we naturally came to two Calabi-Yauthreefolds X and Y of Picard numbers one which are derived equivalent but notbirational to each other. Calabi-Yau manifold X is the so-called three dimensionalReye congruence (whose two dimensional counterpart has been studied in [Co]),and Y is given by a linear section of double quintic symmetroids (see Section 5).In the construction of Y and also in the proof of the derived equivalence to X , birational geometry of the double quintic symmetroids has been worked out indetail in [HoTa3]. It has been found that the birational geometry of symmetroidsitself contains interesting projective geometry of quadrics [Ty].This article is aimed to be a survey of the works [HoTa1,2,3,4] on mirror symme-try and Fourier-Mukai partners of the new Calabi-Yau manifolds of Picard numberone, and also interesting birational geometry of the double quintic symmetroidswhich arises in the constructions. In order to clarify the entire picture of thesubjects, we have included previous works on K3 surfaces and also the Rødland’sexample. Since the expository nature of this article, most of the proofs for thestatements are omitted referring to the original papers. Acknowledgements:
The first named author would like to thank K. Oguiso,B.H. Lian and S.-T. Yau for valuable collaborations on Fourier-Mukai partners ofK3 surfaces. This article is supported in part by Grant-in Aid Scientific Research(C 18540014, S.H.) and Grant-in Aid for Young Scientists (B 20740005, H.T.). irror Symmetry and Projective Geometry of FM partners 3 Fourier-Mukai partners of K3 surfaces
Counting formula of Fourier-Mukai partners.
Let X be a K3 surface,i.e., a smooth projective surface with K X ≃ O X and H ( X, O X ) = 0 . Wehave a symmetric bilinear form ( ∗ , ∗∗ ) on H ( X, Z ) by the cup product. Then ( H ( X, Z ) , ( ∗ , ∗∗ )) is an even unimodular lattice of signature (3 , , which is iso-morphic to L K := E ( − ⊕ ⊕ U ⊕ where U is the hyperbolic lattice ( Z ⊕ Z , ( ) ). Denote by N S X = P ic ( X ) the Picard (Néron-Severi) lattice and set ρ ( X ) = rk N S X . N S X is the primitive sub-lattice in H ( X, Z ) and has signa-ture (1 , ρ ( X ) − . The orthogonal complement T X = ( N S X ) ⊥ in H ( X, Z ) iscalled transcendental lattice. T X has signature (2 , − ρ ( X )) . The extension ˜ H ( X, Z ) = H ( X, Z ) ⊕ H ( X, Z ) ⊕ H ( X, Z ) ≃ E ( − ⊕ ⊕ U ⊕ is called Mukailattice.Let us denote by ω X the nowhere vanishing holomorphic two form of X which isunique up to constant. Then the Global Torelli theorem says that K3 surfaces X and X ′ are isomorphic iff there exists a Hodge isometry, i.e., a lattice isomorphism ϕ : H ( X, Z ) → H ( X ′ , Z ) which satisfies ϕ ( C ω X ) = C ω X ′ . Extending earlierworks by Mukai [Mu1] in 80’, Orlov [Or] has formulated a similar Global Torellitheorem for the derived categories of coherent sheaves on K3 surfaces: Theorem 2.1 ([Mu1][Or]) . K3 surfaces X and X ′ are derived equivalent, D b ( X ) ≃ D b ( X ′ ) , if and only if there exists a Hodge isometry of transcendental lattices ( T X , C ω X ) ≃ ( T X ′ , C ω X ′ ) . Due to the uniqueness theorem of primitive embeddings into indefinite lattices(see Theorem A.1 in Appendix), we note that the Hodge isometry ( T X , C ω X ) ≃ ( T X ′ , C ω X ′ ) above always extends to that of the Mukai lattice ( ˜ H ( X, Z ) , C ω X ) ≃ ( ˜ H ( X ′ , Z ) , C ω X ′ ) , and hence we can rephrase the above theorem in terms of theHodge isometry of Mukai lattices.Consider smooth projective varieties X and Y . Y is called Fourier-Mukai partnerof X if D b ( Y ) ≃ D b ( X ) . We denote the set of Fourier-Mukai partners (up toisomorphisms) of X by F M ( X ) = (cid:8) Y | D b ( Y ) ≃ D b ( X ) (cid:9) / isom.For a K3 surface X , the set F M ( X ) consists of K3 surfaces (see [Hu, Cor.10.2]for example) and its cardinality is known to be finite, i.e. | F M ( X ) | < ∞ in [BM].Studying all possible obstructions for extending a Hodge isometry ( T X , C ω X ) ≃ ( T X ′ , C ω X ′ ) between the transcendental lattices to the corresponding Hodge isom-etry ( H ( X, Z ) , C ω X ) ≃ ( H ( Y, Z ) , C ω Y ) , the following counting formula has beenobtained: Theorem 2.2 ([HLOY2]) . For a K3 surface X , we have | F M ( X ) | = X G ( NS X )= { S ,..,S N } (cid:12)(cid:12) O ( S i ) (cid:31) O ( A S i ) (cid:30) O Hodge ( T X , C ω X ) (cid:12)(cid:12) , where G ( N S X ) is the isogeny classes of the lattice N S X , A S i = ( S ∗ /S, q : S ∗ /S → Q / Z ) is the discriminant of the lattice S i , and O ( S i ) and O ( A S i ) are isometries of S i and A S i . O Hodge ( T X , C ω X ) is the Hodge isometries of ( T X , C ω X ) . We refer to [HLOY2] for the details (see also [HP]). Since the isogeny classes ofa lattice are finite, the counting formula contains the earlier result | F M ( X ) | < ∞ . irror Symmetry and Projective Geometry of FM partners 4 When X is a K3 surface with ρ ( X ) = 1 and deg ( X ) = 2 n , the counting formulacoincides with the result in [Og] (obtained by counting the so-called over-lattices);(2.1) | F M ( X ) | = 2 p ( n ) − (= 12 | O ( A NS X ) | ) , where p ( n ) is the number of prime factors of n (we set p (1) = 1 ). In fact, much isknown by [Mu3] in this case that we have F M ( X ) = {M X ( r, h, s ) | n = rs, ( r, s ) = 1 } , in terms of the moduli space of stable vector bundles E on X with Mukai vector ( r, h, s ) = ch ( E ) √ T d X in H ( X, Z ) ⊕ H ( X, Z ) ⊕ H ( X, Z ) (see also [HLOY3]).We will study in detail the first non-trivial example of | F M ( X ) | 6 = 1 ( n = 6 ) inSubsection 2.7.2.2. Marked M -polarized K3 surfaces. A K3 surface X with a choice of iso-morphism φ : H ( X, Z ) ∼ → L K is called a marked K3 surface ( X, φ ) . Marked K3 sur-faces ( X, φ ) and ( X ′ , φ ′ ) are isomorphic if there exists an isomorphism f : X → X ′ satisfying φ ′ = φ ◦ f ∗ . By the Global Torelli theorem, ( X, φ ) and ( X ′ , φ ′ ) are iso-morphic iff there exists a Hodge isometry ϕ : ( H ( X ′ , Z ) , C ω X ′ ) ∼ → ( H ( X, Z ) , C ω X ) such that φ ′ = φ ◦ ϕ (see [BHPv] for more details of K3 surfaces).Consider a lattice M of signature (1 , t ) and fix a primitive embedding i : M ֒ → L K . A marked K3 surface ( X, φ ) is called marked M -polarized K3 surface if φ − ( M ) ⊂ N S X (where we write φ − ( M ) = ( φ − ◦ i )( M ) for short). Marked M -polarized K3 surfaces ( X, φ ) and ( X ′ , φ ′ ) are isomorphic if there exists a latticeisomorphism ϕ : L K ∼ → L K such that(2.2) H ( X, Z ) ∼ φ / / L K ϕ ≀ (cid:15) (cid:15) M ? _ i o o H ( X ′ , Z ) ∼ φ ′ / / L K M ? _ i o o and the composition ( φ ′ ) − ◦ ϕ ◦ φ : ( H ( X, Z ) , C ω X ) → ( H ( X ′ , Z ) , C ω X ′ ) is aHodge isometry. The lattice isomorphism ϕ in (2.2) is an element of the group Γ( M ) = { g ∈ O ( L K ) | g ( m ) = m ( ∀ m ∈ M ) } . Consider the orthogonal lattice M ⊥ = ( i ( M )) ⊥ . Then there is a natural injectivehomomorphism Γ( M ) → O ( M ⊥ ) . The image is known to be described by thekernel O ( M ⊥ ) ∗ := Ker (cid:8) O ( M ⊥ ) → O ( A M ⊥ ) (cid:9) of the natural homomorphism tothe isometries of the discriminant A M ⊥ (see [Do, Prop.3.3]).A marked K3 surfaces ( X, φ ) determines the period points φ ( C ω X ) in the perioddomain D = { [ ω ] ∈ P ( L K ⊗ C ) | ( ω, ω ) = 0 , ( ω, ¯ ω ) > } . By the surjectivity of theperiod map, D gives a classifying space of the (not necessarily projective) markedK3 surfaces. Then, by the Global Torelli theorem, the quotient D /O ( L K ) classifiesthe isomorphism classes of (not necessarily projective) marked K3 surfaces.From the definition, it is easy to deduce that marked M -polarized K3 surfacesare classified by the period points in the following domain D ( M ⊥ ) := (cid:8) [ ω ] ∈ P ( M ⊥ ⊗ C ) | ( ω, ω ) = 0 , ( ω, ¯ ω ) > (cid:9) , which has two connected components D ( M ⊥ ) = D ( M ⊥ ) + ⊔D ( M ⊥ ) − . Let us define O + ( M ⊥ ) ⊂ O ( M ⊥ ) to be the isometries of M ⊥ which preserve the orientations of irror Symmetry and Projective Geometry of FM partners 5 all positive two spaces in M ⊥ ⊗ R . Then the isomorphisms classes of marked M -polarized K3 surfaces are classified by the following quotient,(2.3) D ( M ⊥ ) /O ( M ⊥ ) ∗ ≃ D ( M ⊥ ) + /O + ( M ⊥ ) ∗ ( ≃ D ( M ⊥ ) − /O + ( M ⊥ ) ∗ ) , where O + ( M ⊥ ) ∗ := O + ( M ⊥ ) ∩ O ( M ⊥ ) ∗ is the monodromy group which acts onthe period points φ ( C ω X ) ∈ D ( M ⊥ ) ± of marked M -polarized K3 surfaces ( X, φ ).2.3. M -polarizable K3 surfaces. Let us fix a primitive lattice embedding i : M ֒ → L K as in the preceding subsection. Following [HLOY1], we call a K3 surface X M - polarizable if there is a marking φ : H ( X, Z ) ∼ → L K such that ( φ − ◦ i )( M ) ⊂ N S X . Two M -polarizable K3 surfaces X and X ′ are defined to be isomorphic ifthere exists lattice isomorphisms ϕ : L K ∼ → L K and g : M ∼ → M which make thefollowing diagram commutative:(2.4) H ( X, Z ) ∼ φ / / L K ϕ ≀ (cid:15) (cid:15) M ? _ i o o g ≀ (cid:15) (cid:15) H ( X ′ , Z ) ∼ φ ′ / / L K M ? _ i o o and the composition ( φ ′ ) − ◦ ϕ ◦ φ : ( H ( X, Z ) , C ω X ) → ( H ( X ′ , Z ) , C ω X ′ ) is aHodge isometry. Note that, as we see in the diagram, the definition of the isomor-phism is slightly generalized for the M -polarizable K3 surfaces. Hence, although M -polarizable K3 surfaces X are obtained by forgetting the marking φ from themarked M -polarized K3 surfaces ( X, φ ) , their isomorphism classes are possiblydifferent. We saw in the last subsection that the isomorphism classes of marked M -polarized K3 surfaces are classified by the quotient D ( M ⊥ ) /O ( M ⊥ ) ∗ . On theother hand, the classifying space of the isomorphism classes of M -polarizable K3surfaces is given by a similar quotient of D ( M ⊥ ) but with a group which residesbetween O ( M ⊥ ) ∗ and O ( M ⊥ ) .2.4. Mirror symmetry of K3 surfaces.
In [Do], Dolgachev defined mirror sym-metry of marked M -polarized K3 surfaces. To summarize his construction/definition,let us fix a primitive embedding i : M ֒ → L K of a lattice M of signature (1 , t ) and assume that the orthogonal lattice M ⊥ has a decomposition M ⊥ = ˇ M ⊕ U , i.e. M ⊕ M ⊥ = M ⊕ U ⊕ ˇ M ⊂ L K , where U is the hyperbolic lattice. Since the signature of ˇ M is (1 , ˇ t ) = (1 , − t ) , theprimitive embedding i : ˇ M ֒ → L K naturally introduces marked ˇ M -polarized K3surfaces. Marked ˇ M -polarized K3 surfaces are classified by D ( ˇ M ⊥ ) , while marked M -polarized K3 surfaces are classified by D ( M ⊥ ) .For a general marked M -polarized K3 surface ( X, φ ) and a general marked ˇ M -polarized K3 surface ( ˇ X, ˇ φ ) , we have the following isomorphisms:(2.5) N S X ≃ M, T X ≃ U ⊕ ˇ M ; N S ˇ X ≃ ˇ M , T ˇ X ≃ U ⊕ M, and observe the exchange of the algebraic and transcendental cycles (up to the factor U ). This exchange is the hallmark of the mirror symmetry of K3 surfaces. Also wesee the so-called “mirror map” [LY] for K3 surfaces in the following isomorphisms irror Symmetry and Projective Geometry of FM partners 6 (see e.g. [GW, Prop.1]):(2.6) V ( M ) ≃ D ( ˇ M ⊥ ) , V ( ˇ M ) ≃ D ( M ⊥ ) , where V ( M ) is the tube domain defined by V ( M ) = { B + iκ ∈ M ⊗ C | ( κ, κ ) > } and similar definition for V ( ˇ M ) . V ( M ) and V ( ˇ M ) are regarded as the tube domainsfor the complexified Kähler moduli spaces of ( X, φ ) and ( ˇ X, ˇ φ ) , respectively, andhence (2.6) describes the mirror isomorphisms between the complex structure and(complexified) Kähler moduli spaces. There are several different ways to definemirror symmetry of K3 surfaces [Ba1, SYZ]. See references [GW, Be], for example,for the relations among them.2.5. Homological mirror symmetry.
There is a slight asymmetry in the ex-change of the Picard lattices and the transcendental lattices in (2.5). This can beremedied by considering the (numerical) Grothendieck group together with a (non-degenerate) pairing ([ E ] , [ F ]) = − χ ( E , F ) where χ ( E , F ) = P ( − i dim Ext i O X ( E , F ) .Namely, we understand the isomorphisms (2.5) as(2.7) T ˇ X ≃ U ⊕ M ≃ ( K ( X ) , ( ∗ , ∗∗ )) , T X ≃ U ⊕ ˇ M ≃ ( K ( ˇ X ) , ( ∗ , ∗∗ )) . Note that the form ( ∗ , ∗∗ ) is symmetric due to the Serre duality for K3 surfaces.Also we note that K ( X ) contains [ O x ] and − [ I x ] , in addition to [ O D ] = [ O X ] − [ O X ( − D )] for D ∈ P ic ( X ) (likewise for K ( ˇ X ) ). By Riemann-Roch theorem, it iseasy to see that [ O x ] and − [ I x ] explain the additional factor U in U ⊕ M . Theabove isomorphisms are consequences of the homological mirror symmetry due toKontsevich [Ko], but we refrain from going into the details about this in this article.2.6. FM(X) and mirror symmetry.
Let us consider the case M n = h n i , i.e., ( Z h, h = 2 n ) in detail. We first note that we can embed the lattice M n into thehyperpolic lattice U by making a primitive embedding h n i ⊕ h− n i ⊂ U . Then,since primitive embedding i : M n ֒ → L K is unique up to isomorphism due toTheorem A.2, we may assume that the embedding i : M n ֒ → L K is given by M n ⊕ M ⊥ n = h n i ⊕ ( U ⊕ ˇ M n ) ⊂ L K where M ⊥ n := ( i ( M n )) ⊥ = h− n i ⊕ U ⊕ ⊕ E ( − ⊕ is the orthogonal lattice and ˇ M n := h− n i ⊕ U ⊕ E ( − ⊕ .Let ( X, φ ) be a marked M n -polarized K3 surface, and h be its polarization ( h = 2 n ) . Then we have | F M ( X ) | = 2 p ( n ) − from the counting formula. Onthe other hand, for a general marked ˇ M n -polarized K3 surface ( ˇ X, ˇ φ ) , we have | F M ( ˇ X ) | = 1 since ρ ( ˇ X ) = 19 and A ˇ M n ≃ Z / n Z (see [HLOY2, Cor.2.6] and also[Mu1, Proposition 6.2]).It has been argued in [HLOY1] that the number | F M ( X ) | = 2 p ( n ) − has anice interpretation from the monodromy group which acts on the period domain D ( ˇ M ⊥ n ) + for the mirror marked polarized ˇ M n -polarized K3 surfaces. Roughlyspeaking, the number | F M ( X ) | appears as the covering degree of the map from D ( ˇ M ⊥ n ) + /O + ( ˇ M ⊥ n ) ∗ to the corresponding quotient for the isomorphism classes of ˇ M n -polarizable K3 surfaces.We have determined, in Subsection 2.2, the monodromy group of the marked ˇ M n -polarized K3 surfaces by O + ( ˇ M ⊥ n ) ∗ = O + ( ˇ M ⊥ n ) ∩ O ( ˇ M ⊥ n ) ∗ . As for the ˇ M n -polarizable K3 surfaces, the corresponding group becomes larger. irror Symmetry and Projective Geometry of FM partners 7 Lemma 2.3 ([HLOY1, Lem.1.14, Def.1.15]) . The monodromy group of the ˇ M n -polarizable K3 surfaces is given by O + ( ˇ M ⊥ n ) / {± id } . By definition, for ˇ M n -polarizable K3 surfaces ˇ X, ˇ X ′ , we have markings φ, φ ′ suchthat ( ˇ X, ˇ φ ) and ( ˇ X ′ , ˇ φ ′ ) are marked ˇ M n -polarized K3 surfaces. Then, the abovelemma can be deduced from the following diagram which describes the isomorphismof ˇ M n -polarizable K3 surfaces:(2.8) H ( ˇ X, Z ) ∼ ˇ φ / / L K ϕ ≀ (cid:15) (cid:15) ˇ M n ? _ i o o g ≀ (cid:15) (cid:15) H ( ˇ X ′ , Z ) ∼ ˇ φ ′ / / L K ˇ M n ? _ i o o Here we sketch the proof of the lemma: Suppose an element h ∈ O ( ˇ M ⊥ n ) is given.Since primitive embedding ˇ M ⊥ n = U ⊕ M n ֒ → L K is unique by Theorem A.2, h extends to an isomorphism ϕ : L K → L K and also determines an isomorphism g : ˇ M n → ˇ M n on the orthogonal complement of ˇ M ⊥ n . By the surjectivity of theperiod map, we see that ϕ extends to an isomorphism of ˇ M n -polarizable K3 surfaces.From the relation D ( ˇ M ⊥ n ) /O ( ˇ M ⊥ n ) ≃ D ( ˇ M ⊥ n ) + /O + ( ˇ M ⊥ n ) and the fact that {± id } has a trivial action on D ( ˇ M ⊥ n ) + , the group O + ( ˇ M ⊥ n ) / {± id } identifies the ˇ M n -polarizable K3 surfaces which are isomorphic to each other. In this sense, we cancall the quotient group O + ( ˇ M ⊥ n ) / {± id } the monodromy group of ˇ M n -polarizableK3 surfaces. (cid:3) Now we can see the FM number | F M ( X ) | = 2 p ( n ) − as the covering degree ofthe map D ( ˇ M ⊥ n ) + /O + ( ˇ M ⊥ n ) ∗ → D ( ˇ M ⊥ n ) + /O + ( ˇ M ⊥ n ) , which we evaluate for n = 1 (see [HLOY1, Theorem 1.18] for details) as [ O + ( ˇ M ⊥ n ) / {± id } : O + ( ˇ M ⊥ n ) ∗ ] = 2 p ( n ) − , where we recall the fact that {± id } acts trivially on the domain. The coveringdegree can be explained by the nontrivial actions of g in the diagram (2.8), whichimplies that ( ˇ X, ˇ φ ) and ( ˇ X ′ , ˇ φ ′ ) are related by Hodge isometries that have non-trivial actions on the Picard lattice. The monodromy group O + ( ˇ M ⊥ n ) ∗ comes fromthe Dehn twists which preserve (the cohomology classes of) generic symplecticforms (Kähler forms) κ ˇ X ([HLOY1, Thm.1.9]). Then the covering group representsisomorphisms of K3 surfaces which do not preserve the (cohomology classes of)generic symplectic forms κ ˇ X . This is the mirror symmetry interpretation of F M ( X ) made in [ibid], where the relation of the Dehn twits to Auteq D b ( X ) has beendiscussed in more detail.2.7. An example due to Mukai.
Here we consider an explicit construction of the M = h i -polarized K3 surfaces due to Mukai [Mu4]. We see general propertiesdiscussed in the last subsections for this specific example, and make an observationthat will be shared with the examples of Calabi-Yau threefolds in the subsequentsections. Note that F M ( X ) = { X, Y } with Y ≃ M X (2 , h, for general M -polarized K3 surfaces X . irror Symmetry and Projective Geometry of FM partners 8 Linear sections of OG ( , ) . Let us consider orthogonal Grassmannian
OG(5 , which parametrizes maximal isotropic subspaces of C with a fixed non-degenerate quadratic form. OG(5 , has two connected components OG ± (5 , ,which are isomorphic to each other. OG + (5 , ≃ OG − (5 , is called spinor vari-ety S (of dimension ), and can be embedded into the projective space P ( S ) ofthe spin representation of SO (10) . OG + (5 , is the Hermitian symmetric space SO(10 , R ) / U(5) , and its Picard group is generated by the ample class of the abovespinor embedding. The projective dual variety (discriminantal variety) S ∗ in thedual projective space P ( S ∗ ) is known to be isomorphic to S . Mukai [Mu4] con-structed a smooth K3 surface of degree (with Picard group Z h ) by consideringa complete linear section X = S ∩ H ∩ ... ∩ H and observed that the modulispace of stable vector bundles M X (2 , h, over X is isomorphic to a K3 surface Y ,which is defined in the dual variety S ∗ in the following way: Let L be a general8-dimensional linear subspace in S ∗ and by L ⊥ its orthogonal space in S . Thenthe K3 surfaces X and Y above are given by the “orthogonal linear sections to eachother”, X = S ∩ P ( L ⊥ ) ⊂ P ( S ) , Y = S ∗ ∩ P ( L ) ⊂ P ( S ∗ ) . Due to the isomorphism Y ≃ M X (2 , h, (see [IM] for a proof), we can writethe equivalence Φ P : D b ( Y ) ≃ D b ( X ) using the universal bundle P over X × Y asthe kernel of the Fourier-Mukai transform Φ P ( − ) = Rπ X ∗ ( Lπ ∗ Y ( − ) ⊗ P ).2.7.2. Mirror family of M -polarized K3 surfaces . Let us consider marked ˇ M -polarized K3 surfaces, which are the mirror K3 surfaces of X as defined in Sub-section 2.4. Their isomorphism classes are classified by the points on the quotient ofthe period domain D ( ˇ M ⊥ ) by the group O ( ˇ M ⊥ ) ∗ . Noting that D ( ˇ M ⊥ ) ≃ V ( M ) consists of two copies of the upper half pane H + and an isomorphism O + ( ˇ M ⊥ ) ∗ ≃ Γ (6) +6 (see [Do, Thm.(7.1), Rem.(7.2)]), we have D ( ˇ M ⊥ ) + /O + ( ˇ M ⊥ ) ∗ ≃ H + / Γ (6) +6 , see Fig.1. On the other hand, we have an isomorphism O + ( ˇ M ⊥ ) / {± id } ≃ Γ (6) + for the monodromy group of the ˇ M ⊥ -polarizable K3 surfaces [ibid] (see also [HLOY1,Thm.5.5]). For these two groups, we have the following presentations:(2.9) Γ (6) + = h (cid:18) (cid:19) , (cid:16) − √ √ (cid:17) , (cid:16) √ √ √ √ (cid:17) i =: h T , S , S S i , Γ (6) +6 = h (cid:18) (cid:19) , (cid:16) − √ √ (cid:17) , (cid:18) (cid:19) i =: h T , S , ( S S ) i , with S = (cid:16) −√ √ − √ √ (cid:17) . Explicit relations of Γ (6) + and Γ (6) +6 to O + ( ˇ M ⊥ ) / {± id } and O + ( ˇ M ⊥ ) ∗ , respectively, are given by fixing an isomorphism ˇ M ⊥ ≃ ( Z ⊕ , Σ ) with Σ = (cid:16) (cid:17) and an anti-homomorphism R : P SL (2 , R ) → SO (2 , , R ) , R : (cid:0) a bc d (cid:1) a − ac − c − ab ad + bc cd − b bd d ! ∈ SO (2 , , R ) , where SO (2 , , R ) = { g ∈ Mat(3 , R ) | t g Σ g = Σ } . Here, we naturally consider O + ( ˇ M ⊥ ) , O + ( ˇ M ⊥ ) ∗ in SO (2 , , R ) (and the image of O + ( ˇ M ⊥ ) → SO (2 , , R ) ,g (det g ) g for O + ( ˇ M ⊥ ) / {± id } ). The group index [Γ (6) + : Γ (6) +6 ] = 2 is irror Symmetry and Projective Geometry of FM partners 9 xa oo a )( x )( x w = ___w t D + D −− D + D −− t a ( ) t ( ) t T −10 a ( ) t S S S T −10 S S i _ _ S S T −10 S S oo ( t ) _ x x x S S S ( ) S S S Fig.2.1.
Fundamental Domain of Γ (6) +6 . The image of themirror map t = t ( x ) [LY] is depicted as the fundamental domain of Γ (6) +6 . The images of , a , a , ∞ have nontrivial isotropy groups,which explain the monodromy around each point. The generatorsof the isotropy groups are shown at each point.obvious from (2.9) and this is the mirror interpretation of | F M ( X ) | = 2 in thiscase.We can actually construct a family of (marked) ˇ M -polarized K3 surfaces ˇ X = (cid:8) ˇ X x (cid:9) x ∈ P parametrized by P [LY, PS], whose Picard-Fuchs differential equationfor period integrals has the following form with θ x = x ddx :(2.10) (cid:8) θ x − x (2 θ x + 1)(17 θ x + 17 θ x + 5) + x ( θ x + 1) (cid:9) ω ( x ) = 0 , where ω ( x ) = ´ γ Ω( ˇ X x ) is the period integrals of nowhere vanishing holomorphic2 form ω ˇ X x = Ω( ˇ X x ) with respect to a transcendental cycle γ ∈ H ( ˇ X x , Z ). In[HLOY1], the corresponding P family of ˇ M -polarizable K3 surfaces has beenstudied in detail.2.7.3. Monodromy calculations . As we see in Fig.1, there are two cusps in H + / Γ (6) +6 . By Proposition 2.4 below, we see that these two are identified bythe action of an element Γ (6) + \ Γ (6) +6 . In fact, these cusps correspond to themaximally unipotent monodromy (MUM) points at x = 0 and x = ∞ of (2.10),which we read in the following Riemann’s P scheme: a a ∞
12 12 with a := 17 − √ , a := 17 + 12 √ (see [Mo] for a general definition of MUMpoints). The relation of these cusps becomes explicit by constructing an integralbasis of the solutions of the Picard-Fuchs equation (2.10) which is compatible withthe mirror isomorphism T ˇ X ≃ ( K ( X ) , − χ ( ∗ , ∗∗ )) in (2.7). Since the construction isgeneral for other K3 surfaces [Ho] and also parallel to that for Calabi-Yau threefolds(see [HoTa1, Secti.2]), we briefly sketch it here. Firstly, we set up the local solutionsabout the MUM point x = 0 of the form w ( x ) = 1 + O ( x ) and w ( x ) = w ( x ) log( x ) + w reg ( x ) ,w ( x ) = − w ( x )(log x ) + 2 w ( x ) log x + w reg ( x ) irror Symmetry and Projective Geometry of FM partners 10 requiring the forms w reg ( x ) = c x + O ( x ) and w reg ( x ) = c x + O ( x ) . We make sim-ilar solutions ˜ w k ( z ) ( z = x ) around z = 0 requiring ˜ w ( z ) = z (1 + O ( z )) , ˜ w reg ( z ) = z (˜ c z + O ( z )) and ˜ w reg ( z ) = z (˜ c z + O ( z )) . Using these, we set the followingansatz for the integral basis:(2.11) Π( x ) = N x (cid:18) − deg X (cid:19) (cid:16) n w n w n w (cid:17) , ˜Π( z ) = N z (cid:18) − deg X (cid:19) (cid:18) n ˜ w n ˜ w n ˜ w (cid:19) , where N x and N z are unknown constants and n k := πi ) k . These forms are ex-pected in general to give an integral basis which represents the mirror isomorphism T ˇ X ≃ ( K ( X ) , − χ ( ∗ , ∗∗ )) with the bilinear form Σ n = (cid:16) n
01 0 0 (cid:17) (deg X = 2 n ) . Theconstants N x, N z are determined by the Griffiths tansversalities;(2.12) t ΠΣ n Π = t ΠΣ n ddx Π = 0 , t ΠΣ n d dx Π = − πi ) C xx , t ˜ΠΣ n ˜Π = t ˜ΠΣ n ddz ˜Π = 0 , t ˜ΠΣ n d dz ˜Π = − πi ) C xx (cid:0) dxdz (cid:1) , where C xx = 12 / ((1 − x + x ) x ) is the Griffiths-Yukawa coupling [CdOGP]normalized by deg X = 12 . The following results are parallel to those in [HoTa1,Prop.2.10]: Proposition 2.4. (1)
The ansatz (2.11) with N x = N z = 1 satisfies (2.12). (2) The two local solutions are related under an analytic continuation along a paththrough the upper half plane by Π( x ) = U xz ˜Π( z ) with U xz = (cid:16) −
21 5 − − −
12 3 (cid:17) . (3) Monodromy matrices M c of Π( x ) ( ˜ M c of ˜Π( z )) around each singular point x = c of (2.10) are given by x = 0 a a ∞ M c (cid:16) - -
12 1 (cid:17) (cid:16) (cid:17) (cid:16) -
24 120 25 -
10 49 1025 - - (cid:17) (cid:16) - - - − -
54 180 25 (cid:17) ˜ M c (cid:16)
25 120 - - -
71 14 - -
252 49 (cid:17) (cid:16) - -
120 2510 49 - - (cid:17) (cid:16) (cid:17) (cid:16) - -
12 1 (cid:17) and satisfy M M a M a M ∞ = id and ˜ M c = U − xz M c U xz with U − xz = (cid:16) − − − − (cid:17) . (4) M c ’s and U xz are given in terms of generators of Γ (6) + in (2.9) by M = R ( T − ) , M a = − R ( S ) , M a = − R ( S S S ) , U xz = R ( S S ) . In particular M , M a , M a ∈ O ( ˇ M ⊥ ) ∗ and U xz ∈ O ( ˇ M ⊥ ) \ O ( ˇ M ⊥ ) ∗ with thesymmetric form Σ . In Fig. 2.1, we see that the modular action of the element S S ∈ Γ (6) + \ Γ (6) +6 on H + identifies the image of D + with that of D − by exchanging the two cusp points.2.7.4. FM functor Φ P and Auteq D b ( X ) . We can read more from the mirrorisomorphism T ˇ X ≃ ( K ( X ) , − χ ( ∗ , ∗∗ )) which comes from the monodromy calcu-lations. Let us note that the integral basis Π( x ) = t (Π , Π , Π ) in Proposition2.4 implicitly determines the corresponding basis ( γ , γ , γ ) of the transcendentallattice T ˇ X . As for the basis of the lattice ( K ( X ) , − χ ( ∗ , ∗∗ )) , we may take ([ E ] , [ E ] , [ E ]) = ([ O x ] , [ O h ] + 6[ O x ] , − [ I x ]) , with → O X ( − h ) → O X → O h → , and O x the skyscraper sheaf and I x the idealsheaf of a point x ∈ X . Note that we choose [ E ] so that ch ([ O h ] + 6[ O x ]) = h , and irror Symmetry and Projective Geometry of FM partners 11 hence we can verify ( − χ ([ E i ] , [ E j ])) = Σ by Riemann-Roch theorem. Identifyingthese two basis, we have an explicit isomorphism T ˇ X ≃ ( K ( X ) , − χ ( ∗ , ∗∗ )) (this canbe done in general [Ho, Sect.2.4]).Actually, the identification of the two basis above is somehow canonical from theviewpoint of homological mirror symmetry, since we can show that the topology of γ is isomorphic to the real two torus, i.e. γ ≈ T . The identification of such toruscycle with O x is justified from many aspects of the homological mirror symmetry D b F uk ( ˇ X ) ≃ D b ( X ) (see [Ko, SYZ]). Note also that γ is isotropic in T ˇ X andchoosing such a vector in T ˇ X determines (almost uniquely, i.e., up to signs) otherbases with the specified intersection numbers in the entries of Σ . Similar construc-tion of the basis of ˜Π( z ) (or the cycles ˜ γ , ˜ γ , ˜ γ ) and the identification ˜ γ ≈ T with O y are valid for ( K ( Y ) , − χ ( ∗ , ∗∗ )) . We denote by h ′ the polarization of Y .Now recall that the Fourier-Mukai functor Φ P : D b ( Y ) ≃ D b ( X ) is defined by thekernel P , the universal bundle over X × Y = X × M X (2 , h, , and hence we have Φ P ( O y ) = P y with the Mukai vector ch ( P y ) √ T odd X = 2 + h + 3 v ( v := ch ( O x )) .From this, we have ch (Φ P ( O y )) = ch ( P y ) = 2 + h + v = 3 v + h + 2(1 − v )= 3 ch ([ E ]) + ch ([ E ]) − ch ([ E ]) , and identify this in the 1st column of the connection matrix U xz = R ( S S ) (notethat we identify ˜ γ with O y ). This leads us to a conjecture that the continuationof the cycles ˜ γ , ˜ γ , ˜ γ to γ , γ , γ corresponds to the Fourier-Mukai functor Φ P : D b ( Y ) ≃ D b ( X ) . Note that the analytic continuation of Π( x ) connects cycles in thefibers around x = 0 and those around x = ∞ , but actually it comes from a Dehntwist of ˇ X because the local family around x = 0 and x = ∞ are isomorphic as thefamily of ˇ M -polarizable K3 surfaces. Dehn twists around x = 0 , a , a , ∞ are easyto be identified from the standard forms of the monodromy matrices M , M a , ˜ M a and ˜ M ∞ . They can be identified, respectively, with the following Fourier-Mukaifunctors (see e.g. [ST]): ( − ) ⊗ O X ( h ) , Φ I ∆( X ) , Φ P ◦ Φ I ∆( Y ) ◦ Φ − P and Φ P ◦ (cid:0) ( − ) ⊗ O Y ( h ′ ) (cid:1) ◦ Φ − P , where I ∆( X ) (resp. I ∆( Y ) ) is the ideal sheaf of the diagonal ∆ ⊂ X × X (resp. ∆ ⊂ Y × Y ) and h ′ is the polarization of Y . From the above considerations, andtaking the monodromy relation into account, we naturally come to a conjecturethat the group Auteq D b ( X ) is generated by the shift functor and the followingFourier-Mukai functors: ( − ) ⊗ O X ( h ) , Φ I ∆( X ) and Φ P ◦ Φ I ∆( Y ) ◦ Φ − P . Some other aspects.
From the example in the previous subsection, one mayexpect some relation between the Fourier-Mukai numbers | F M ( X ) | and the num-bers of MUM points in D ( ˇ M ⊥ ) /O ( ˇ M ⊥ ) ∗ . In fact, S. Ma [Ma] (see also [Ha])showed that the counting formula in Theorem 2.2 allows such interpretation if weidentify MUM points with the standard cusps in the Baily-Satake compactificationof D ( ˇ M ⊥ ) /O ( ˇ M ⊥ ) ∗ . From this viewpoint, we can read the counting formula as thenumber of non-isomorphic decompositions of ˇ M ⊥ into ˇ M ⊥ = U ⊕ M modulo the The correspondence between the Chern characters ch ( P y ) = ch (Φ P ( O y )) for P = P Y → X ( Y ∈ F M ( X )) and the elements in Γ ( n ) + \ Γ ( n ) + n in general has been worked in [Kaw]. irror Symmetry and Projective Geometry of FM partners 12 actions of O ( ˇ M ⊥ ) ∗ . Non-standard cusps are 0-dimensional boundary points whichcorrespond to the decompositions ˇ M ⊥ = U ( m ) ⊕ M ( m > . In ref. [Ma], thecounting formula has been generalized to incorporate non-standard cusps, and ithas been shown that the generalized formula counts the number of twisted Fourier-Mukai partners, i.e., K3 surfaces Y satisfying D b ( X ) ≃ D b ( Y, α ) where α is anelement of the Brauer Group Br ( Y ) . See references [HS, Ca] for the derived cate-gories of twisted sheaves on Y .3. Fourier-Mukai partners of Calabi-Yau threefolds I
We define Calabi-Yau 3-folds by smooth, projective, three dimensional varieties X over C which satisfy K X ≃ O X , H ( X, O X ) = H ( X, O X ) = 0 . It is known, dueto Bridgeland [Br2], that birational Calabi-Yau 3-folds X, Y are derived equivalent,i.e., D b ( X ) ≃ D b ( Y ) . Except this general theorem, however, not much is knownabout the Fourier-Mukai partners of Calabi-Yau 3-folds. Here and in the nextsection, we focus on two examples of pairs of Calabi-Yau 3-folds with Picard numberone which are Fourier-Mukai partners but not birational to each other. In bothcases, some similarity to the example of Mukai in the last section will be observed inthe fact that suitable projective dualities play important roles in their constructionsand also their derived equivalences.3.1. Grassmannian and Pfaffian Calabi-Yau threefolds.
The first example isCalabi-Yau 3-folds due to Rødland. Let
G(2 , be the Grassmannian of two dimen-sional subspaces in C . Consider the Plücker embedding of G(2 , into P ( ∧ C ) .Then the projective dual of G(2 , is the Pfaffian variety Pf(4 , in the dual pro-jective space P ( ∧ ( C ∗ ) ) , i.e., the locus (cid:8) [ c ij ] ∈ P ( ∧ ( C ∗ ) ) | rank ( c ij ) ≤ (cid:9) . Letus consider general 7 dimensional linear subspace L ⊂ ∧ ( C ∗ ) and its orthogonalsubspace L ⊥ ⊂ ∧ C . Then, similarly to the construction in Subsection 2.7.1, wedefine X = G(2 , ∩ P ( L ⊥ ) ⊂ P ( ∧ C ) , Y = Pf(4 , ∩ P ( L ) ⊂ P ( ∧ ( C ∗ ) ) .X and Y , respectively, are called Grassmannian and Pfaffian Calabi-Yau 3-folds. Proposition 3.1 (Rødland [Ro]) . When L is general, both X and Y are smoothCalabi-Yau 3-folds with Picard number one and the following invariants: H X = 42 , c ( X ) .H X = 84 h , ( X ) = 1 , h , ( X ) = 50 H Y = 14 , c ( Y ) .H Y = 56 h , ( Y ) = 1 , h , ( Y ) = 50 where H X and H Y are the ample generators of the Picard groups, respectively. As for the smoothness, it is further known that X is smooth if and only if Y issmooth [BC]. The equal Hodge numbers might indicate a possibility that X and Y were birational to each other [Ba2]. However, looking the degrees H X = 42 and H Y = 14 together with ρ ( X ) = ρ ( Y ) = 1 , we see that this is not the case. irror Symmetry and Projective Geometry of FM partners 13 In [Ro], Rødland studied mirror symmetry of Pfaffian Calabi-Yau threefold Y and constructed a mirror family Y = (cid:8) ˇ Y x (cid:9) x ∈ P by the so-called orbifold mirrorconstruction. His construction starts with a special family of Pfaffian Calabi-Yau3-folds which admits a Heisenberg group action [GrPo]. By finding a suitablesubgroup of the Heisenberg group as the orbifold group, and making a crepantresolutions for the singularities in the orbifold mirror construction, the desiredmirror Calabi-Yau 3-folds ˇ Y with Hodge numbers h , ( ˇ Y ) = 50 , h , ( ˇ Y ) = 1 wasobtained. Independently, mirror symmetry of Grassmannian Calabi-Yau 3-folds X was studied in [BCKvS] by the method of toric degeneration of Grassmannians.It was recognized by these authors that the Picard-Fuchs differential equations forthese two families have exactly the same form but they are distinguished by twodifferent MUM points of the equation, as we have witnessed in the equation (2.11).In particular, it was observed that Gromov-Witten invariants ( g = 0 ) calculatedfrom the two MUM points ( x = 0 and x = ∞ in Subsection 3.4) match to those for X and Y, respectively.Later, in [HK], the calculation of Gromov-Witten invariants ( g = 0) have beenextended to higher genus ( g ≤ solving the so-called BCOV holomorphic anomalyequation discovered in [BCOV1,2].3.2. Derived equivalence D b ( X ) ≃ D b ( Y ) . As described in the previous subsec-tion, there are similarities in their constructions between the example of Fourier-Mukai partners in Subsection 2.7 and the Grassmannian and Pfaffian Calabi-Yau3-folds X and Y. It is natural to expect that X and Y are derived equivalent.In fact, the derived equivalence is supported from the analysis of Gauged LinearSigma Model (GLSM) in physics [HT]. The derived equivalence has been provedmathematically in [BC] and [Ku2] (see also [BDFIK, ADS] for recent progresses).Let Y be the Pfaffian variety Pf(4 , . Y is singular along Y sing = { [ c ij ] | rk c ≤ } and has a natural (Springer-type) resolution(3.1) ˜ Y = { ([ c ] , [ w ]) | w ⊂ ker c } ⊂ Y × G(3 , . Since it is easy to see that all the fibers of the projection ρ : ˜ Y →
G(3 , areisomorphic to P , ˜ Y is smooth. Let us denote G(2 , by X . Then we have X = X ∩ P ( L ⊥ ) and also we can write Y = ˜ Y ∩ P ( L ) since Y sing is away from P ( L ) for general L . Let us summarize our settings into the following diagram:(3.2) X ˜ YY G(2 ,
7) G(3 , π (cid:15) (cid:15) ρ x x rrrrrr The proofs of the derived equivalence in [BC] and [Ku2] uses a natural incidencecorrespondence between the two Grassmannians in the diagram, which is given by ∆ = { ([ ξ ] , [ w ]) | dim( ξ ∩ w ) ≥ } ⊂ G(2 , × G(3 , . To sketch the proofs, let us consider the ideal sheaf I ∆ of ∆ and define its pullback I := (id × ρ ) ∗ I ∆ on X × ˜ Y . The restriction I := I| X × Y is an ideal sheaf on X × Y .We regard I as an object in D b ( X × Y ) and defines the Fourier-Mukai functor Φ I ( − ) := Rπ X ∗ ( Lπ ∗ Y ( − ) ⊗ I ) , where π X and π Y are projections to X and Y .Then, Borisov and Caldararu proved the following irror Symmetry and Projective Geometry of FM partners 14 Theorem 3.2 ([BC, Theorem 6.2]) . Φ I ( − ) : D b ( Y ) → D b ( X ) is an equivalence. The proof of the above theorem is based on the following theorem for smoothprojective varieties
X, Y and a Fourier-Mukai functor Φ P ( − ) = Rπ X ∗ ( Lπ ∗ Y ( − ) ⊗P ) with an object P ∈ D b ( X × Y ) (see [BO, Thm.1.1], [Br2, Thm.1.1], [Hu, Cor. 7.5,Prop. 7.6]): Theorem 3.3. If P a coherent sheaf on X × Y flat over Y , then Φ P : D b ( Y ) → D b ( X ) is fully faithful if and only if the following two conditions are satisfied: (i) For any point x ∈ X , it holds Hom( P x , P x ) ≃ C , and (ii) if x = x , then Ext i ( P x , P x ) = 0 for any i.Under these conditions, Φ P is an equivalence if and only of dim X = dim Y and P ⊗ π ∗ X ω X ≃ P ⊗ π ∗ Y ω Y . It has been proved that the ideal sheaf I is flat over Y , and in fact, defines a flatfamily of curves parametrized by Y [BC, Prop. 4.4]. The condition Hom( I y , I y ) ≃ C follows from a general property of ideal sheaves of subschemes of dimension ≤ in smooth projective 3-folds [ibid,Prop. 4.5]. Hence, verifying the cohomologyvanishings(3.3) Ext • ( I y , I y ) = 0 ( y = y ) is the main part of the proof given in [ibid].Kuznetsov formulates the derived equivalence as a consequence of the homo-logical projective duality (HPD) between G(2 , and Pf(4 , (precisely, the non-commutative resolution of Pf(4 , ). In the proof given in [Ku2], the followinglocally free resolution of the ideal sheaf I on X × ˜ Y plays an important role:(3.4) → S U ⊠ O ˜ Y → U ⊠ ˜ Q → O X ⊠ ∧ ˜ Q → I ⊗ O X × ˜ Y (1 , (1 , → , where U is the universal bundle on G(2 , , ˜ Q is the universal quotient bundle on G(3 , and O X × ˜ Y (1 , (1 , O X (1) ⊠ ρ ∗ O G(3 , (1)) (see [ibid, Lemma 8.2]). Therestriction of (3.4) to X × { y } is nothing but the Eagon-Northcot complex whichwas used for the proof of the vanishings (3.3) in [BC, Prop. 3.6]. Although wedo not go into the details of HPD, but for the comparison with the correspondingresults in another example in the next section it is useful to summarize some ofthe main results in [Ku2]. For that, let us introduce the following notation for thesheaves that appear in (3.4): E = S U , E = U , E = O X ; F = O ˜ Y , F = ˜ Q, F ′ = ∧ ˜ Q, and define the following full subcategories A i ⊂ D b ( X ) ( i = 0 , ..., and B k ⊂ D b ( ˜ Y ) ( k = 0 , ..., :(3.5) h E , E , E i = A = A = · · · = A ⊂ D b ( X ) , h F ∗ , F ∗ , F ∗ i = B = B = · · · = B ⊂ D b ( ˜ Y ) , where we set F := F ′ / O ˜ Y (1 , − with O ˜ Y ( a, b ) = ρ ∗ O G(3 , ( a ) ⊗ π ∗ O Y ( b ) . Theorem 3.4 ([Ku2, Theorem 4.1]) . Denote by A i ( a ) , B i ( a ) the twists of A i , B i by O X ( a ) and π ∗ O Y ( a ) , respectively. Then (i) hA , A (1) , · · · , A (6) i is a Lefschetz decomposition of D b ( X ) , and (ii) hB ( − , · · · , B ( − , B i is a dual Lefschetz decomposition of ˜ D b ( Y ) , irror Symmetry and Projective Geometry of FM partners 15 where ˜ D b ( Y ) ⊂ D b ( ˜ Y ) is a full subcategory which is equivalent to D b ( Y , R ), thebounded derived category of coherent sheaves of right R -modules on Y with R = π ∗ E nd ( O ˜ Y ⊕ ρ ∗ ˜ U ) and ˜ U the universal bundle on G(3 , . A (dual) Lefschetz decomposition is a special form of a semi-orthogonal decom-position of a triangulated category [BO]. In our case, the vanishings
Hom • D b ( ˜ Y ) ( B i ( − i ) , B j ( − j )) = 0 ( i < j ) , which are implied in (ii) of the above theorem, entail the desired vanishings (3.3).3.3. BPS numbers.
As noted in the previous subsection, the ideal sheaf I y ( y ∈ Y )defines a family of curves on X . It can be shown by explicit calculations withMacaulay2 that Proposition 3.5.
For a general point y ∈ Y , the ideal sheaf I y defines a smoothcurve on X of genus and degree 14. Expecting some relations to the moduli problems of ideal sheaves on X , such asDonaldson-Thomas invariants of X [PT] or BPS numbers [HST], it is interesting toseek a possibly related number in the table of the BPS numbers calculated in [HK].The relevant part of the table to the curves of Proposition 3.5 reads as follows (with d = 14) :(3.6) g · · · n Xg ( d ) × · · · n X (14) = 123676 is rather large to find a relationto the curve defined by I y . However, as noted in [HoTa1, (4-1.6)], we can observethat n (14) = 7 counts a well-known family of curves studied by Mukai, i.e., curvesthat are linear sections of G(2 , . Such curves appear in our setting as G(2 , ∩ P ( L ⊥ ) ⊂ G(2 , ∩ P ( L ⊥ ) = X, and hence they are naturally parametrized by P ≃ { G(2 , ⊂ G(2 , } . Generalmembers of this family are smooth and of genus and degree . Then, followingthe counting “rule” of BPS numbers [GV], we explain the number n (14) = 7 as n (14) = ( − dim P e ( P ) = 7 . The counting “rule” also tells us that such a generically smooth family of curves ofgenus g contributes to the numbers n h ( d ) ( h ≤ g ) in a specified way [ibid]. Thus ourobservation above indicates that there are contributions from at least two differentfamilies of (generically) smooth curves in the BPS numbers { n h (14) } h ≤ in (3.6).3.4. Mirror symmetry.
Consider the mirror family ˇ Y = (cid:8) ˇ Y x (cid:9) x ∈ P obtained fromthe orbifold mirror construction [Ro]. The Picard-Fuchs differential equation satis-fied by the period integrals w ( x ) = ´ γ Ω( ˇ Y x ) ( γ ∈ H ( ˇ Y , Z )) has been determined irror Symmetry and Projective Geometry of FM partners 16 by Rødland as D x w ( x ) = 0 with D x =9 θ x − x (15 + 102 θ x + 272 θ x + 340 θ x + 173 θ x ) − x (1083 + 4773 θ x + 7597 θ x + 5032 θ x + 1129 θ x )+ 2 x (6 + 675 θ + 2353 θ x + 2628 θ x + 843 θ x ) − x (26 + 174 θ x + 478 θ x + 608 θ x + 295 θ x ) + x ( θ x + 1) , and θ x = x ddx . As described in Subsection 3.1, the operator D x is the same as thatof X in [BCKvS, ES] and Gromov-Witten invariants of X and Y are calculated,respectively, from the MUM points at x = 0 and z = x = 0 . Although the geometryof the family is rather complicated (cf. Subsection 4.4), monodromy calculationsproceeds in a similar way to Subsection 2.7. The Riemann’s P -scheme is α α α ∞ , where α k are the (real) roots of the ’discriminant’ − x − x + x = 0 and x = 3 is an apparent singularity with no monodromy (with order α < < α < < α ) .The symplectic and integral basis of the solution can be obtained by making ansatzsimilar to those in Subsection 2.7.3 (see also [DM, ES]). In fact, its full details arecompletely parallel to [HoTa1, (2-5.1)-(2-5.7)] assuming two local solutions of theforms, Π( x ) = N x β a κ/ γ β − κ/ ! n w ( x ) n w ( x ) n w ( x ) n w ( x ) ! , ˜Π( z ) = N z β ˜ a ˜ κ/ γ ˜ β − ˜ κ/ ! n ˜ w ( z ) n ˜ w ( z ) n ˜ w ( z ) n ˜ w ( z ) ! . Here we summarize only the results of the monodromy matrices.
Proposition 3.6. (1)
When N x = N z = 1 , a = ˜ a = 0 and ( κ, β, γ ) = (cid:16) H X , − c .H X , − ζ (3) e ( X )(2 πi ) (cid:17) , ˜( κ, ˜ β, ˜ γ ) = (cid:16) H Y , − c .H Y , − ζ (3) e ( Y )(2 πi ) (cid:17) , the solutions Π( x ) and ˜Π( z ) are integral and symplectic with respect to the sym-plectic form S = (cid:18) − − (cid:19) . These are analytically continued along a path in theupper-half plane as Π x ( x ) = U xz ˜Π( z ) by a symplectic matrix U xz = (cid:18) − − − −
14 0 − (cid:19) with its inverse U − xz = (cid:18) − − − −
40 5 0 114 0 3 70 −
14 0 − (cid:19) . (2) The monodromy matrices M c of Π( x ) ( ˜ M c of ˜Π( z )) around each singular point c are symplectic with respect to S , and they are given by ( with ˜ M c = U − xz M c U xz ) x = 0 α α α ∞ M c (cid:18) - - - (cid:19) (cid:18) (cid:19) (cid:18) -
14 2 47 - -
49 8 14 -
49 49 - - (cid:19)(cid:18) -
196 1 -
420 0 0 1 (cid:19)(cid:18) -
14 16 426 - -
322 7 - - -
35 28 - - (cid:19) ˜ M c (cid:18) -
27 322 - -
125 4 - -
308 1 - -
42 385 -
13 155 (cid:19)(cid:18) -
196 1 -
700 0 0 1 (cid:19) (cid:18) -
27 0 - -
80 0 1 0 -
49 0 -
14 29 (cid:19) (cid:18) (cid:19) (cid:18) - - - (cid:19) irror Symmetry and Projective Geometry of FM partners 17 and satisfy M a M M a M a M ∞ = id . As before, the integral basis Π( x ) = (Π , Π , Π , Π ) implicitly determines thecorresponding integral cycles γ i , likewise for ˜Π( z ) with the corresponding integralcycles ˜ γ i ( i = 1 , .., . From the geometry of the family, one can see that γ ≈ ˜ γ ≈ T and also γ ≈ ˜ γ ≈ S about the topologies of the cycles. Form thehomological mirror symmetry, these cycles may be identified with the skyscrapersheaves O x , O y ( x ∈ X, y ∈ Y ) and the structure sheaves O X , O Y as was the case inSubsection 2.7.4. Unfortunately we do not see directly the relation ch (Φ I ( O y )) = ch ( I y ) in the 1st column of U xz as before. However, we believe that if we takesuitable auto-equivalences into account, in other words, if we change the path ofthe analytic continuation, we can identify the Chern character in the connectionmatrix. Recently, precise analysis of the co-called hemi-sphere partition functionsof GLSMs [HR] have been developed. The analysis provides a concrete recipe toconnect the cycles to the objects in derived category (of matrix factorizations),and also reproduces the connection matrix of the analytic continuation [EHKR].We expect that the new method provides us new insights into more details of theabove problem. Also, the significant progresses made in refs [Ha, BDFIK, DS] inthe mathematical aspects of GLSMs are expected to provide us powerful tools tolook into the derived categories of Fourier-Mukai partners and also their mirrorsymmetry.4. Fourier-Mukai partners of Calabi-Yau threefolds II
Here we continue our exposition by the second example which was found recentlyby the present authors [HoTa1,2,3,4].4.1.
Reye congruences Calabi-Yau 3-folds and double coverings.
In [HoTa1],we have found that Rødland’s construction of a pair of Calabi-Yau 3-folds has a nat-ural counterpart in the projective space of symmetric matrices P ( S C ) . Hereafter,we will fix V = C and denote by V k a k -dimensional subspace of V .We have found in [ibid] that the tower of secant varieties of v ( P ( V )) in P ( S V ) and the corresponding (reversed) tower in P ( S V ∗ ) entail a similar duality of Calabi-Yau 3-folds. For the construction, we start with S P ( V ) , i.e., the symmetric productof P ( V ) as the counterpart of the Grassmannian G(2 , ⊂ P ( ∧ C ) . S P ( V ) isthe first secant variety of v ( P ( V )) and can be considered as the rank 2 locus ofsymmetric matrices [ c ij ] ∈ P ( S V ) . It is singular along the v ( P ( V )) , i.e., the rank1 locus. The precise definition of the Pfaffian counterpart will be introduced inthe next section, but here we only describe the resulting Calabi-Yau 3-fold startingwith the rank 4 locus in the dual projective space P ( S V ∗ ) , H := (cid:8) [ a ij ] ∈ P ( S V ∗ ) | det( a ij ) = 0 (cid:9) . irror Symmetry and Projective Geometry of FM partners 18 H is singular along the locus H with H k := { rk ( a ij ) ≤ k } . As before, we considera general five dimensional linear subspace L ⊂ S V ∗ and its orthogonal linearsubspace L ⊥ ⊂ S V . Then we define X = S P ( V ) ∩ P ( L ⊥ ) ⊂ P ( S V ) , H = H ∩ P ( L ) ⊂ P ( S V ∗ ) . Proposition 4.1 (Hosono-Takagi [HoTa1]) . (1) When L is general, X is a smoothCalabi-Yau 3-fold with P ic ( X ) ≃ Z ⊕ Z and the following invariants: H X = 35 , c .H X = 50 , h , ( X ) = 1 , h , ( X ) = 51 , where H X is the generator of the free part of P ic ( X ) . (2) When L is general, H is a determinantal quintic hypersurface in P ( L ) ≃ P ,which is singular along a smooth curve C H of genus 26 and degree 20 with A typesingularities. (3) There is a double covering Y → H branched along C H . Furthermore, Y is asmooth Calabi-Yau 3-fold with P ic ( Y ) = Z H Y and H Y = 10 , c .H Y = 40 , h , ( Y ) = 1 , h , ( Y ) = 51 . If we do parallel constructions with V = C , we obtain an Enriques surface for X . From historical reasons, this Enriques surface X is called Reye congruence, ormore precisely, Cayley model of Reye congruence (see [Co]). In our case of V = C ,Reye congruence X is a Calabi-Yau 3-fold and is paired with another Calabi-Yau3-fold Y as above. It is easy to see that Y is not birational to X by the samearguments as described below Proposition 3.1. In addition to this, we can show[HoTa4] the derived equivalence D b ( X ) ≃ D b ( Y ) , which will be sketched in thenext subsection. Here it should be worth while noting the following interestingproperties of X and Y ([HoTa3, Prop. 3.5.3, 4.3.4], [HoTa4, Prop.3.2.1]): Proposition 4.2. (1) π ( X ) ≃ Z . (2) π ( Y ) ≃ and the Brauer group of Y contains a non-trivial 2-torsion element. As argued in [ibid.,Sect.9.2], one can show an exact sequence, → Z → Br( Y ) → Br( X ) → . If Br( Y ) ≃ Z , then Br( X ) ≃ and this indicates the invariance of the product of(abelianization of) π and the Brauer group, but not each factor, under the derivedequivalence (see [Ad, S] for details).4.2. Derived equivalence D b ( X ) ≃ D b ( Y ) . Here we sketch our proof of thederived equivalence. As we saw in the preceding subsection, our construction of thepair ( X, Y ) is parallel to Rødland’s construction of Grassmannian-Pfaffian Calabi-Yau manifolds. We can pursue this parallelism toward the proof of the derivedequivalence, although the projective geometries become more involved, and we haveonly partial results about the HPD (corresponding to Theorem 3.4) in our case.4.2.1. Resolutions . Let X := S P ( V ) . X is defined by a linear section of X as X = X ∩ P ( L ⊥ ) . We see that X plays a similar role of G(2 , in Rødland’sexample, however there is a difference in that X is singular along the Veroneseembedding of P ( V ) , v ( P ( V )) ⊂ X ⊂ P ( S V ) . For this singularity, we have the irror Symmetry and Projective Geometry of FM partners 19 following natural resolution, ˇ X := Hilb P ( V ) X G(2 , V ) , f | | ③③③③③③ g " " ❉❉❉❉❉❉ where Hilb P ( V ) is the Hilbert scheme of two points on P ( V ) and f is the Hilbert-Chow morphism. The morphism g sends points x ∈ ˇ X to the points g ( x ) ∈ G(2 , V ) representing the lines determined by x . The fiber over [ V ] ∈ G(2 , V ) is g − ([ V ]) ≃ S P ( V ) ≃ P . By our genericity assumption of L , X = X ∩ P ( L ⊥ ) is smooth (seeProposition 4.1) and hence P ( L ⊥ ) is away from the singularity of ˇ X , therefore wemay consider our linear intersection in ˇ X , i.e., X = ˇ X ∩ P ( L ⊥ ) . Again, by thesame reasoning, we have g ( X ) ≃ X , i.e., we have isomorphic image g ( X ) of X in G(2 , V ) . Historically, the image g ( X ) ⊂ G(2 , V ) is called a Reye congruence. H is singular along the rank ≤ locus H . Expecting a (partial) resolutionof the singularity, we consider the following (Springer-type) pairing of singularquadrics and planes therein (cf. (3.1)): Z := { ([ Q ] , [Π]) | P (Π) ⊂ Q } ⊂ H × G(3 , V ) , where [ Q ] ∈ H represents the point corresponding to a singular quadric Q . It iseasy to see that all the fibers of the projection Z → G(3 , V ) are isomorphic to P since they consist of quadrics that contain a fixed plane P (Π) ⊂ P ( V ) . Hence, wesee that Z is smooth. However we have dim Z = 6 + 8 = 14 , while dim H =dim P ( S V ) − , and hence Z → H can not be a resolution of H thatwe expect. To remedy the situation, we consider the Stein factorization Y of themorphism Z → H as follows:(4.1) ZYH ⊂ P ( S V ∗ ) , G(3 , V ) P -bundle / / π Z connected fibers (cid:15) (cid:15) ρ Y (cid:15) (cid:15) where π Z : Z → Y has connected fibers and ρ Y is a finite morphism by definition.From the above dimension counting, the connected fibers generically have dimension dim Z − dim H = 1 . As for the finite morphism ρ Y , looking into the families ofplanes in a singular quadric, it is easy to see that ρ Y is generically and hasits ramification along the singular locus Sing( H ) = H . This corresponds to thecovering we observed in (3) of Proposition 4.1. In fact, about the singular locusof Y , we can see Sing ( Y ) = H [HoTa3, Prop.5.7.2] where we identify the inverseimage ρ − Y ( H ) in Y with H . Hence the covering Y changes the singular locus of H to a smaller one. If the linear subspace L is general, then since P ( L ) ∩ H = ∅ ,the singularities in the linear section H = H ∩ P ( L ) is removed by ρ Y . This isexactly the smooth double covering Y in (3) of Proposition 4.1. We write the doublecover of H simply by Y = Y ∩ P ( L ) with understanding the pullback of P ( L ) to Y . A natural resolution f Y → Y follows by studying geometries of singularquadrics H [HoTa3], which is interesting by itself from the projective geometry of irror Symmetry and Projective Geometry of FM partners 20 quadrics [Ty]. Birational geometry of Y and f Y will be described in Section 5 byintroducing other birational models of Y .It would be helpful now to write our X and Y in terms of the resolutions ˇ X and f Y as X = ˇ X ∩ P ( L ⊥ ) , Y = f Y ∩ P ( L ) . The derived equivalence follows from certain ideal sheaf on f Y × ˇ X constructedin a parallel way to the Grassmannian-Pfaffian Calabi-Yau 3-folds. The followingproposition is a part of the birational geometry of Y (see Fig. 5.2): Proposition 4.3. (1)
There exists a resolution ρ e Y : f Y → Y . (2) There exists a blow-up Y → f Y , and over Y there is a generically conic bundle π ′ : Z → Y that admits a morphism µ : Z → G(3 , V ) . We summarize the resolutions and morphisms as follows (cf. (3.2) ):(4.2) Z Y f YY G(3 , V ) G(2 , V ) ˇ X π ′ (cid:15) (cid:15) ˜ ρ o o ρ f Y o o µ ❇❇❇❇❇❇ g (cid:2) (cid:2) ✆✆✆✆✆ Incidence relation ∆ . In the diagram (4.2), we introduce the followingincidence relation ∆ : ∆ = { ([ V ] , [ V ]) | V ⊃ V } ⊂ G(3 , V ) × G(2 , V ) , and consider its ideal sheaf I ∆ . Pulling this back to Z × ˇ X , we obtain I ∆ =( µ × g ) ∗ I ∆ . Since the variety ∆ is nothing but the flag variety F (2 , , V ) , wehave locally free resolution,(4.3) → ∧ ( W ∗ ⊠ F ) → ∧ ( W ∗ ⊠ F ) → ∧ ( W ∗ ⊠ F ) → W ∗ ⊠ F → I ∆ → , where → U → V ⊗ O G(3 ,V ) → W → and → F → V ⊗ O G(2 ,V ) → G → are the universal sequences on the Grassmannians G(3 , V ) and G(2 , V ) ( rk U =3 , rk F = 2 ), respectively. Roughly speaking, the direct image (˜ ρ × id) ∗ ◦ ( π ′ × id) ∗ I ∆ is the ideal sheaf I on f Y × ˇ X which corresponds to the one used in theGrassmannian-Pfaffian case in [BC] and [Ku2]. In actual calculation of the directimage, however, we need to use the structure of the conic bundle. Hence we firstrestrict the generically conic bundle to a conic bundle π o ′ : Z o → Y o := Y \ P σ ,where P σ is a certain subvariety of dimension , and define I o := (˜ ρ o × id) ∗ ◦ ( π o ′ × id) ∗ I ∆ with the corresponding restriction ˜ ρ o : Y o → f Y o . Then I = ι ∗ I o underthe inclusion ι : f Y o ֒ → f Y is the precise definition of the ideal sheaf I .4.2.3. Derived equivalence . The proof of derived equivalence in [HoTa4] proceedsby constructing the Fourier-Mukai functor with the kernel I = I| Y × X as in Sub-section 3.2. In the paper [ibid], we have obtained a locally free resolution of theideal sheaf I starting with (4.3). To describe the results, we introduce locally freesheaves on f Y . irror Symmetry and Projective Geometry of FM partners 21 Proposition 4.4.
There exists locally free sheaves ˜ S L , ˜ T , ˜ Q on f Y which satisfy π ′ ∗ (cid:8) µ ∗ O G(3 ,V ) (1) (cid:9) ≃ ˜ ρ ∗ ˜ S ∗ L , π ′ ∗ ( µ ∗ W ) ≃ ˜ ρ ∗ ˜ T ,π ′ ∗ (cid:8)(cid:0) µ ∗ S W (cid:1) ⊗ µ ∗ O G(3 ,V ) ( − (cid:9) ≃ ˜ ρ ∗ (cid:0) ˜ Q ⊗ O e Y ( − M e Y ) (cid:1) , where M e Y is the divisor corresponding to ρ ∗ e Y ◦ ρ ∗ Y O H (1) .Proof. See [HoTa4, Prop.5.6.4] and [HoTa3, Prop.6.1.2,6.2.3]. (cid:3)
We denote by L ˇ X (resp. H ˇ X ) the divisor on ˇ X corresponding to g ∗ O G(2 ,V ) (1) (resp. g ∗ O X (1) ). Then, we have Theorem 4.5 ([HoTa4, Theorem 5.1.3]) . We have the following locally free reso-lution: → ˜ S L ⊠ O ˇ X → ˜ T ∗ ⊠ g ∗ F ∗ → (cid:0) O e Y ⊠ g ∗ S F ∗ (cid:1) ⊕ (cid:0) ˜ Q ∗ ( M e Y ) ⊠ O ˇ X ( L ˇ X ) (cid:1) → I ⊗ (cid:0) O e Y ( M e Y ) ⊠ O ˇ X (2 L ˇ X ) (cid:1) → . Extracting each term of the above resolution of I , we define the following nota-tion: ( E , E , E a , E b ) =( ˜ S L , ˜ T ∗ , O e Y , ˜ Q ∗ ( M e Y )) , ( F , F , F ′ a , F b ) =( O ˇ X , g ∗ F ∗ , g ∗ S F ∗ , O ˇ X ( L ˇ X )) , and set F a = F ′ a / O ˇ X ( − H ˇ X + 2 L ˇ X ) . Now corresponding to (3.5) in Subsection3.2, we define the following full-subcategories hE , E , E a , E b i = A = A = · · · = A ⊂ D b ( f Y ) , hF ∗ b , F ∗ a , F ∗ , F ∗ i = B = B = · · · = B ⊂ D b ( ˇ X ) . Theorem 4.6 ([HoTa3, Theorem 3.4.5, 8.1.1]) . Denote by A i ( a ) , B i ( b ) the twistsof A i , B i by O e Y ( aM e Y ) and O ˇ X ( bH ˇ X ) , respectively. Then (i) hA , A (1) , · · · , A (9) i is a Lefschetz collection in D b ( f Y ) , and (ii) hB ( − , · · · , B ( − , B i is a dual Lefschetz collection in D b ( ˇ X ) .In particular the following vanishings hold: Hom • D b ( e Y ) ( A i ( i ) , A j ( j )) = 0 ( i > j ) , Hom • D b ( ˇ X ) ( B i ( − i ) , B j ( − j )) = 0 ( i < j ) . Although it is implicit in the above theorem, the (dual) Lefschetz collections (i)and (ii) above indicate that there exist some non-commutative resolutions of Y and X , respectively, and furthermore, they are expected to be HPD with each other.This should be contrasted to Theorem 3.4 where non-commutative resolution hasappeared only for the Pfaffian variety Y . Of course, this difference is due to thefact that both Y and X are singular varieties in our case. See [Ku3] for a recentsurvey about known examples of HPDs.As in Subsection 3.2, the derived equivalence follows from the flatness of theideal sheaf I = I| Y × X over X and the vanishing properties in Theorem 3.3. Theorem 4.7 ([HoTa4, Theorem 8.0.3]) . The restriction I = I| Y × X defines ascheme C flat over X , and an equivalence Φ I : D b ( Y ) → D b ( X ) with Φ I ( − ) = Rπ X ∗ ( Lπ ∗ Y ( − ) ⊗ I ) . The proof given in [HoTa4, Sect.8] proceeds in a similar way to [BC] and onlyuses the vanishing properties in Theorem 4.6. irror Symmetry and Projective Geometry of FM partners 22
BPS numbers.
The ideal sheaf I describes a family of curves on Y parametrizedby x ∈ X . In particular, in [HoTa4], an interesting relation of them to some BPSnumber of Y has been observed. Here we start with the following proposition: Proposition 4.8 ([HoTa4, Sect.3, Prop.7.2.2]) . The ideal sheaf I = I | Y × X definesa flat family { C x } x ∈ X whose general members are smooth curves of genus 3 anddegree 5 in Y . The curve C x appears from the incidence relation ∆ in G(3 , V ) × G(2 , V ) .Recall X = ˇ X ∩ P ( L ⊥ ) and the morphism g : ˇ X → G(2 , V ) . Then g ( x ) ( x ∈ X ) determines a line l x = P ( V ,x ) . Then we have ∆ | G(3 ,V ) ×{ g ( x ) } = { [Π] ∈ G(3 , V ) | l x ⊂ P (Π) } . Now let us recall the definition of Y in (4.1) and Y = Y ∩ P ( L ) . We define Z x := { ([ Q ] , [Π]) | l x ⊂ P (Π) ⊂ Q } ⊂ Z and γ x := Z x ∩ π − Z ( Y ) = { ([ Q ] , [Π]) | l x ⊂ P (Π) ⊂ Q, [ Q ] ∈ P ( L ) } . When Y is smooth, then Y = f Y ∩ P ( L ) = Y ∩ P ( L ) , i.e., ρ − Y ( P ( L )) is awayfrom the singular locus Sing ( Y ) = H . On the other hand, over Y \ Sing ( Y ) theStein factorization Z → Y has the structure of a conic bundle which is isomorphicto the generically conic bundle Z → Y over Y \ (˜ ρ ◦ ρ e Y ) − ( Sing ( Y )) (see[ibid,Sect.2.3] and also the next section). Therefore we have C x = π Z ( γ x ) for thefamily of curves on Y . We can further study the following properties: Proposition 4.9. (1) ¯ γ x = ρ Y ◦ π Z ( γ x ) = ρ Y ( C x ) is a plane quintic curve in H = H ∩ P ( L ) with 3 nodes and arithmetic genus 6 for general x ∈ X . (2) When x ∈ X is general, ¯ γ x is away from the branch locus C H ⊂ H and C x → ¯ γ x is the normalization map. (3) For general x ∈ X , there exists a ’shadow’ curve C ′ x of genus 3 and degree 5with the properties ρ − Y (¯ γ x ) = C x ∪ C ′ x and C x ∩ C ′ x = ρ − Y ( ¯ γ x ) . We refer to [ibid Sect. 3, Fig.1] for details, but only remark that the plane curve ¯ γ x can be written explicitly by ¯ γ x = { [ Q ] ∈ H | l x ⊂ Q } . Considering the condition l x ⊂ Q under x ∈ ˇ X ∩ P ( L ⊥ ) , we see easily that ¯ γ x is a plane curve H ∩ P x with P x = (cid:8) [ a ij ] ∈ P ( L ) | t zAz = t wAw = 0 ( ∀ [ z ] , [ w ] ∈ l x ) (cid:9) ≃ P , where A = ( a ij ) is the symmetric matrix corresponding to a point [ a ij ] . Note that x ∈ ˇ X ∩ P ( L ⊥ ) implies t zAw = 0 , which is one of the three conditions for l x ⊂ Q .We depict the claims in Proposition 4.9 in Fig. 4.1.As claimed in Proposition 4.9, there are two (distinct) families of curves { C x } x ∈ X and { C ′ x } x ∈ X in Y parametrized X . These two are smooth curves of genus 3 anddegree 5 for general x ∈ X , and interestingly, can be identified in the BPS numberscalculated in [HoTa,1]. The relevant part of the table of BPS numbers reads asfollows:(4.4) g n Yg ( d ) d = 5 . As discovered in [ibid], we can exactly identify the two families in theBPS number n Y (5) = 100 as n Y (5) = ( − dim X e ( X ) × − ( − × irror Symmetry and Projective Geometry of FM partners 23 ●● ● HY C x C ′ x γ x C H P x ≃ P HY Fig.4.1. Shadow curve C ′ x . Two intersecting curves C x and C ′ x in Y covers the plane quintic curve γ x in H . C H is the curve of thebranch locus.following the counting “rule” described in Subsection 3.3. This indicates that theBPS numbers, which are preferred in physics interpretations [GV] to other math-ematical invariants such as Donaldson-Thomas invariants, has a nice moduli in-terpretation in some cases although their mathematical definition (as invariants ofmanifolds) is difficult in general [HST].4.4. Mirror symmetry.
In Subsection 3.4, we have only described the mon-odromy properties of Picard-Fuchs differential equation for the mirror family ofRødland’s Pfaffian Calabi-Yau 3-fold. This is partially because the geometry of themirror family is rather involved. Our second example of FM partners { X, Y } of ρ = 1 has a nice feature from this perspective. We have a rather simple descriptionfor the mirror family of Reye congruence Calabi-Yau 3-folds X in terms of specialform of determinantal quintic hypersurfaces in P .Recall the definition X = S P ( V ) ∩ P ( L ⊥ ) ⊂ P ( S V ) . Using the fact S P ( V ) = P ( V ) × P ( V ) / Z , it is easy to see the isomorphism X ≃ ˜ X/ Z with(4.5) ˜ X = (cid:18) P | P | (cid:19) , , where the superscripts , represent the Hodge numbers h , and h , , respectively.The r.h.s of (4.5) is a common notation in physics literatures to represent completeintersections of five (generic) (1 , -divisors in P × P . In our case, we should readthis as the complete intersection of five generic and symmetric (1 , -divisors whichcorrespond to five linear forms in P ( S V ) determined by L ⊂ S V ∗ . Note thatwhen L is taken in general position, X is smooth which means that the Z actionon ˜ X is free.For concreteness, let us take a basis of L by A k = ( a ( k ) ij ) ( k = 1 , .., . Then thedefining equations of ˜ X are given by f = f = ... = f = 0 with f k = P i,j z i a ( k ) ij w j and ([ z ] , [ w ]) ∈ P × P . If we introduce a notation A ( z ) = (cid:16)P i z i a ( k ) ij (cid:17) ≤ k,j ≤ forthe × matrix defined by A k , then we have ˜ X = (cid:8) ([ z ] , [ w ]) ∈ P × P | A ( z ) w = 0 (cid:9) . irror Symmetry and Projective Geometry of FM partners 24 It is easy to deduce that the projection of ˜ X to the first factor of P × P is adeterminantal quintic hypersurface, Z = (cid:8) [ z ] ∈ P | det A ( z ) = 0 (cid:9) . Proposition 4.10 ([HoTa1]) . (1) When the linear subspace L ⊂ S V ∗ is general,the quintic hypersurface Z is singular at ordinary double points(ODPs) where rk A ( z ) = 3 . (2) The morphism π : ˜ X → Z is a small resolution of the 50 ODPs. Details can be found in [ibid, Prop.3.3]. Here we summarize properties of X, ˜ X and Z in the left of the following diagrams:(4.6) ˜ XX Z ˜ X ∗ X ∗ ˜ X sp Z sp/ Z (cid:15) (cid:15) / Z (cid:15) (cid:15) OPD’s ●●●●●●●●● / / (cid:31) (cid:31) ❄❄❄❄❄❄❄ For the construction of mirror family of X , we invoke the orbifold mirror con-struction, which schematically described in the right diagram of (4.6). Namely,we start with a certain special form A sp ( z ) of A ( z ) (or the linear subspace L ) to define Z sp = { det A sp ( z ) = 0 } . Z sp is singular in general, and so is ˜ X sp := { A sp ( z ) w = 0 } ⊂ P × P . Finding a suitable crepant resolution ˜ X ∗ → ˜ X sp , whichis compatible with the Z action of exchanging the two factors of P × P , we obtaina mirror family of X by the quotient X ∗ = ˜ X ∗ / Z . In the final process, we usuallyneed to find a suitable finite group G orb (called orbifold group) to arrive at the de-sired properties h , ( X ) = h , ( X ∗ ) and h , ( X ) = h , ( X ∗ ) , however interestinglyit turns out that G orb = { id } in our case.The special form A sp ( z ) found in [HoTa2] corresponds to a linear subspace L = h A , A , · · · , A i with A , A , ..., A in order given by a a ! , a a ! , a
00 0 a ! , a a ! , a a ! . Using these special form of A k , we have Z sp ( a ) := { det A sp ( z ) = 0 } ⊂ P where(4.7) det A sp ( z ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z + az az z + az az z + az az
00 0 0 z + az az az z + az (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = a z z z z z + ( z + az )( z + az )( z + az )( z + az )( z + az ) . By coordinate change, it is easy to see that { Z sp ( a ) } a defines a family of Calabi-Yauthreefolds over P by [ − a , ∈ P . Proposition 4.11. (1)
When a is general ( a = − , − a + a = 0) , Z sp ( a ) is singular along 5 lines of A singularities and 10 lines of A singularities. (2) ˜ X sp ( a ) := { ([ z ] , [ w ]) | A sp ( z ) z = 0 } partially resolves the singularities in (1) to20 lines of A singularities. (3) There exists a crepant resolution ˜ X ∗ ( a ) → ˜ X sp ( a ) . And ˜ X ∗ ( a ) for general a is a smooth Calabi-Yau 3-fold with Hodge numbers h , = 52 , h , = 2 . More details of the singularities and their resolutions can be found in [ibid]. Forgeneral a , we can see that ˜ X ∗ ( a ) admits a free Z action, and hence X ∗ ( a ) =˜ X ∗ ( a ) / Z is a Calabi-Yau 3-fold with Hodge numbers h , = 26 , h , = 1 . We havethen a family X ∗ := { X ∗ ( a ) } [ − a , ∈ P of Calabi-Yau 3-folds over P . irror Symmetry and Projective Geometry of FM partners 25 Proposition 4.12 ([HoTa2, Prop.6.9]) . X ∗ is a mirror family of Reye congruenceCalabi-Yau 3-fold X . We omit the monodromy calculations which correspond to those in Subsection3.4, since they are reported in [ibid, Prop.2.10].
Remark. (1) Set x = − a , then from the defining equation (4.7) we observe thatboth x = 0 and x = ∞ are MUM points. In [HoTa1], Gromov-Witten invariants ( g ≤ ) of Reye congruence X have been calculated from the MUM degeneration at x = 0 and the invariants of Fourier-Mukai partner Y from x = ∞ . We believe thatour mirror family X ∗ provides us a nice example to study the geometry of mirrorsymmetry [SYZ, GrS1, GrS2, RuS] when non-trivial Fourier-Mukai partners exist.It is interesting, although accidental, that in (4.7) we come across to the geometryof quintic from which the study of mirror symmetry started [Ge, GP, CdOGP].(2) If we focus on the form of Picard-Fuchs differential operators in [AESZ,ES],[DM], there are many other examples which exhibit two MUM points. Amongthem, a nice example has been identified in [Mi] with the mirror family of theCalabi-Yau 3-fold given by general linear sections of a Schubert cycle in the Cayleyplane E /P . It is expected that this Calabi-Yau 3-fold has a non-trivial Fourier-Mukai partner [ibid][Ga]. Also the mirror family of the Calabi-Yau 3-folds givenby the intersection of two copies of Grassmannians X = G(2 , ∩ G(2 , ⊂ P [Kan, Kap] shows two MUM points whose interpretation seems slightly differentfrom those we have seen in this article. The two MUM points seems to correspondFourier-Mukai partners which are diffeomorphic but not bi-holomorphic. It wouldbe interesting to investigate these new examples in more detail.(3) In [Hor], the pair of Reye congruence Calabi-Yau 3-fold X and its Fourier-Mukai partner Y have been understood in the language of Gauged linear sigmamodes along the arguments used for the Grassmannian-Pfaffian example. Extend-ing these arguments, many other examples have been worked out in [HK] by calcu-lating the so-called “two sphere partition” in physics [JKLMR]. irror Symmetry and Projective Geometry of FM partners 26 Birational Geometry of the Double Symmetroid Y We describe the birational geometry of the double (quintic) symmetroid Y andits resolution f Y . We will see intensive interplay of the projective geometry ofquadrics and that of relevant Grassmannians. In this section, we fix V = C andretain all the notations introduced in the last section. This section is an expositionof the results whose details are contained in [HoTa3, HoTa4].5.1. Generically conic bundle Z → Y . We describe the (connected) fibers of Z → Y of the Stein factorization Z → Y → H in (4.1). Recall the definition H = (cid:8) [ a ij ] ∈ P ( S V ∗ ) | det a = 0 (cid:9) and Z := { ([ Q ] , [Π]) | P (Π) ⊂ Q } ⊂ H × G(3 , V ) , i.e., Z consists of pairs of singular quadric and (projective) plane therein. Thenotation [ Q ] ∈ H above indicates that we identify points [ a ij ] ∈ H with thecorresponding quadrics Q in P ( V ) . Since dim Z − dim H = 1 , we have genericallyone dimensional fibers for π Z : Z → Y . It is easy to deduce the fibers of π Z : Z → Y from those of Z → H :The fibers of Z → H over a point [ Q ] consists of planes contained in thequadric Q . In Fig. 5.1 , depending on the rank of [ Q ] = [ a ij ] , the correspondingquadric Q is depicted schematically. Let us define reduced quadric ¯ Q to be thesmooth quadric naturally defined in P ( V /
Ker ( a ij ) ). Then, as is clear in Fig. 5.1, ¯ Q ≃ P × P , a smooth conic, two points and one point depending on rk Q =4,3, 2 and 1, respectively. Singular quadrics Q are then described by the conesover the reduced quadric ¯ Q with the vertex Ker Q := P (Ker ( a ij )) . The fibers of π Z : Z → Y over y ∈ Y are given by connected families of planes contained inthe quadric Q y = ρ Y ( y ) . We summarize the connected fibers:(a) When rk Q y = 4 , the fiber is the P -families of planes which corresponds toone of the two possible rulings of ¯ Q y ≃ P × P .(b) When rk Q y = 3 , the fiber is the P -family of planes parametrized by the conic ¯ Q y . (c) When rk Q y = 2 , the fiber is the planes parametrized by ( P ) ∗ ⊔ pt ( P ) ∗ where ( P ) ∗ parametrizes planes in P and A ⊔ pt B represents the union with a ∈ A and b ∈ B (one point from each) are identified.(d) When rk Q y = 1 , the fiber is the planes parametrized by ( P ) ∗ .We remark that, in the case of (a), one of the two possible P -families of planesis specified (by the definition of Stein factorization) when we take y ∈ Y . Thisand the other cases explain the finite morphism ρ Y : Y → H which is over H \ H and branched over H . We say that a point y ∈ Y has rank i if rank a y = i for ρ Y ( y ) = [ a y ] , and define G Y := { y ∈ Y | rk y ≤ } . Note that dim G Y = dim H = 8 . Proposition 5.1. (1)
Sing H = H and Sing Y = G Y (= H ) . (2) π Z : Z → Y is a generically conic bundle with the conics in G(3 , V ) .Proof. (1) Sing H = H follows from the basic properties of secant varieties. Forthe latter claim Sing Y = G Y , we refer to [HoTa3, Prop.5.7.2]. irror Symmetry and Projective Geometry of FM partners 27 .. . P ( P ) ∗
111 0 0
11 0 0 0 P ⊔ P ( P ) ∗ ⊔ pt ( P ) ∗ ZYH
Fig.5.1. Quadrics and planes therein.
Quadrics Q are de-picted for each rank, rk Q = 4 , , , . When rk Q = 4 , there aretwo connected fibers of Z → H .(2) Over Y \ G Y , the fibers of π Z : Z → Y consists of smooth P -families ofplanes in G(3 , V ) . As we see in the next subsection, it is easy to see that these aresmooth conics on G(3 , V ) . (cid:3) Birational model Y of Y . Let us consider a quadric Q of rank 4 and 3, inorder, and a P -family of planes in Q .First, for a quadric Q of rank 4, let us denote the vertex of Q (the kernel of ( a ij )) by h v i . Then, one of the P -family of plane described in (a) in Subsection 5.1 takesthe following form: { [Π s,t ] } := (cid:8) h c ( s, t ) , d ( s, t ) , v i | [ s, t ] ∈ P (cid:9) , where c ( s, t ) , d ( s, t ) ∈ V are linear in s, t and span the h c ( s, t ) , d ( s, t ) i ≃ P whichgives the ruling ¯ Q ≃ P × P . One of the key observations is that for such a P -familyof plane we have a conic q in P ( ∧ V ) by q := (cid:8) [ c ∧ d ∧ v ] = [Λ s + Λ st + Λ t ] | [ s, t ] ∈ P (cid:9) , which actually defines a conic in G(3 , V ) by the Plücker embedding G(3 , V ) ⊂ P ( ∧ V ) . We note that conic q resides in the plane P q which is uniquely determinedby the P -family, P q := h Λ , Λ , Λ i ⊂ P ( ∧ V ) . When rk Q = 3 , we start with { [Π s,t ] } = (cid:8) h d ( s, t ) , v , v i | [ s, t ] ∈ P (cid:9) with v , v being bases of Ker ( a ij ) and d ( s, t ) = s v + stv + t v parametrizing the conic ¯ Q in P ( V /
Ker ( a ij )) . Again, we have the corresponding conic q in G(3 , V ) and alsothe plane P q ⊂ P ( ∧ V ) which contains the conic q .The conics q above explain the generically conic bundle Z → Y claimed inProposition 5.1. The planes P q ⊂ P ( ∧ V ) and conics q will play central roles in thedescription of the resolution f Y → Y . Here noting that the planes P q above have aspecific forms, we define the following subset of planes in P ( ∧ V ) : Y = (cid:8) [ U ] ∈ G(3 , ∧ V ) | U = ¯ U ∧ v for some v ∈ P ( V ) (cid:9) , where we regard ¯ U as an element in P ( ∧ ( V /V )) with V = C v . To introducea (reduced) scheme structure on the subset Y , we consider a linear morphism irror Symmetry and Projective Geometry of FM partners 28 ϕ : S ( ∧ V ) → V by the composition of the following natural linear morphisms:(5.1) ϕ : S ( ∧ V ) → S ( ∧ V ∗ ) → ∧ V ∗ ≃ V. We define ϕ U := ϕ | S U to be the natural restriction of ϕ for a fixed subspace [ U ] ∈ G(3 , ∧ V ) . Then, we have the following proposition: Proposition 5.2. (1) U ⊂ ∧ V decomposes as U = ¯ U ∧ v if and only if rk ϕ U ≤ . (2) The scheme (cid:8) [ U ] ∈ G(3 , ∧ V ) | rk ϕ U ≤ (cid:9) is nonreduced along the singularlocus of its reduced structure. The proof of the above proposition follows by writing the rank condition ex-plicitly for the matrix representing ϕ U under suitable bases (see [HoTa3, Sub-sect.5.3, 5.4]). Hereafter, we consider Y as the scheme with the reduced structureon (cid:8) [ U ] ∈ G(3 , ∧ V ) | rk ϕ U ≤ (cid:9) . Proposition 5.3. Y and Y are birational.Proof. By definition of the Stein factorization, points y ∈ Y are specified by theconnected fibers of Z → Y , which are generically given by conics q in G(3 , V ) .Hence we can write general points y ∈ Y by y = ([ Q y ] , q y ) where [ Q y ] = ρ Y ( y ) and the corresponding conic q y which is a P -family of planes contained in Q y .Rational map Y Y has been described already above by y = ([ Q y ] , q y ) → P q y for y ∈ Y \ G Y . To describe the inverse rational map Y Y , we note that thefollowing isomorphism for U = ¯ U ∧ v ∈ ∧ V :(5.2) P ( U ) ∩ G(3 , V ) in P ( ∧ V ) ≃ P ( ¯ U ) ∩ G(2 , V /V ) in P ( ∧ ( V /V )) , where V = C v . Since G(2 , V /V ) ≃ G(2 , is the Plücker quadric, when U isgeneral, the r.h.s. determines a smooth conic on G(2 , V /V ) and in turn a smoothconic on G(3 , V ) . We can see that this is the inverse rational map. (cid:3) Obviously, the inverse rational map Y Y is not defined when P ( U ) ∩ G(3 , V ) = P ( U ) , i.e. P ( U ) ⊂ G(3 , V ) . There are two cases where P ( U ) ⊂ G(3 , V ) occurs for [ U ] ∈ Y : The first one is when P ( U ) is given by the Plücker image ofthe plane P V := { [Π] | V ⊂ Π ⊂ V } ≃ P in G(3 , V ) for some V . The second one is given by the Plücker image of the plane P V V := { [Π] | V ⊂ Π ⊂ V } ≃ P in G(3 , V ) for some V and V . The plans of the form P V and P V V , respectively,are called ρ -planes and σ -planes. These planes determine the following loci in Y :(5.3) P ρ := (cid:8) [ U ] | V ⊂ V, U = V /V ∧ ( ∧ V ) (cid:9) , P σ := (cid:8) [ U ] | V ⊂ V ⊂ V, U = ∧ ( V /V ) ∧ V (cid:9) . Note that P ρ ≃ G(2 , V ) and P σ ≃ F (1 , , V ) . irror Symmetry and Projective Geometry of FM partners 29 Sing Y and resolutions of Y . We consider the reduced structure on Y asdescribed in the preceding subsection. Then writing the condition rk ϕ U ≤ , wecan study the singularities of Y explicitly. Proposition 5.4. (1) Y is singular along P ρ ≃ G(2 , V ) . (2) Define Y := (cid:8) ([ U ] , [ V ]) | U = ¯ U ∧ V (cid:9) ⊂ Y × P ( V ) , then the natural projec-tion Y → Y is a resolution of the singularity. (3) Y is isomorphic to the Grassmannian bundle G(3 , ∧ T P ( V ) ( − over P ( V ) . (4) The singularities of Y are the affine cone over P × P along P ρ , and there isa (anti-)flip to another resolution f Y → Y which fits into the following diagram: (5.4) Y . Y Y f YY ρ f Y ❆❆❆❆❆❆ ~ ~ ⑥⑥⑥⑥⑥⑥ ❆❆❆❆❆❆ / / ~ ~ ⑥⑥⑥⑥⑥⑥ (anti-)flip / / ❴❴❴❴❴ Proof. (1) and (4) follow directly by writing the condition rk ϕ U ≤ , see [HoTa3,Prop.5.4.2, 5.4.3]. Global descriptions of the blow-up Y → Y will be given inProposition 5.10. (3) We consider ¯ U ≃ C as a subspace in ∧ ( V /V ) . Then claimis clear since T P ( V ) ( − | [ V ] ≃ V /V . (2) follows from (3). (cid:3) We denote by P ρ the exceptional set (which is contracted to P ρ ) of the resolution Y → Y and by P σ ≃ P σ the proper transform of P σ . It is easy to observe thefollowing isomorphisms:(5.5) P ρ ≃ F (1 , , V ) ≃ P ( T P ( V ) ( − , P σ ≃ F (1 , , V ) ≃ P ( T P ( V ) ( − ∗ ) . These loci P ρ and P σ in G(3 , ∧ T P ( V ) ( − will be interpreted in the next section.In the diagram (5.4), we have included the content of the following theorem: Theorem 5.5.
There is a morphism ρ e Y : f Y → Y which contracts an exceptionaldivisor F e Y to the singular locus G Y of Y . The above theorem is one of the main results of [HoTa3]. We refer to [ibid,Subsect. 5.7, and Fig.2] for details. Also, for the proof of Theorem 4.6, we used anatural flattening of the fibers of F e Y → G Y constructed in [ibid,Section 7]. Below,we describe the construction of the morphism ρ e Y briefly.5.4. The resolution ρ e Y : f Y → Y . We formulate a rational map ϕ DS : Y H which extends to ˜ ϕ DS : f Y → H . Then the Stein factorization of ˜ ϕ DS gives theclaimed morphism ρ e Y : f Y → Y [ibid,Prop.5.6.1].The key relation for the construction is the following decomposition:(5.6) ∧ ( ∧ ( V /V )) = Σ (3 , , , ( V /V ) ⊕ Σ (2 , , , ( V /V ) ≃ S ( V /V ) ⊕ S ( V /V ) ∗ , as irreducible so ( ∧ V /V ) ≃ sl ( V /V ) -modules, where Σ α represents the Schur func-tor. We called this double spin decomposition since the r.h.s. is V λ s ⊕ V λ ¯ s withthe spinor and conjugate spinor weights λ s and λ ¯ s , respectively. G(3 , ∧ ( V /V )) consists of 3-spaces in ∧ ( V /V ) . We have also OG(3 , ∧ ( V /V )) which consists ofisotropic 3-spaces with respect to the natural symmetric form ∧ ( V /V ) ×∧ ( V /V ) → irror Symmetry and Projective Geometry of FM partners 30 ∧ ( V /V ) ≃ C . We denote by OG ± (3 , ∧ ( V /V )) the connected components of OG(3 , ∧ ( V /V )) .If we consider the above decomposition fiberwise for ∧ T P ( V ) ( − , then we havethe following embedding:(5.7) i : Y = G(3 , ∧ T P ( V ) ( − ֒ → P ( S T P ( V ) ( − ⊗ ∧ T P ( V ) ( − ⊕ S T P ( V ) ( − ∗ ⊗ ( ∧ T P ( V ) ( − ⊗ ) . Proposition 5.6.
The following properties hold for the loci P ρ and P σ in Y : (1) i ( P ρ ) = v ( P ( T P ( V ) ( − , i ( P σ ) = v ( P ( T ( − ∗ )) . (2) P ρ = OG + (3 , ∧ T P ( V ) ( − , P σ = OG − (3 , ∧ T P ( V ) ( − ∗ ) .Proof. (1) The claimed relations follow from the isomorphisms (5.5) and the formof the embedding (5.7). We can also verify the claim explicitly by writing thedecomposition (5.6) (see Appendix B). (2) The points [ V , V ] ∈ F (1 , , V ) ≃ P ρ determine the corresponding points ([ ¯ U ] , [ V ]) ∈ P ρ with [ ¯ U ] = [( V /V ) ∧ ( V /V )] ∈ G(3 , ∧ ( V /V )) . Then we verify ¯ U ∧ ¯ U = 0 . Similarly, points ([ ¯ U ] , [ V ]) ∈ P σ havethe forms [ ¯ U ] = [ ∧ ( V /V )] for some V . Again, we have ¯ U ∧ ¯ U = 0 . The claimsfollow since all maximally isotropic subspaces in ∧ ( V /V ) take either of these twoforms. (cid:3) Now we consider the following sequence of (rational) morphisms:(5.8) Y i ֒ → P ( S T ( − ⊗ O P ( V ) (1) ⊕ S T ( − ∗ ⊗ O P ( V ) (2)) P ( S T ( − ∗ ) ֒ → P ( S V ∗ ⊗ O P ( V ) ) → P ( S V ∗ ) , where we use ∧ T P ( V ) ( −
1) = O P ( V ) (1) , and (here and hereafter) we write T ( − for T P ( V ) ( − to simplify formulas. In the middle, we consider the projection tothe second factor. The injection in the right is defined by considering the dual ofthe surjection V ⊗ O P ( V ) → T ( − → , and P ( S V ∗ ⊗ O P ( V ) ) → P ( S V ∗ ) is thenatural projection for P ( S V ∗ ⊗ O P ( V ) ) = P ( S V ∗ ) × P ( V ) . Since the image of thecomposition is in H ⊂ P ( S V ∗ ) , we have a rational map, φ DS : Y H . Proposition 5.7. (1)
The rational map φ DS defines a morphism φ DS : Y \ P ρ ≃ Y \ P ρ → H . In particular, it induces a rational map ϕ DS : Y H whoseindeterminacy locus is P ρ . (2) φ DS ( P σ ) = ϕ DS ( P σ ) = H . (3) The rational map ϕ DS : Y H extends to a morphism ˜ ϕ DS : f Y → H .Proof. (1) and (2) follow from the claim (1) in Theorem 5.6 and the definition ϕ DS with the Plücker embedding (5.7). We can verify (3) explicitly by writingthe rational map ϕ DS and extending it to the blow-up f Y → Y (see [HoTa3,Prop.5.5.3]). (cid:3) Theorem 5.8. ˜ ϕ DS : f Y → H factors as f Y → Y ρ Y → H with the morphism ρ Y : Y → H in (4.1). This defines the resolution ρ e Y : f Y → Y .Proof. The claim basically follows from the Stein factorization. In [ibid, Section5.6, Fig.2], the fibers of ˜ ϕ DS : f Y → H have been described completely, and theclaim is clear from the results there. (cid:3) irror Symmetry and Projective Geometry of FM partners 31 Remark.
We describe the inverse image of the rational map φ DS . Let us fix [ a ] ∈ H . When we fix (a choice of) V ⊂ Ker a , we have a “reduced matrix” [ a V ] ∈ P ( S ( V /V ) ∗ ) representing the quadric in P ( V /V ) . Consider the restriction φ V := φ DS | π − ([ V ]) of φ DS to the fiber π − ([ V ]) = G(3 , ∧ ( V /V )) of π : Y → P ( V ) ,and also similar restriction i V : G(3 , ∧ ( V /V )) ֒ → P ( S ( V /V ) ⊕ S ( V /V ) ∗ ) of thePlücker embedding (5.7). Then, over the fiber π − ([ V ]) , the rational map φ DS : Y H (5.8) is basically given by the projection P ( S ( V /V ) ⊕ S ( V /V ) ∗ ) P ( S ( V /V ) ∗ ) sending [ v ij , w kl ] to [ w ij ] . The ideal of the Plücker embedding interms coordinate [ v ij , w kl ] turns out to have a rather nice form as shown in AppendixB. Using the results listed in Appendix B, we can prove the following properties ofthe inverse image of φ DS :1) When rk a = 4 , V is unique and we have i V ◦ φ − V ( a ) = [ ± p det a V a − V , a V ] .
2) When rk a = 3 , for any V ⊂ Ker a , we have i V ◦ φ − V ( a ) = ∅ .3) When rk a = 2 , for each choice of V ⊂ Ker a , we have i V ◦ φ − V ( a ) ≃ P × P .
4) When rk a = 1 , for each choice of V ⊂ Ker a , we have i V ◦ φ − V ( a ) ≃ P (1 , . Let us denote by G ρ the exceptional set of the resolution ˜ ϕ DS : f Y → Y . Then,since Y \P ρ ≃ Y \P ρ ≃ f Y \ G ρ , we can identify φ DS , ϕ DS and ˜ ϕ DS with each otherover these complement sets. Then the above results indicate that ˜ ϕ − DS ( a ) (rk a = 3) is contained in the exceptional set G ρ (and this is indeed the case [ibid, Lemma5.6.2]). Note also that from 3) and 4) and dim G Y = 8 ( G Y ≃ H ) , we see that ˜ ϕ − DS ( G Y ) is a divisor in f Y , which is nothing but the divisor F e Y that appeared inTheorem 5.5. Full details of 1)–4) can be found in [ibid, Section 5.6] (see also [ibid,Fig.2]). (cid:3) Generically conic bundles.
We describe the generically conic bundle π ′ : Z → Y which has appeared in (4.2). The basic idea is the same as that we usedin the proof of Proposition 5.3, i.e., to consider the intersection P ( U ) ∩ G(3 , V ) ≃ P ( ¯ U ) ∩ G(3 , V /V ) for U = ¯ U ∧ V .5.5.1. Generically conic bundle Z → Y . Let us fix the embedding
G(3 , V ) ⊂ P ( ∧ V ) . We recall the definition Y = (cid:8) [ U ] ∈ G(3 , ∧ V ) | U = ¯ U ∧ V for some V ⊂ V (cid:9) . Then from the isomorphism (5.2), we have generically conic bundle by Z := (cid:8) ([ c ] , [ U ]) | [ c ] ∈ P ( U ) ∩ G(3 , V ) , [ U ] ∈ Y (cid:9) ⊂ G(3 , V ) × Y , with the natural projection Z → Y . As explained in Subsection 5.2, the fibers P ( U ) ∩ G(3 , V ) over point [ U ] are conics for [ U ] ∈ Y \ ( P ρ ∪ P σ ) while they are ρ -planes and σ -planes ( ≃ P ( U ) ) for [ U ] ∈ P ρ and [ U ] ∈ P σ , respectively.5.5.2. Generically conic bundle Z → Y . The generically conic bundle Z → Y naturally extends to Z → Y by the isomorphism P ( U ) ∩ G(3 , V ) ≃ P ( ¯ U ) ∩ G(2 , V /V ) for U = ¯ U ∧ V . To describe it, let us introduce the universal bundlesfor the Grassmannian bundle π : Y = G(3 , ∧ T ( − → P ( V ) , → S → π ∗ ∧ T ( − → Q → . irror Symmetry and Projective Geometry of FM partners 32 Denote by P ( S ) the universal planes over Y , whose fiber over ([ ¯ U ] , [ V ]) is P ( ¯ U ) .Now, consider Grassmannian bundle π G : G(2 , T ( − → P ( V ) , and define Z := G(2 , T ( − × P ( V ) Y , with the natural projections π G ′ : Z → G(2 , T ( − and π ′ : Z → Y . Bydefinition, the fiber of π ′ over the points ([ ¯ U ] , [ V ]) ∈ Y \ ( P ρ ∪ P σ ) is G(2 , V /V ) ∩ P ( ¯ U ) , which are conics isomorphic to P ( U ) ∩ G(3 , V ) with U = ¯ U ∧ V , i.e., the fibersof Z → Y over [ U ] . As before the fibers over P ρ and P σ are the ρ -planes and σ -planes, respectively.Noting the isomorphism G(2 , T ( − ≃ F (1 , , V ) , the following lemma is clear: Lemma 5.9.
There is a natural morphism ρ G : G(2 , T ( − → G(3 , V ) . Z Y f Z f Y G ρ P σ P ( U ) ∩ G(3 , V ) P ( U ) P ( U ) Y [ U ] G ( , V ) ZY P σ P ρ F ρ P ρ P σ Z Y P ( ¯ U )([ ¯ U ] , [ V ]) P ( ¯ U ) Y G ( , T ( − )) P ( ¯ U ) ∩ G(2 , V/V ) P σ Fig.5.2. Generically conic bundles.
Generically conic bundlesin the text are schematically described. The proper transforms of P σ are written by the same letter P σ for simplicity.5.5.3. Generically conic bundle Z → Y . As described in Proposition 5.4, Y is given as the blow-up of Y along P ρ . We denote the exceptional divisor of theblow-up by F ρ (note that F ρ is a divisor). Proposition 5.10. (1)
We have N P ρ / Y = S S ∗ ⊗ π ∗ O P ( V ) (1) | P ρ for the normalbundle of P ρ ⊂ Y , and hence F ρ = P ( S S ∗ | P ρ ) . irror Symmetry and Projective Geometry of FM partners 33 (2) The fibers of F ρ → P ρ can be identified with the conics in the ρ -planes parametrizedby P ρ .Proof. (1) We have seen in Proposition 5.6 that P ρ = OG + (3 , ∧ T ( − , i.e., oneof the connected component of OG(3 , ∧ T ( − ⊂ G(3 , ∧ T ( − . The orthogonalGrassmannian consists maximally isotropic subspaces with respect to the symmetricform on the universal bundle S induced from ∧ T ( − × ∧ T ( − → ∧ T ( − ≃ O P ( V ) (1) . Hence it is given by the zero locus of the section of the bundle S S ∗ ⊗ π ∗ O P ( V ) (1) over G(3 , ∧ T ( − .(2) The points ([ ¯ U ] , [ V ]) ∈ P ρ determine the ρ -planes P ( ¯ U ) ⊂ P ( ∧ ( V /V )) . Wecan evaluate the fiber over a point ([ ¯ U ] , [ V ]) ∈ P ρ as P ( S S ∗ | ([ ¯ U ] , [ V ]) ) = P ( S ¯ U ∗ ) , which we identity with the conics in the ρ -plane. (cid:3) Proposition 5.11.
Let ρ ′ : Z → Z be the blow-up of Z along π − ′ ( P ρ ) , and E ρ be its exceptional divisor. Then E ρ → F ρ is the universal family of ρ -conicsparametrized by F ρ .Proof. This follows by considering the normal bundle of π − ′ ( P ρ ) in Z carefully.We refer to [HoTa4, Prop.4.3.4] for the proof. (cid:3) Now we summarize the above results into
Proposition 5.12.
The natural morphism π ′ : Z → Y between the blow-ups Z and Y is a generically conic bundle. Precisely, the fibers over Y \ P σ are conicsand the fibers over P σ are σ -planes ( where we use the same notation P σ for theproper transform of P σ in Y ) . We may summarize generically conic bundles into the following diagram:(5.9) Z Z Y Y P ( V ) P ( V )G(2 , T ( − , V ) f Y Y π G ′ z z ✉✉✉✉✉ ρ G y y ttttt ρ ′ o o π ′ (cid:15) (cid:15) π ′ (cid:15) (cid:15) ρ o o ˜ ρ $ $ ❏❏❏❏❏❏❏ π (cid:15) (cid:15) π (cid:15) (cid:15) π G $ $ ■■■■■ ρ f Y / / In the above diagram, we have included all the morphisms claimed in Proposition4.3. In Fig. 5.2, we schematically have depicted the generically conic bundles, Z → Y , Z → Y , Z → Y and also f Z → f Y which is deduced from Z → Y and Z → Y . irror Symmetry and Projective Geometry of FM partners 34 Appendix
A. Two Theorems on Indefinite LatticesWe summarize two theorems on indefinite lattices which we use in Section 2.
Theorem A.1 ([Ni, Theorem 1.14.2]) . Let L be an indefinite lattice and ℓ ( L ) bethe minimal number of generators of L ∨ /L . If rk L ≥ ℓ ( L ) , then the isogenyclasses of L consists of L itself, G ( L ) = { L } and the natural group homomorphism O ( L ) → O ( A L ) is surjective. Theorem A.2 ([Ni, Theorem 1.14.4]) . Let L be an even unimodular lattice withsignature ( l + , l − ) and M be an even lattice with signature ( m + , m − ) . If (i) sgn( L ) − sgn( M ) > l + − m + > , l − − m − > and (ii) rk L − rk M ≥ l ( A M ) hold,then primitive embedding L ← ֓ M is unique up to automorphism of L . Appendix
B. Plücker Ideal of
G(3 , Let us fix a 4-dimensional space V and write the double spin decomposition(5.6) as ∧ ( ∧ V ) = Σ (3 , , , V ⊕ Σ (2 , , , V ≃ S V ⊕ S V ∗ . We fix a basis of V and write the corresponding bases of ∧ V in terms of the indexset I = {{ i, j } | ≤ i < j ≤ } (where we regard { i, j } as an ordered set). Thenwe introduce the standard Plücker coordinate by [ p IJK ] ∈ P ( ∧ ( ∧ V )) . On theother hand, we introduce the homogeneous coordinate (which may be called doublespin coordinate) by [ v ij , w kl ] ∈ P ( S V ⊕ S V ∗ ) with × symmetric matrices v = ( v ij ) , w = ( w kl ) . Writing the isomorphism of the above decomposition, wehave a linear relation between [ p IJK ] and [ v ij , w kl ] . Then the Plücker ideal I G ofthe embedding G(3 , ∧ V ) ⊂ P ( S V ⊕ S V ∗ ) follows from that of the standardembedding G(3 , ∧ V ) ⊂ P ( ∧ ( ∧ V )) .Let us introduce some notations. We define the signature function ǫ IJ ( I, J ∈ I )by the signature of the permutation of the “ordered” union I ∪ J , e.g., { , } ∪{ , } = { , , , } . We also define the dual index ˇ I ∈ I of I ∈ I by the property ˇ I ∪ I = { , , , } (here ∪ is the standard union). Proposition B.1 ([HoTa3, Appendix A]) . The Plücker ideal I G of the embedding G(3 , ∧ V ) ⊂ P ( S V ⊕ S V ∗ ) is generated by (B.1) | v IJ | − ǫ I ˇ I ǫ J ˇ J | w ˇ I ˇ J | ( I, J ∈ I ) , ( v.w ) ij , ( v.w ) ii − ( v.w ) jj ( i = j, ≤ i, j ≤ , where | v IJ | , | w IJ | represent the × minors of v, w with the rows and columnsspecified by I and J . ( v.w ) ij is the ij -entry of the matrix multiplication v.w . For [ v, w ] ∈ V ( I G ) ≃ G(3 , , we have1) det v = det w ,2) v.w = ±√ det w id ,3) rk v = 3 and rk w = 3 ,4) rk v = 2 ⇔ rk w = 2 , and5) rk v ≤ ⇔ rk w ≤ . These are easy consequences from (B.1). irror Symmetry and Projective Geometry of FM partners 35
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