Homological mirror symmetry of C P n and their products via Morse homotopy
aa r X i v : . [ m a t h . S G ] A ug HOMOLOGICAL MIRROR SYMMETRY OF C P N AND THEIRPRODUCTS VIA MORSE HOMOTOPY
MASAHIRO FUTAKI AND HIROSHIGE KAJIURA
Abstract.
We propose a way of understanding homological mirror symmetry whena complex manifold is a smooth compact toric manifold. So far, in many example, thederived category D b ( coh ( X )) of coherent sheaves on a toric manifold X is compared withthe Fukaya-Seidel category of the Milnor fiber of the corresponding Landau-Ginzburgpotential. We instead consider the dual torus fibration π : M → B of the complementof the toric divisors in X , where ¯ B is the dual polytope of the toric manifold X . Anatural formulation of homological mirror symmetry in this set-up is to define F uk ( ¯ M )a variant of the Fukaya category and show the equivalence D b ( coh ( X )) ≃ D b ( F uk ( ¯ M )).As an intermediate step, we construct the category Mo ( P ) of weighted Morse homotopyon P := ¯ B as a natural generalization of the weighted Fukaya-Oh category proposed byKontsevich-Soibelman [14]. We then show a full subcategory Mo E ( P ) of Mo ( P ) generates D b ( coh ( X )) for the cases X is a complex projective space and their products. Contents
1. Introduction 22. Toric manifolds and T n -invariant manifolds 32.1. Dual torus fibrations 32.2. Toric manifolds and T n -invariant manifolds 73. Lagrangian submanifolds and holomorphic vector bundles 83.1. Lagrangian submanifolds in M M C P n and the corresponding Lagrangians 114. Homological mirror symmetry set-up 124.1. DG-category V associated to ˇ M F associated to M F and V M.F. is supported by Grant-in-Aid for Scientific Research (C) (18K03269) of the Japan Society forthe Promotion of Science. H.K. is supported by Grant-in-Aid for Scientific Research (C) (18K03293) ofthe Japan Society for the Promotion of Science.
Date : September 1, 2020. F and the Fukaya category F uk ( M ) 164.5. The category Mo ( P ) of weighted Morse homotopy 165. Homological mirror symmetry of C P n DG ( C P n ) of line bundles over C P n C P n C P m × C P n Introduction
In this paper, we propose a way of understanding homological mirror symmetry for thecase of smooth compact toric manifolds. So far, in many example, the derived category D b ( coh ( X )) of coherent sheaves on a toric manifold X is compared with the Fukaya-Seidel category of the Milnor fiber of the corresponding Landau-Ginzburg potential. Inthis paper, we consider the dual torus fibration π : M → B , in the sense of Strominger-Yau-Zaslow construction [18], of the complement of the toric divisors in X , where P = ¯ B is the dual polytope of the toric manifold X . In [6], Fang discusses homological mirrorsymmetry of C P n along this line. There, he starts with considering line bundles on C P n and the corresponding Lagrangians in the mirror dual side. His idea of discussing thehomological mirror symmetry is to consider the category of constructible sheaves as anintermediate step. We instead apply Kontsevich-Soibelman’s approach [14] to our caseand consider a category Mo ( P ) of Morse homotopy. Namely, a natural formulation ofhomological mirror symmetry in our situation is to define a variant of Fukaya category F uk ( ¯ M ) and show the equivalence D b ( coh ( X )) ≃ D b ( F uk ( ¯ M )). As an intermediatestep, we construct the category Mo ( P ) of weighted Morse homotopy on P as a naturalgeneralization of the weighted Fukaya-Oh category proposed in [14]. We then show afull subcategory Mo E ( P ) of Mo ( P ) generates D b ( coh ( X )) for the case X is a complexprojective space and their products. For more general X , we may consider the Lagrangiansections discussed in [5], where Chan discusses the correspondence between holomorphicline bundles over projective toric manifolds and Lagrangian sections in the mirror dual.The relation of such an approach with Abouzaid’s one [1] is also mentioned there.As a formulation of the homological mirror symmetry for non-Calabi-Yau situationsbased on the SYZ construction, Fukaya discusses in [8] how to treat the singular fibersof the torus fibrations where the K¨ahler metrics go zero at singular fibers. In our set-up,the K¨ahler metrics go infinity at the boundaries ∂ ( P ). We believe our category Mo ( P )may be a correct candidate for such a situation. MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 3
This paper is organized as follows. In section 2, we recall the SYZ torus fibration set-up [18] following [16, 15]. There, a pair of dual torus fibrations M → B and ˇ M → B isdefined. We use this set-up by identifying ˇ M with the complement of the toric divisorsin a toric manifold X . In section 3, we recall the correspondence of Lagrangian sectionsof M → B and holomorphic line bundles on ˇ M , again, following [16, 15]. In the lastsubsection, we demonstrate a Lagrangian section to be derived from the line bundle O ( k )restricted on the complement of the toric divisors for X = C P n . by the correspondenceabove. In section 4, we first recall DG-categories F ( M ) and V ( ˇ M ) associated to M andˇ M , respectively, in [11]. Kontsevich-Soibelman’s approach for the homological mirrorsymmetry [14] proposes an intermediate category Mo ( B ) and the existence of an A ∞ -equivalence F uk ( M ) ≃ Mo ( B ) ∼ → F ( M ) . This Mo ( B ) is called the weighted Fukaya-Oh category or the category of weighted Morsehomotopy on B . In subsection 4.5, we propose a modification Mo ( P ) of Mo ( B ) where P = ¯ B is the dual polytope of a smooth compact toric manifold X . In the last section,we discuss the correspondence between D b ( coh ( X )) and Mo ( P ) when X is a complexprojective space or their products. In particular, we see that we can take strongly excep-tional collections E of D b ( coh ( X )) consisting of line bundles and the corresponding fullsubcategory Mo E ( P ) of Mo ( P ) so that Tr ( Mo E ( P )) ≃ D b ( coh ( X ))where Tr is the Bondal-Kapranov-Kontsevich construction of triangulated categories from A ∞ -categories. Acknowledgements
We are grateful to Akira Ishii for introducing Orlov’s paper [17]to us. 2.
Toric manifolds and T n -invariant manifolds Dual torus fibrations.
In this subsection, we briefly review the SYZ torus fibrationset-up [18]. For more details see [16, 15]. We follow the convention of [11].Throughout this section, we consider an n -dimensional tropical Hessian manifold B ,which we will define shortly, as the base space of a torus fibration. A smooth manifold B is called affine if B has an open covering { U λ } λ ∈ Λ such that the coordinate transformationis affine. This means that, for any U λ and U µ such that U λ ∩ U µ = ∅ , the coordinate systems x ( λ ) := ( x λ ) , . . . , x n ( λ ) ) t and x ( µ ) := ( x µ ) , . . . , x n ( µ ) ) t are related to each other by(1) x ( µ ) = ϕ λµ x ( λ ) + ψ λµ with some ϕ λµ ∈ GL ( n ; R ) and ψ λµ ∈ R n . If in particular ϕ λµ ∈ GL ( n, Z ) for any U λ ∩ U µ ,then B is called tropical affine . (If in addition ψ λµ ∈ Z n , B is called integral affine . ) MASAHIRO FUTAKI AND HIROSHIGE KAJIURA
For simplicity, we take such an open covering { U λ } λ ∈ Λ so that the open sets U λ and theirintersections are all contractible. It is known that B is an affine manifold iff the tangentbundle T B is equipped with a torsion free flat connection. When B is affine, then itstangent bundle T B forms a complex manifold. This fact is clear as follows. For each openset U = U λ , let us denote by ( x , . . . , x n ; y , . . . , y n ) the coordinates of U × R n ≃ T B | U sothat a point P ni =1 y i ∂∂x i | x ∈ T x B ⊂ T B corresponds to ( x , . . . , x n ; y , . . . , y n ) ∈ U × R n .We locally define the complex coordinate system by ( z , . . . , z n ), where z i := x i + i y i with i = 1 , . . . , n . By the coordinate transformation (1), the bases are transformed by ∂∂x ( µ ) = (cid:0) ϕ tλµ (cid:1) − ∂∂x ( λ ) , ∂∂x := ( ∂∂x , . . . , ∂∂x n ) t , and hence the corresponding coordinates are transformed by y ( µ ) = ϕ λµ y ( λ ) , y := ( y , . . . , y n ) t so that the combination P i y i ∂∂x i is independent of the coordinate systems. This showsthat the transition functions for the manifold T B are given by (cid:18) x ( µ ) y ( µ ) (cid:19) = (cid:18) ϕ λµ ϕ λµ (cid:19) (cid:18) x ( λ ) y ( λ ) (cid:19) + (cid:18) ψ λµ (cid:19) , and hence the complex coordinate systems are transformed holomorphically: z ( µ ) = ϕ λµ z ( λ ) + ψ λµ . On the other hand, for any smooth manifold B , the cotangent bundle T ∗ B has a(canonical) symplectic form ω T ∗ B . For each U λ = U , when we denote the coordinates of T ∗ B | U ≃ U × R n by ( x , . . . , x n ; y , . . . , y n ), ω T ∗ B is given by ω T ∗ B := d ( n X i =1 y i dx i ) = n X i =1 dx i ∧ dy i . This is actually defined globally since the coordinate transformations on T ∗ B are inducedfrom the coordinate transformations of { U λ } λ ∈ Λ . Actually, one has dx ( λ ) = ϕ λµ dx ( µ ) and the corresponding coordinates are transformed by(2) ˇ y ( λ ) = (cid:0) ϕ tλµ (cid:1) − ˇ y ( µ ) , ˇ y := ( y , . . . , y n ) t so that the combination P ni =1 y i dx i ∈ T ∗ B is independent of the coordinates. From this,it follows that the symplectic form ω T ∗ B = d ( P ni =1 y i dx i ) is defined globally.By choosing a metric g on a smooth manifold B , one obtains a bundle isomorphismbetween T B and T ∗ B . For each b ∈ B , this isomorphism T B → T ∗ B is defined by ξ g ( ξ, − ) for ξ ∈ T b B . This actually defines a bundle isomorphism since g is nondegenerate MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 5 at each point b ∈ B . This bundle isomorphism also induces a diffeomorphism from T B to T ∗ B . In this sense, hereafter we sometimes identify T B and T ∗ B . By this identification, y i and y i is related by y i = n X j =1 g ij y j , g ij := g ( ∂∂x i , ∂∂x j ) . When an affine manifold B is equipped with a metric g which is expressed locally as g ij = ∂ φ∂x i ∂x j for some local smooth function φ , then ( B, g ) is called a
Hessian manifold . When B isa Hessian manifold, then T B ≃ T ∗ B is equipped with the structure of K¨ahler manifoldas we explain below. In this sense, a Hessian manifold is also called an affine K¨ahlermanifold.First, when B is affine, then T B is already equipped with the complex structure J T B .We fix a metric g and set a two-form ω T B on T B as ω T B := n X i,j =1 g ij dx i ∧ dy j . This ω T B is nondegenerate since g is nondegenerate. Furthermore, ω T B is closed iff (
B, g )is Hessian, where ω T B coincides with the pullback of ω T ∗ B by the diffeomorphism T B → T ∗ B . Thus, a Hessian manifold ( B, g ) is equipped with the complex structure J T B andthe symplectic structure ω T B . A metric g T B on T B is then given by g T B ( X, Y ) := ω T B ( X, J
T B ( Y ))for X, Y ∈ Γ( T ( T B )). This is locally expressed as g T B = n X i,j =1 ( g ij dx i dx j + g ij dy i dy j ) . This shows that g T B is positive definite. To summarize, for a Hessian manifold (
B, g ),(
T B, J
T B , ω
T B ) forms a K¨ahler manifold, where g T B is the K¨ahler metric.In order to define a K¨ahler structure on T ∗ B , we employ the dual affine coordinateson B . Since P nj =1 g ij dx j is closed if ( B, g ) is Hessian, for each i , there exists a function x i := φ i of x such that dx i = n X i =1 g ij dx j . We thus obtain the dual coordinate systemˇ x ( λ ) := ( x ( λ )1 , . . . , x ( λ ) n ) t MASAHIRO FUTAKI AND HIROSHIGE KAJIURA for each λ . As above, we just denote x i instead of ˇ x i whenever no confusion occurs. Thedual coordinates then define another affine structure on B . Actually, the local descriptionof the metric is changed by g ( λ ) = { ( g ( λ ) ) ij } i,j =1 ,...,n = (cid:0) ϕ tλµ (cid:1) − g ( µ ) ϕ − λµ , so one has d ˇ x ( λ ) = (cid:0) ϕ tλµ (cid:1) − d ˇ x ( µ ) and then(3) ˇ x ( λ ) = (cid:0) ϕ tλµ (cid:1) − ˇ x ( µ ) + ˇ ψ λµ for some ˇ ψ λµ ∈ R n . Thus, the combinations z i := x i + i y i , i = 1 , . . . , n , form a complexcoordinate system on T ∗ B , and T ∗ B forms a complex manifold. Actually, by eq.(2) and(3), one has the holomorphic coordinate transformationˇ z ( µ ) = (cid:0) ϕ tλµ (cid:1) − ˇ z ( λ ) + ˇ ψ λµ , ˇ z := ( z , . . . , z n ) t . Using this dual coordinates, the symplectic form ω T ∗ B is expressed locally as ω T ∗ B = n X i,j =1 g ij dx i ∧ dy j , where g ij is the ( i, j ) element of the inverse matrix of { g ij } . Then, we set a metric on T ∗ B by g T ∗ B ( X, Y ) := ω T ∗ B ( X, J T ∗ B ( Y ))for X, Y ∈ Γ( T ( T ∗ B )), which is locally expressed as g T ∗ B = n X i,j =1 ( g ij dx i dx j + g ij dy i dy j ) . These structures define a K¨ahler structure on T ∗ B .For a tropical Hessian manifold B , we consider two T n -fibrations over B obtained by aquotient M of T B and a quotient ˇ M of T ∗ B by fiberwise Z n action as follows.For T B , we locally consider
T B | U and define a Z n -action generated by y i y i + 2 π for each i = 1 , . . . , n . For T ∗ B , we again locally consider T ∗ B | U and define a Z n -actiongenerated by y i y i + 2 π for each i = 1 , . . . , n . Both Z n -actions are well-defined globallysince B is tropical affine, i.e., the transition functions of n -dimensional vector bundles T B and T ∗ B belong to GL ( n ; Z ).Then, M := T B/ Z n is a K¨ahler manifold whose symplectic structure ω M and complexstructure J M are those naturally induced from ω T B and J T B on T B . Similarly, ˇ M := T ∗ B/ Z n is a K¨ahler manifold whose symplectic structure ω ˇ M and complex structure J ˇ M are those induced from ω T ∗ B and J T ∗ B , respectively. The fibrations π : M → B and π ˇ : ˇ M → B are often called semi-flat torus fibrations or T n -invariant manifolds . See[16, 15] and also [8]. Since M and ˇ M are dual to each other, we can construct them in the MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 7 opposite way. That is, if we consider the coordinate systems ˇ x ( λ ) λ for B , then the tangentbundle over B is T ∗ B above, and the cotangent bundle is T B . Following [16, 15], we treat M as a symplectic manifold and ˇ M as a complex manifold and discuss the homologicalmirror symmetry.2.2. Toric manifolds and T n -invariant manifolds. The set-up in the previous sub-section is originally applied to the mirror symmetry of compact Calabi-Yau manifolds M, ˇ M . We would like to extend this set-up to the case ˇ M is the complement of the toricdivisors of a smooth compact toric manifold X . The complement ˇ M is actually a trivialtorus fibration π ˇ : ˇ M → B where the base B is identified with the interior of the dualpolytope P of X .What may be more interesting is that B is actually tropical affine in this situation.Of course, since B is a contractible open set, B = Int( P ) has an open covering by itself,which means that B is tropical affine. However, what we meant is something stronger inthe following sense. The natural open covering of the smooth compact toric manifold X induces an open covering { ˜ U λ } λ ∈ Λ of P . Then we see that the coordinate transformationsare tropical affine (though ˜ U λ ∩ B = B for any λ . ) This seems important since we needto include some information from the boundary ∂ ( B ) of B when we discuss homologicalmirror symmetry of X and its mirror dual.Let us see the above construction explicitly for X = C P n . For C P n = { [ t : · · · : t n ] } , the natural open covering is { ˜ U λ } λ =0 , ,...,n where˜ U λ = { [ t : · · · : t n ] | t λ = 0 } . The corresponding local coordinates are ( w ( λ )1 , . . . , w ( λ ) n ) where(4) w ( λ )1 = t /t λ , ..., w ( λ ) λ = t λ − /t λ , w ( λ ) λ +1 = t λ +1 /t λ , ..., w ( λ ) n = t n /t λ . We identify ˇ M with the complement of the toric divisors of C P n :ˇ M = { [ t : · · · : t n ] | t · t · · · t n = 0 } , where π ˇ: ˇ M → B is given by π ˇ([ t ; · · · : t n ]) := [ | t | : · · · : | t n | ] . So we have U λ := ˜ U λ ∩ ˇ M = ˇ M for any λ . We further denote U λ := π ˇ( U λ ). For each U λ ,we express w ( λ ) i =: e z ( λ ) i = e x ( λ ) i + i y ( λ ) i . Since the coordinate transformation between z ( λ ) and z ( µ ) is tropical affine by (4), so isthe coordinate transformation between x ( λ ) and x ( µ ) . MASAHIRO FUTAKI AND HIROSHIGE KAJIURA
Hereafter we consider U := U (since U = U = · · · = U n = B ) and drop the upperindex (0) ; for instance w (0) i =: w i and x (0) i =: x i . The K¨ahler metric is then expressed in U := ( π ˇ) − ( U ) as ω = − i d (cid:18) ¯ w dw + · · · + ¯ w n dw n w w + · · · + ¯ w n w n (cid:19) . When we express this as ω = P ni,j =1 dx i g ij dy j , we have g ij = ∂ ˇ φ∂x i ∂x j , ˇ φ = log(1 + e x + · · · + e x n ) . Thus, B is a Hessian manifold. The dual coordinates ( x , . . . , x n ) is obtained by dx i = n X j =1 ∂ ˇ φ∂x i ∂x j dx j = d (cid:18) ∂ ˇ φ∂x i (cid:19) , so(5) x i = ∂ ˇ φ∂x i = 2 e x i e x + · · · + e x n . By this ( x , . . . , x n ), B is expressed as B = { ( x , . . . , x n ) | x > , . . . , x n > , x + · · · + x n < } . We can regard M , the dual torus fibration of ˇ M , as a subset of ( C × ) n , where ( e x + i y , . . . , e x n + i y n )is the coordinate system of ( C × ) n . Note that ω diverges at the boundary ∂ ( M ) = π − ( ∂ ( B )).3. Lagrangian submanifolds and holomorphic vector bundles
In the first two subsections, we first recall the construction of line bundles on ˇ M asso-ciated to Lagrangian sections of M → B discussed in [16, 15]. Then, in subsection 3.3,we apply this construction to the case ˇ M is the complement of the toric divisors of C P n .3.1. Lagrangian submanifolds in M . We fix a tropical affine open covering { U λ } λ ∈ Λ .Let s : B → M be a section of M → B . Locally, we may regard s as a section of T B ≃ T ∗ B . Then, s is locally described by a collection of functions as y i ( λ ) = s i ( λ ) ( x )on each U λ .On U λ ∩ U µ , these local expressions are related to each other by(6) s ( µ ) ( x ) = s ( λ ) ( x ) + I λµ MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 9 for some I λµ ∈ Z n . Here, x may be identified with either x ( λ ) or x ( µ ) . Also, s ( λ ) ( x ) and s ( µ ) ( x ) are expressed by the common coordinates y ( λ ) or y ( µ ) . This transformation ruleautomatically satisfies the cocycle condition(7) I λµ + I µν + I νλ = 0for U λ ∩ U µ ∩ U ν = ∅ . We denote by s such a collection { s ( λ ) : U λ → T B | U λ } λ ∈ Λ which isequipped with the transformation rule (6) satisfying the cocycle condition (7).Now we discuss when the graph of s forms a Lagrangian submanifold in M . By defini-tion, an n -dimensional submanifold L in a 2 n -dimensional symplectic manifold ( M, ω M )is Lagrangian iff ω M | L = 0. This is a local condition. Thus, in order to discuss whetherthe graph of a section s : B → M is Lagrangian or not, we may check the conditionlocally and in particular in T ∗ B .It is known (as shown easily by taking the basis) that the graph of P ni =1 y i dx i with localfunctions y i is Lagrangian in T ∗ B iff there exists a local function f such that P ni =1 y i dx i = df . Now, a section s : B → M is locally regarded as a section of T ∗ B by setting y i = P nj =1 g ij y j = P nj =1 g ij s j , from which one has n X i =1 y i dx i = n X i =1 ( n X j =1 g ij s j ) dx i = n X j =1 s j dx j . Thus, the graph of the section s : B → M is Lagrangian iff there exists a local function f such that P nj =1 s j dx j = df .Note that y = s ( x ) defines a special Lagrangian submanifold if s is affine with respectto x i . (Thus, the zero section of M → B is a special Lagrangian submanifold. )The gradient vector field is of the form:(8) grad( f ) := X i,j ∂f∂x j g ji ∂∂x i = X i ∂f∂x i ∂∂x i . Holomorphic vector bundles on ˇ M . Consider a section s : B → M and expressit as a collection s = { s ( λ ) } λ ∈ Λ of local functions. We define a line bundle V with a U (1)-connection on the mirror manifold ˇ M associated to s . We set the covariant derivativelocally as (9) D := d − i n X i =1 s i ( x ) dy i , We switch the sign of the connection one form compared to that in [11] so that the mirror corre-spondence of objects fits with the one in homological mirror symmetry of tori as in [13] and referencestherein. whose curvature is D = i n X i,j =1 ∂s i ∂x j dx j ∧ dy i . The (0 , ∂s i ∂x j is symmetric, which is the case when thereexists a function f locally such that df = P ni =1 s i dx i . Thus, the condition that D definesa holomorphic line bundle on ˇ M is equivalent to that the graph of s is Lagrangian in M .This covariant derivative D is in fact defined globally. Suppose that D is given locallyon each ˇ M | U λ of the T n -fibration ˇ M → B with a fixed tropical affine open covering { U λ } λ ∈ Λ . Namely, we continue to employ { U λ } λ ∈ Λ for local trivializations of the linebundle associated to a section s : B → M . The transition functions for ( V, D ) are definedas follows. Recall that the section s : B → M is expressed locally as y i ( λ ) = s i ( λ ) ( x )on each U λ , where, on U λ ∩ U µ , the local expression is related to each other by s ( µ ) ( x ) = s ( λ ) ( x ) + I λµ for some I λµ ∈ Z n (see eq.(6)). Correspondingly, the transition function for the line bundle V with the connection D is given by ψ ( µ ) = e i I λµ · ˇ y ψ ( λ ) for local expressions ψ ( λ ) , ψ ( µ ) of a smooth section ψ of V , where I λµ · ˇ y := P nj =1 i j y j for I λµ = ( i , . . . , i n ). We see the compatibility( Dψ ( λ ) ) ( µ ) = D ( ψ ( µ ) )holds true since the left hand side turns out to be e i I λµ · ˇ y (( d − i s ( λ ) ( x ) · dy ) e − i I λµ · ˇ y ψ ( µ ) )= e i I λµ · ˇ y e − i I λµ · ˇ y (( d − i ( s ( λ ) ( x ) + I λµ ) · dy ) ψ ( µ ) )= ( d − i s ( µ ) ( x ) · dy ) ψ ( µ ) . Since (
V, D ) is locally-trivialized by { ˇ M | U λ } λ ∈ Λ , for each x ∈ B , ψ ( x, · ) gives a smoothfunction on the fiber T n . Thus, on each U λ , ψ ( x, y ) can be Fourier-expanded as ψ ( x, y ) | U λ = X I ∈ Z n ψ λ,I ( x ) e i I · ˇ y , MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 11 where I · ˇ y := P nj =1 i j y j for I = ( i , . . . , i n ). Note that each coefficient ψ λ,I is a smoothfunction on U λ . In this expression, the transition function acts to each ψ λ,I as X I ∈ Z n ψ µ,I e i I · ˇ y = e i I λµ · ˇ y X I ∈ Z n ψ λ,I e i I · ˇ y = X I ∈ Z n ψ λ,I e i ( I + I λµ ) · ˇ y = X I ∈ Z n ψ λ,I − I λµ e i I · ˇ y and hence ψ µ,I = ψ λ,I − I λµ .3.3. Holomorphic line bundles on C P n and the corresponding Lagrangians. Inthe previous subsections, we assign a line bundle on ˇ M to each Lagrangian section in M → B . In this subsection, we start from a line bundle on on C P n . We identify ˇ M withthe complement of the toric divisors of C P n , and restrict the line bundle to ˇ M . We seethat, by twisting it with an appropriate isomorphism, the result actually comes from aLagrangian section in M → B . In this way, we construct a Lagrangian section in M → B corresponding to O ( a ) on C P n for any a ∈ Z .We continue the convention in subsection 2.2. The complement of the toric divisors of C P n is ˇ M = { [ t : · · · : t n ] | t · t · · · t n = 0 } , where e x i + i y i = w i = t i /t . A connection one-form of O ( a ) is given by the one which is expressed locally on ˇ M as A a = − a ¯ w dw + · · · + ¯ w n dw n w w + · · · + ¯ w n w n = − a e x ( dx + i dy ) + · · · + e x n ( dx n + i dy n )1 + e x + · · · + e x n . (10)We twist this by Ψ a := (1 + e x + · · · + e x n ) a/ , and then obtain(11) Ψ − a ( d + A a )Ψ a = d − i a e x dy + · · · + e e xn dy n e x + · · · + e x n . By the previous subsections, this is the line bundle on ˇ M which corresponds to theLagrangian section L a in M → B expressed as y ... y n = s a ... s na = a e x e x + ··· + e xn ... e xn e x + ··· + e xn = a x ... x n where x i > i = 1 , . . . , n and x + · · · + x n < f a = a e x + · · · + e x n ) − log 2)= − a − x − x − · · · − x n )satisfies df a = P ni =1 s ia dx i . The corresponding gradient vector field isgrad( f a ) = n X i =1 ∂f a ∂x i ∂∂x i = ax ∂∂x + · · · + ax n ∂∂x n by (8). Remark 3.1.
This Lagrangian section L a is a special Lagrangian since it is expressedlocally as the graph of linear functions y i ( x ) of ( x , . . . , x n ).Furthermore, we see that L a includes a critical point of the corresponding Landau-Ginzburg potential. In fact, the Landau-Ginzburg potential W is W ( z , z , . . . , z n ) := z + · · · + z n + e − z z · · · z n . The critical points are given by( z , . . . , z n ) = (cid:16) e − n +1 ω a , . . . , e − n +1 ω a (cid:17) =: c a , a = 0 , , . . . , n where ω = (1) n +1 is the ( n + 1)-th root of unity. Thus, we see that each critical point c a ∈ ( C × ) n is included in L a .4. Homological mirror symmetry set-up
In this section, we first recall DG-categories F ( M ) and V ( ˇ M ) associated to M and ˇ M respectively, following [11]. Kontsevich-Soibelman’s approach for the homological mirrorsymmetry [14] introduces an intermediate category Mo ( B ) and the existence of an A ∞ -equivalence F uk ( M ) ≃ Mo ( B ) ∼ → F ( M ) . MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 13
This Mo ( B ) is called the weighted Fukaya-Oh category or the category of weighted Morsehomotopy on B . In subsection 4.5, we propose a modification Mo ( P ) of Mo ( B ) where P = ¯ B is the dual polytope of a smooth compact toric manifold X .4.1. DG-category V associated to ˇ M . We define a DG-category V = V ( ˇ M ) of holo-morphic line bundles over ˇ M as follows. The objects are holomorphic line bundles V with U (1)-connections D associated to lifts s of sections as we defined in subsection3.2. We sometimes label these objects as s instead of ( V, D ). For any two objects s a = ( V a , D a ) , s b = ( V b , D b ) ∈ V , the space V ( s a , s b ) of morphisms is defined by V ( s a , s b ) := Γ( V a , V b ) ⊗ C ∞ ( ˇ M ) Ω , ∗ ( ˇ M ) , where Ω , ∗ ( ˇ M ) is the space of anti-holomorphic differential forms and Γ( V a , V b ) is the spaceof homomorphisms from V a to V b . The space V ( s a , s b ) is a Z -graded vector space, wherethe grading is defined as the degree of the anti-holomorphic differential forms. The degree r part is denoted by V r ( s a , s b ). We define a linear map d ab : V r ( s a , s b ) → V r +1 ( s a , s b ) asfollows. We decompose D a into its holomorphic part and anti-holomorphic part D a = D (1 , a + D (0 , a , and set 2 D (0 , a =: d a . Then, for ψ ∈ V r ( s a , s b ), we set d ab ( ψ ) := d b ψ − ( − r ψd a ∈ V r +1 ( s a , s b ) . Note that d ab = 0 since each ( V a , D a ) is holomorphic, i.e., ( d a ) = 0.The product structure m : V ( s a , s b ) ⊗ V ( s b , s c ) → V ( s a , s c ) is defined by the compo-sition of homomorphisms of line bundles together with the wedge product for the anti-holomorphic differential forms. More precisely, for ψ ab ∈ V r ab ( s a , s b ) and ψ bc ∈ V r bc ( s b , s c ),we set m ( ψ ab , ψ bc ) := ( − r ab r bc ψ bc ∧ ψ ab (= ψ ab ∧ ψ bc ) , where ∧ denotes the operation consisting of the composition and the wedge product.Then, we see that V forms a DG-category. In order to construct another equivalent curved DG-category, we rewrite this DG-category V more explicitly. For an element ψ ∈ V r ( s a , s b ), we Fourier-expand this locallyas ψ (ˇ x, ˇ y ) = X I ∈ Z n ψ I (ˇ x ) e i I · ˇ y , Here we again make a minor change of the formulation of the DG category compared to [11] due tothe change of sign in (9). In [11], we construct a curved DG category DG ˇ M where the objects are not necessarily holomorphic.The relation is given by V = DG ˇ M (0). where ψ I is a smooth anti-holomorphic differential form of degree r . Namely, it is ex-pressed as ψ I = X i ,...,i r ψ I ; i ··· i r d ¯ z i ∧ · · · ∧ d ¯ z i r with smooth functions ψ I ; i ··· i r . Let us express the transformation rules for s a and s b as( s a ) ( µ ) = ( s a ) ( λ ) + I a , ( s b ) ( µ ) = ( s b ) ( λ ) + I b with I a = I a ; λµ ∈ Z n , I b = I b ; λµ ∈ Z n . The transition function is then given by ψ ( µ ) = e i ( I b − I a ) · ˇ y ψ ( λ ) and hence ψ ( µ ) ,I = ψ ( λ ) ,I + I a − I b . The differential d ab is expressed locally as follows. Since D a = d − i n X j =1 s ja ( x ) dy j = n X j =1 (cid:18) ∂∂x j dx j + ( ∂∂y j − i s ja ) dy j (cid:19) = 12 n X j =1 ( ∂∂x j − i ( ∂∂y j − i s ja )) dz j + 12 n X j =1 ( ∂∂x j + i ( ∂∂y j − i s ja )) d ¯ z j , one has d a = 2 D (0 , a = n X j =1 ( ∂∂x j + s ja + i ∂∂y j ) d ¯ z j and then(12) d ab ( ψ ) = 2 ¯ ∂ ( ψ ) − n X i =1 ( s a − s b ) i d ¯ z i ∧ ψ. DG-category F associated to M . We define a DG-category F = F ( M ) consistingof Lagrangian sections in M as follows. As we shall see, we construct it so that it iscanonically isomorphic to the previous DG-category V . We fix a tropical affine opencovering { U λ } λ ∈ Λ of B .The objects are the same as those in V , that is, lifts s of sections of M → B . For anytwo objects s a , s b ∈ F , we express the transformation rules for s a and s b as( s a ) ( µ ) = ( s a ) ( λ ) + I a , ( s b ) ( µ ) = ( s b ) ( λ ) + I b This F corresponds to DG M (0) in [11]. MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 15 as we did in the previous subsection. For each λ ∈ Λ and I ∈ Z n , let Ω λ,I ( s a , s b ) be thespace of complex valued smooth differential forms on U λ . The space F ( s a , s b ) is then thesubspace of Y λ ∈ Λ Y I ∈ Z n Ω λ,I ( s a , s b )such that • φ λ,I ∈ Ω λ,I ( s a , s b ) satisfies φ µ,I | U λ ∩ U µ = φ λ,I + I a − I b | U λ ∩ U µ for any U λ ∩ U µ = ∅ and • the sum P I ∈ Z n φ λ,I e i I · ˇ y converges as smooth differential forms on each M | U λ .The space F ( s a , s b ) is a Z -graded vector space, where the grading is defined as the degreeof the differential forms. The degree r part is denoted F r ( s a , s b ). We define a linear map d ab : F r ( s a , s b ) → F r +1 ( s a , s b ) which is expressed locally as d ab ( φ λ,I ) := d ( φ λ,I ) − n X j =1 ( s ja − s jb + i j ) dx j ∧ φ λ,I for φ λ,I ∈ Ω λ,I ( s a , s b ) with I := ( i , . . . , i n ) ∈ Z n , where d is the exterior differential on B .We have d ab = 0.The composition of morphisms m : F ( s a , s b ) ⊗ F ( s b , s c ) → F ( s a , s c ) is defined by m ( φ ab ; λ,I , φ bc ; λ,J ) := φ ab ; λ,I ∧ φ bc ; λ,J ∈ Ω λ,I + J ( s a , s c )for φ ab ; λ,I ∈ Ω λ,I ( s a , s b ) and φ bc ; λ,J ∈ Ω λ,I ( s b , s c ). These structures define a DG-category F . Note that this F is believed to be A ∞ -equivalent to the corresponding full subcategoryof the Fukaya category F uk ( M ). (Compare this F with what is called the deRham modelfor the Fukaya category in Kontsevich-Soibelman [14], in particular a construction in theAppendix (Section 9.2). ) In subsection 4.4, we shall explain the outline of how tocompare F with the Fukaya category.4.3. Equivalence between F and V . The DG-category F is canonically isomorphic tothe DG-category V . In fact, we see that the objects in F are the same as those in V .The spaces of morphisms in F and in V are also identified canonically as follows. For amorphism φ ab = { φ ab ; λ,I } ∈ F r ( s a , s b ), each φ ab ; λ,I is expressed as φ ab ; λ,I = X i ,...,i r φ ab ; λ,I ; i ··· i r dx i ∧ · · · ∧ dx i r . To this, we correspond an element in V r ( s a , s b ) which is locally given as X i ,...,i r ( φ ab ; λ,I ; i ··· i r e i I · ˇ y ) d ¯ z i ∧ · · · ∧ d ¯ z i r on U λ . We denote this correspondence by f : F → V . It is easily seen that our constructionguarantees the following fact.
Proposition 4.1 ([11, Proposition 4.1.]) . The functor f : F → V is a DG-isomorphism.
The DG-category F and the Fukaya category F uk ( M ) . The DG-category F isexpected to be A ∞ -equivalent to the Fukaya category F uk ( M ) [7] of Lagrangian sections.The idea discussed in [14] to relate them is to apply homological perturbation theoryto the DG-category F (as an A ∞ -category) in an appropriate way so that the induced A ∞ -category coincides with (the full subcategory of) the Fukaya category F uk ( M ). Moreprecisely, what should be induced directly from F is the category Mo ( B ) of weightedMorse homotopy or the Fukaya-Oh category for the torus fibration M → B introducedin section 5.2 of [14]. Here, the Fukaya-Oh category means the A ∞ -category of Morsehomotopy on B introduced in [7]. It is shown in [9] that the Fukaya-Oh category isequivalent to the Fukaya category F uk ( T ∗ B ) consisting of the corresponding objects.The Fukaya-Oh category for the torus fibration M → B is a generalization of the Fukaya-Oh category on B so that it corresponds to the Fukaya category F uk ( M ) instead of F uk ( T ∗ B ). Thus, a natural way of obtaining the A ∞ -equivalence F ≃
F uk ( M ) is tointerpolate the category Mo ( B ) so that F uk ( M ) = Mo ( B ) ∼ → F , where an A ∞ -equivalence Mo ( B ) → F is expected to be obtained by the homological per-turbation theory. There are some technical difficulties in proceeding this story precisely.See subsection 5.5 of [11].4.5. The category Mo ( P ) of weighted Morse homotopy. If we start with a toricmanifold X and set ˇ M as the complement of the toric divisors, we obtain M as a torusfibration over the interior B of the dual polytope P . As we discuss in the next section, fromthe homological mirror symmetry viewpoint, what we should discuss is not F uk ( M ) buta kind of Fukaya category F uk ( ¯ M ) of a torus fibration over P = ¯ B . As an intermediatestep, we consider the category Mo ( P ) of weighted Morse homotopy for the dual polytope P . This Mo ( P ) is a generalization of the weighted Fukaya-Oh category given in [14] tothe case where the base manifold has boundaries and critical points may be degenerate.In the present paper we consider affine Lagrangians only. However we do not limitourselves to that cases in this subsection as the framework should work in more generaltoric cases. The detailed discussion in full generality should be carried out elsewhere. Definition of Mo ( P ) The definition is as follows. The objects of Mo ( P ) are Lagrangiansections of π : M → B satisfying certain conditions (see for instance [5]). We extend eachLagrangian section on B to that on ¯ B smoothly. We say that two objects L, L ′ intersects MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 17 cleanly if there exists an open set e B such that ¯ B ⊂ e B and L, L ′ over B can be extendedto graphs of smooth sections over e B so that they intersect cleanly. We assume that anytwo objects L, L ′ intersect cleanly.For each L , we take a “Morse” function f L on e B so that L is the graph of df L . For agiven ordered pair ( L, L ′ ), we assign a grading | V | for each connected component V ofthe intersection π ( L ∩ L ′ ) in P = ¯ B as the dimension of the stable manifold S v ⊂ e B ofthe gradient vector field − grad( f L − f L ′ ) with a point v ∈ V . This does not depend onthe choice of the point v ∈ V . The space Mo ( P )( L, L ′ ) of morphisms is then set to be the Z -graded vector space spanned by the connected components V of π ( L ∩ L ′ ) ∈ P suchthat there exists a point v ∈ V which is an interior point of S v ∩ P ⊂ S v . Now let us consider an ( l + 1)-tuple ( L , . . . L l +1 ), l ≥
2, and take a generator V i ( i +1) ∈ Mo ( P )( L i , L i +1 ) for each i and V l +1) ∈ Mo ( P )( L , L l +1 ). We denote by GT ( v , . . . , v l ( l +1) ; v l +1) )the set of gradient trees starting at v , . . . , v l ( l +1) , where v i ( i +1) ∈ V i ( i +1) , and endingat v l +1) ∈ V l +1) . Here, a gradient tree γ ∈ GT ( v , . . . , v l ( l +1) ; v l +1) ) is a continuousmap γ : T → P with a rooted trivalent l -tree T . Regarding T as a planar tree, theleaf external vertices from the left to the right are mapped to v , . . . , v l ( l +1) , and theroot external vertex is mapped to v l +1) by γ . Furthermore, for each edge e of T , therestriction γ | e is a gradient trajectory of the corresponding gradient vector field. See [14].We then denote GT ( V , . . . , V l ( l +1) ; V l +1) ) := [ ( v ,...,v l ( l +1) ; v l +1) ) ∈ V ×···× V l ( l +1) × V l +1) GT ( v , . . . , v l ( l +1) ; v l +1) ) . We say that two gradient trees γ, γ ′ ∈ GT ( V , . . . , V l ( l +1) ; V l +1) ) is C ∞ -homotopic toeach other if γ : T → P is homotopic to γ ′ : T → P so that γ | e is C ∞ -homotopic to γ ′ | e for each edge of T . We further denote HGT ( V , . . . , V l ( l +1) ; V l +1) ) := { [ γ ] | γ ∈ GT ( V , . . . , V l ( l +1) ; V l +1) ) } , where [ γ ] is the C ∞ -homotopy class of γ .We in particular consider the case where | V l +1) | = | V | + · · · + | V l ( l +1) | + 2 − l ,then assume that HGT ( V , . . . , V l ( l +1) ; V v l +1) ) is a finite set. For each element in γ ∈GT ( V , . . . , V l ( l +1) ; V v l +1) ), we can assign the weight e − A ( γ ) where A ( γ ) ∈ [0 , ∞ ] is thesymplectic area of the piecewise smooth disk in π − ( γ ( T )) as is done in Kontsevich-Soibelman [14]. This weight is invariant with respect to a C ∞ -homotopy. Then, we definea multilinear product m l : Mo ( P )( L , L ) ⊗ Mo ( P )( L , L ) ⊗ · · · ⊗ Mo ( P )( L l , L l +1 ) → Mo ( P )( L , L l +1 ) We consider the Morse cohomology degree instead of the Morse homology degree. of degree 2 − l by(13) m l ( V , . . . , V l ( l +1) ) = X V l +1) X [ γ ] ∈HGT ( V ,...,V l ( l +1) ; V l +1) ) ± e − A ( γ ) V l +1) , where V l +1) are the bases of Mo ( P )( L , L n +1 ) of degree | V | + · · · + | V l ( l +1) | + 2 − l , andthe sign ± is given by the formula of the homological perturbation lemma of a (minimal) A ∞ -structure.For a given set E of objects of Mo ( P ), we denote by Mo E ( P ) the full subcategory of Mo ( P ) consisting of objects in E . In this paper, we consider Mo E ( P ) so that E formsa strongly exceptional collection in Tr ( Mo E ( P )) where we need only m , with all higher m k ’s vanish because of the degree reasons. As we shall explain later, we take E to be theset of Lagrangian sections L a corresponding to O ( a ) for the X = C P n case, and take E to be their products for the X = C P m × C P n case. Therefore we postpone to show thewell-definedness of the whole Mo ( P ) for general P and that { m k } k =2 , ,... defines a minimal A ∞ -structure. A strong minimality assumption
As above, we defined m l for l ≥ A ∞ -structure. In order for it to be minimal naturally, for each pair( L, L ′ ), the differentials of a Morse-Bott version of the Floer complex on Mo ( P )( L, L ′ )should be trivial. In our construction, we rather impose a stronger assumption for theclass of objects as follows. For any pair ( L, L ′ ) and any two distinct elements of the basis V, W ∈ Mo ( P )( L, L ′ ) , there does not exist any gradient flow starting at a point in V andending at a point in W . The identity morphism
For each L ∈ Mo ( P ), the space Mo ( P )( L, L ) of morphismsis generated by P itself which is of degree zero. When the above Mo ( P ) is well-definedand forms a minimal A ∞ -category, we believe that P is the strict unit. We see that P ∈ Mo ( P )( L, L ) forms at least the identity morphism with respect to m under thestrong minimality assumption above. In order to show that it is the strict unit, we needto show that Mo ( P ) is obtained by applying homological perturbation theory to a DGcategory. Thus, that Mo ( P ) forms a minimal A ∞ -category and that Mo ( P ) is strictlyunital should be shown at the same time. More explicit expression
For each L , let us choose a local expression s : B → T B aswe did for F or V . (A different choice leads to an isomorphic object. ) This enables usto assign each generator V of a morphism space a Z n -grading (which is different from thegrading | V | above). For instance, for lifts s a : B → T B and s b : B → T B of L a and L b ,consider s b,I : B → T B defined by y j = s jb,I = s jb − i j , MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 19 where I = ( i , . . . , i n ) ∈ Z n . Denote by M o I ( P )( s a , s b ) the space generated by thegenerators of Mo ( P )( s a , s b ) which are included in the image of the intersection graph( s a ) ∩ graph( s b ; I ) by T B → B . Then, we have the decomposition Mo ( P )( s a , s b ) = a I ∈ Z n Mo I ( P )( s a , s b ) . In this way, each generator of Mo ( P )( s a , s b ) is assigned a Z n -grading. It turns out thatthe multilinear product (13) preserves these gradings. Connection to DG ( X ) We end with this subsection by explaining why we expect this Mo ( P ) to be a candidate of the category on the mirror dual of X . We start with theDG category DG ( X ) of holomorphic line bundles on X , as is constructed explicitly insubsection 5.1, and remove the toric divisors of X to obtain ˇ M . Then, DG ( X ) should beregarded as a subcategory of V ( ˇ M ). In particular, the subcategory is not full since thesmoothness condition at the removed toric divisors is imposed in V ( ˇ M ). The cohomologiesof the morphism spaces in DG ( X ) then differs from those in V ( ˇ M ). They can be largerthan those in V ( ˇ M ) though the morphism spaces in DG ( X ) are smaller at the cochainlevel.In the original set-up where M and ˇ M are supposed to be compact Calabi-Yau mani-folds, the cohomologies of the morphism spaces in V ( ˇ M ) are in one-to-one correspondencewith those in F ( M ) since F ( M ) and V ( ˇ M ) are isomorphic DG-category (subsection 4.4).Furthermore, the cohomologies of the morphism spaces F ( M )( s a , s b ) are isomorphic to F uk ( M )( s a , s b ) at least when s a and s b define Lagrangian sections L a and L b which aretransversal to each other. Namely, the cohomologies H ( V ( s a , s b )) are spanned by baseswhich are associated with connected components of L a ∩ L b . We would like to keep thisrelation even when ˇ M is noncompact. Then, if the smoothness condition at the removedtoric divisors produces additional generators in H ( DG ( X )( s a , s b )) from H ( V ( ˇ M )( s a , s b )),we would like to enlarge M so that there exist the corresponding additional connectedcomponents of the intersections of the Lagrangians. Our feeling is that it seems to gowell if we add the boundary of M and consider a kind of Fukaya category F uk ( ¯ M ) or thecorresponding category Mo ( P = ¯ B ) of weighted Morse homotopy. In the next section, weexplicitly proceed this story successfully for X the projective spaces and their products.5. Homological mirror symmetry of C P n In this section, we discuss a version of homological mirror symmetry of C P n as the com-plex side by explicitly proceeding the story described in the last subsection. In subsection5.1, we construct the DG category DG ( C P n ) of holomorphic line bundles on C P n , and re-call the structure of its cohomologies. Then, we discuss the homological mirror symmetryfor C P n in subsection 5.2. We extend the story to C P m × C P n in subsection 5.3. DG category DG ( C P n ) of line bundles over C P n . We first construct the DGcategory DG ( C P n ) consisting of holomorphic line bundles O ( a ), a ∈ Z . The space DG ( C P n )( O ( a ) , O ( b )) of morphisms is defined as the Dolbeault resolution of Γ( O ( a ) , O ( b )).Namely, it is the graded vector space, each graded piece of which is given by DG r ( C P n )( O ( a ) , O ( b )) := Γ( O ( a ) , O ( b )) ⊗ Ω ,r ( C P n )with Γ( O ( a ) , O ( b )) being the space of smooth bundle morphism from O ( a ) to O ( b ). Thecomposition of morphisms is defined in a similar way as that in V ( ˇ M ) in subsection 4.1.Each O ( a ) is associated with the connection D a , which is expressed locally as D a = d − a ¯ w dw + · · · + ¯ w n dw n w w + · · · + ¯ w n w n on U = U (eq.(10)), and the differential d ab : DG r ( C P n )( O ( a ) , O ( b )) → DG r +1 ( C P n )( O ( a ) , O ( b ))is defined by d ab ( ˜ ψ ) := 2 (cid:16) D , b ˜ ψ − ( − r ˜ ψD , a (cid:17) . This differential satisfies the Leibniz rule with respect to the composition. Thus, DG ( C P n )is a DG category.The generator of H ( DG ( C P n ))( O ( a ) , O ( a + 1)) is given by(14) 1 , w , w , . . . , w n locally on U . These generate H ( DG ( C P n ))( O ( a ) , O ( b )) as products of functions, so H ( DG ( C P n ))( O ( a ) , O ( b )) is represented by polynomials in ( w , . . . , w n ) of degree equalto or less than b − a . In particular, H ( DG ( C P n ))( O ( a ) , O ( b )) = 0 for a > b . It isknown by [2] that E := ( O ( q ) , . . . , O ( q + n )) forms a full strongly exceptional collectionof D b ( coh ( C P n )) for each q ∈ Z . That E forms a strongly exceptional collection means H ( DG ( C P n ))( O ( a ) , O ( a )) ≃ C ,H ( DG ( C P n ))( O ( a ) , O ( b )) = 0 , a > bH r ( DG ( C P n ))( O ( a ) , O ( b )) = 0 , r = 0for any a, b = { q, q +1 , . . . , q + n } . Let DG E ( C P n ) be the full DG subcategory of DG ( C P n )consisting of E . Then the strongly exceptional collection E is full means that it generates D b ( coh ( C P n )) in the sense that Tr ( DG E ( C P n )) ≃ D b ( coh ( C P n )) , where Tr is the Bondal-Kapranov construction [4]. MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 21
Note that H n ( DG ( C P n ))( O ( q + n + 1) , O ( q )) = 0; it includes an element representedby d ¯ w · · · d ¯ w n (1 + w ¯ w + · · · + w n ¯ w n ) . Homological mirror symmetry of C P n . First, we identify the DG category DG ( C P n ) with a (non-full) subcategory V ′ of the DG category V = V ( ˇ M ) consistingof the same objects O ( a ), a ∈ Z , whereˇ M = C P n \{ [ t : t : · · · : t n ] | t · t · · · t n = 0 } . For a given morphism ˜ ψ ∈ DG ( C P n )( O ( a ) , O ( b )), we express it locally on U (see subsec-tion 2.2), and remove the origin (corresponding to t = 0). We send this to V ′ ( O ( a ) , O ( b ))using (11): ˜ ψ ψ := Ψ − b ◦ ˜ ψ ◦ Ψ a . Clearly, this map is compatible with the differentials and the compositions in both sides.In this way, we obtain a functor I : DG ( C P n ) → V of DG-categories. We see that I is faithful. However, I is not full since ˜ ψ is smooth atthe points { [ t : t : · · · : t n ] | t · t · · · t n = 0 } . Thus, the image V ′ := I ( DG ( C P n ))is a non-full DG subcategory of V .The local expression for morphisms are transformed by I as follows. By w i = e x i + i y i = r i e i y i , we Fourier-expanded ˜ ψ as˜ ψ = X I ∈ Z n ˜ ψ I ( r ) e i I ˇ y , r := ( r , . . . , r n ) . Now, recall (5) and then x + · · · + x n = 2( e x + · · · + e x n )1 + e x + · · · + e x n = 2 −
21 + e x + · · · + e x n , so we have Ψ a = (cid:0) e x + · · · + e x n (cid:1) a = (cid:18) − x − x − · · · − x n (cid:19) a , and r i = e x i = (cid:18) x i e x + · · · + e x n (cid:19) = (cid:18) x i − x − x − · · · − x n (cid:19) . Then, ψ = I ( ˜ ψ ) turns out to be X I ∈ Z n ˜ ψ I ( r ) (cid:18) − x − x − · · · − x n (cid:19) b − a e i I ˇ y = X I ∈ Z n ψ ( x ) e i I ˇ y , so Fourier-componentwisely we have the transformation ψ I ( x ) = ˜ ψ I ( r ( x )) (cid:18) − x − x − · · · − x n (cid:19) b − a . We can bring the generators (14) of H ( DG ( C P n )( O ( a ) , O ( a + 1))) over C to those of H ( V ′ )( O ( a ) , O ( a + 1)), which are given by(15) "r − x − x − · · · − x n and(16) "r x e i y , "r x e i y , . . . , "r x n e i y n . The above bases (15) and (16) generate the whole space H ( V ′ )( O ( a ) , O ( b )) as prod-ucts of these functions. Explicitly, the bases e ab ; I , I = ( i , . . . , i n ), of the vector space H ( V ′ )( O ( a ) , O ( b )) are(17) e ab ; I = c ab ; I · r − x − x − · · · − x n ! b − a −| I | r x e i y ! i · · · r x n e i y n ! i n , where i ≥ , . . . , i n ≥ | I | := i + · · · + i n ≤ b − a , and we attach c ab ; I so thatmax x ∈ P | e ab ; I ( x ) | = 1. Note that this is valid for a = b , too, where we only have I =(0 , . . . ,
0) =: 0 and e aa ;0 is the identity element; e aa ;0 ( x ) = 1 for any x ∈ P .By direct calculations, we have the following lemma. Lemma 5.1.
For a fixed a < b and e ab ; I ∈ H ( V ′ )( O ( a ) , O ( b )) , the set { x ∈ P || e ab ; I ( x ) | = 1 } consists of a point v ab ; I := (cid:18) i b − a , . . . , i n b − a (cid:19) , which is the intersection V ab ; I ⊂ π ( L a ∩ L b ) with label I . This correspondence then givesa quasi-isomorphism ι : Mo ( P )( L a , L b ) → V ′ ( O ( a ) , O ( b )) of cochain complexes. (cid:3) MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 23
For each a < b and I , we later employ a function f ab ; I on P defined uniquely by(18) n X j =1 ( s ja − s jb + i j ) dx j = df ab ; I , f ab ; I ( v ab ; I ) = 0 . Remark 5.2.
In the original set-up where B is compact, this f ab ; I is a Morse function,where v ab ; I is the critical point of degree zero. However, now in our case, v ab ; I may be atthe boundary ∂ ( P ). Even if we extend P to ˜ B naturally, v ab ; I ∈ ∂ ( P ) may not a criticalpoint since the symplectic form on M diverges at the boundary.For each a , the space Mo ( P )( L a , L a ) is generated by P . The two conditionsmax x ∈ P | e aa ;0 ( x ) | = 1 , { x ∈ P | | e aa ;0 ( x ) | = 1 } = P are clearly satisfied. We define a quasi-isomorphism ι : Mo ( P )( L a , L a ) → V ′ ( O ( a ) , O ( a ))by ι ( P ) = e aa ;0 .For a > b , both the space M o ( P )( L a , L b ) and the cohomology H ( V ′ ( O ( a ) , O ( b ))) aretrivial. Thus, the zero map ι : Mo ( P )( L a , L b ) → V ′ ( O ( a ) , O ( b )) is a quasi-isomorphism.Now, let us fix q ∈ Z and consider E := ( O ( q ) , O ( q + 1) , . . . , O ( q + n )). We denotethe corresponding full subcategories by DG E ( C P n ) ⊂ DG ( C P n ), V ′E ⊂ V ′ and Mo E ( P ) ⊂ Mo ( P ). It is known that E (with any q ) forms a full strongly exceptional collection in Tr ( DG E ( C P n )) ≃ D b ( coh ( C P n )) [3]. Recall that an A ∞ -equivalence is an A ∞ -functorwhich induces a category equivalence on the corresponding cohomology categories. Theorem 5.3.
For each q ∈ Z , the quasi-isomorphisms ι : Mo ( P )( L a , L b ) → V ′ ( O ( a ) , O ( b )) with a, b ∈ { q, ..., q + n } extend to a linear A ∞ -equivalence ι : Mo E ( P ) ∼ → V ′E . Now, we have the DG isomorphism DG E ( C P n ) ≃ I ( DG E ( C P n )) = V ′E . Since a DGfunctor is a linear A ∞ -functor, we immediately obtain the following. Corollary 5.4.
One has a linear A ∞ -equivalenceMo E ( P ) → DG E ( C P n ) . (cid:3) Corollary 5.5.
One has an equivalence of triangulated categoriesTr ( Mo E ( P )) ≃ D b ( coh ( C P n )) . (cid:3) In the rest of this subsection, we show Theorem 5.3 by computing the structure of Mo E ( P ). We already see that nontrivial morphisms in Mo ( P ) are of degree zero only.This implies, by degree counting, that the higher A ∞ -products of Mo ( P ) are trivial.Thus, what remains to show the theorem is to construct the product m and show thecompatibility of the products with respect to ι . Lemma 5.6.
For a < b < c and bases V ab ; I ab ∈ Mo ( P )( L a , L b ) , V bc ; I bc ∈ Mo ( P )( L b , L c ) ,we have (19) ι m ( V ab ; I ab , V bc ; I bc ) = e ab ; I ab · e bc ; I bc . proof. Recall that each base consists of a point; V ab ; I ab = { v ab ; I ab } and so on. We takethe function f ab ; I ab defined by (5.2). Since its gradient vector field is of the form − grad( f ab ; I ab ) = ( b − a )2 (cid:18) ( x − i ab ;1 ) ∂∂x + · · · + ( x n − i ab ; n ) ∂∂x n (cid:19) , its gradient trajectories starting from v ab ; I ab go straight. Similarly, gradient trajectoriesof − grad( f b − f c ) starting from v bc ; I bc go straight. On the other hand, the only gradienttrajectory of − grad( f a − f c ) ending at v ac ; I ac , I ac := I ab + I bc is the one staying at v ac ; I ac since it is of degree zero. This means that these three gradient trajectories should meetat v ac ; I ac . Thus we obtained the gradient tree γ defining the product m ( V ab ; I ab , V bc ; I bc )explicitly. (The result is that v ac ; I ac sits on the straight line segment v ab ; I ab v bc ; I bc in allcases. ) Now, A ( γ ) turns out to be A ( γ ) = f ab ; I ab ( v ac ; I ac ) + f bc ; I bc ( v ac ; I ac ) . Here, f ab ; I ab ( v ac ; I ac ) is the symplectic area of the triangle disk whose edges belong to s a ( γ ( T )), s b ( γ ( T )) and π − ( v ac ; I ac ). Similarly, f bc ; I bc ( v ac ; I ac ) is the symplectic area of thecorresponding triangle disk. We thus obtain the weight + e − A ( γ ) .Next, we look at the product in V ′ side. We can express the bases as e ab ; I ab = e f ab ; Iab · e i I ab ˇ y and e bc ; I bc = e f bc ; Ibc · e i I bc ˇ y . We have e ab ; I ab · e bc ; I bc = e f ab ; Iab + f bc ; Ibc · e i I ac ˇ y . The right hand side is guaranteed to be proportional to e ac ; I ac , whose absolute value takesthe maximal value at v ac ; I ac . Namely, we have e ab ; I ab · e bc ; I bc = e f ab ; Iab ( v ac ; Iac )+ f bc ; Ibc ( v ac ; Iac ) · e ac ; I ac . This shows that the compatibility (19) holds true. (cid:3)
We need to show the compatibility (19) for any a ≤ b ≤ c . If a = b , then V ab ; I ab = P .IF b = c , then V bc ; I bc = P . Now, we see that Mo E ( P ) satisfies the strong minimality as-sumption in subsection 4.5, which implies that P forms the identity morphism in Mo E ( P ). MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 25
Since we already know that ι ( P ) is the identify morphism in V ′E , the compatibility (19)follows and the proof of Theorem 5.3 is competed. (cid:3) Remark 5.7.
The product m and the linear A ∞ -equivalence ι can be induced by ap-plying homological perturbation theory to DG ( C P n ) in a suitable way. As the higher A ∞ -products of Mo ( P ) are trivial, the induced A ∞ -equivalence turns out to be linear bydegree counting since nontrivial cohomologies of morphisms are of degree zero only.As a biproduct of the proof, we see that Mo E ( P ) has the following properties. Proposition 5.8.
For any L a , L b ∈ Mo E ( P ) such that L a = L b , V ab = π ( L a ∩ L b ) belongsto the boundary ∂ ( P ) .For given bases V ab ∈ Mo E ( L a , L b ) and V bc ∈ Mo E ( L b , L c ) , the image γ ( T ) by anygradient tree γ ∈ GT ( V ab , V bc ; V ac ) belongs to the boundary ∂ ( P ) unless L a = L b = L c . (cid:3) Remark 5.9. If L a = L b = L c , then V ab = P , V bc = P and V ac = P . Then γ ∈GT ( V ab , V bc ; V ac ) is a constant map to a point in P . If L a = L b = L c , then V ab = P and V bc = { v bc } = V ac . Then γ ∈ GT ( V ab , V bc ; V ac ) is the constant map to the point v bc ∈ ∂ ( P ).Similarly, if L a = L b = L c , then γ ∈ GT ( V ab , V bc ; V ac ) is the constant map to the point v ab ∈ ∂ ( P ).We expect that for many other toric Fano manifold X and (strongly) exceptional col-lection E , Mo E ( P ) may satisfy these properties.We also believe that there exists an A ∞ -equivalence Mo ( P ) → DG ( C P n )between the whole categories. However, it is not easy to show directly that the wholecategory M o ( P ) is well-defined as an A ∞ -category since there are infinitely many gradienttrees for which we should check whether our assumption hold or not. In particular, if Mo ( P ) is well-defined, it should have nontrivial higher A ∞ -products. We would comeback to see this problem elsewhere.5.3. Homological mirror symmetry of C P m × C P n . In this subsection we shall seehow the framework presented in the last subsections works for the case of the product ofprojective spaces. The point here is that we need not only transversal but clean intersec-tions of Lagrangians in the symplectic side. That’s why we included clean intersections inthe definition of Mo ( P ) in subsection 4.5. We still do not need higher products m , m ,...because we can pick up full strongly exceptional collections on both sides (see remark atthe end of subsection 4.5). Let X = C P m × C P n , and ˇ M be the complement of the toric divisors. For X p | | ②②②②②②②② p " " ❊❊❊❊❊❊❊❊ C P m C P n we denote O ( a, b ) := p ∗ O ( a ) ⊗ p ∗ O ( b ). Then E := {O ( a, b ) } a =0 , ,...,m, b =0 , ,...,n with thelexicographic order forms a strongly exceptional collection. According to Orlov [17] thesemi orthogonal ordered set of admissible subcategories ( D , ..., D n ) where D is the imageof D b ( coh ( C P m )) in D b ( coh ( C P m × C P n )) under the pull-back functor p ∗ and its twistsalong C P n generates D b ( coh ( C P m × C P n )), which means that the collection of O ( a, b )’sabove is full. We consider the DG category DG ( X ) of these line bundles, and the cor-responding DG-category V ( ˇ M ). Just in a similar way as in the previous subsection, wehave a DG subcategory I ( DG ( X )) = V ′ ⊂ V = V ( ˇ M ) so that DG ( X ) ≃ V ′ . Their fullsubcategories consisting of E are denoted DG E ( X ) and V ′E .Then, the parallel statements to the case X = C P n holds. Theorem 5.10.
There exists a linear A ∞ -equivalence ι : Mo E ( P ) ∼ → V ′E such that for each generator V ∈ Mo E ( P )( L, L ′ ) we have max x ∈ P | ι ( V )( x ) | = 1 and V = { x ∈ P | | ι ( V )( x ) | = 1 } . Corollary 5.11.
We have a linear A ∞ -equivalenceMo E ( P ) ≃ DG E ( C P m × C P n ) . (cid:3) Corollary 5.12.
We have an equivalence of triangulated categoriesTr ( Mo E ( P )) ≃ D b ( coh ( C P m × C P n )) . (cid:3) Proposition 5.13. If L = L ′ , any generator V ∈ Mo E ( P )( L, L ′ ) belongs to the boundary ∂ ( P ) .For bases V ∈ Mo E ( L, L ′ ) and V ′ ∈ Mo E ( L ′ , L ′′ ) such that L = L ′ and L ′ = L ′′ , anygradient tree γ ∈ GT ( V, V ′ ; V ′′ ) with V ′′ ∈ Mo E ( P )( L, L ′′ ) belongs to the boundary ∂ ( P ) .proof of Theorem 5.10 The bases of the space H ( V ′ )( O ( a , a ) , O ( b , b )) are e a b ; I ⊗ e a b ; J MS OF C P N AND THEIR PRODUCTS VIA MORSE HOMOTOPY 27 where e a b ; I and e a b ; J are the bases of the corresponding zero-cohomology spaces ofmorphisms defined in (17) for C P m and C P n , respectively, so I = ( i , . . . , i m ) and J =( j , . . . , j n ) run over i ≥ , . . . , i m ≥ , | I | ≤ b − a ,j ≥ , . . . , j n ≥ , | J | ≤ b − a . Each base satisfies max x ∈ P | ( e a b ; I ⊗ e a b ; J )( x ) | = 1. Let us denote by L ( a ,a ) ∈ Mo ( P )the object corresponding to O ( a , a ). The base corresponding to e a b ; I ⊗ e a b ; J is then { x ∈ P | | ( e a b ; I ⊗ e a b ; J )( x ) | = 1 } = V a b ; I × V a b ; J ∈ Mo ( P )( L ( a ,a ) , L ( b ,b ) ) . It consists of the point ( v a b ; I , v a b ; J ) if a < b and a < b . Otherwise, we have V a b ; I × V a b ; J = P × { v a b ; J } for a = b and a < b , V a b ; I × V a b ; J = { v a b ; I } × P for a < b and a = b , and V a b ; I × V a b ; J = P × P = P for a = b and a = b , where P and P are the dual polytope of C P m and C P n ,respectively.For V a b ; I × V a b ; J ∈ Mo ( P )( L ( a ,a ) , L ( b ,b ) ) and V b c ; K × V b c ; L ∈ Mo ( P )( L ( b ,b ) , L ( c ,c ) ),the equation ιm ( V a b ; I × V a b ; J , V b c ; K × V b c ; L ) = ( e a b ; I ⊗ e a b ; J ) · ( e b c ; K ⊗ e b c ; L )follows immediately from looking at the structure of the gradient tree γ defining theproduct m ( V a b ; I × V a b ; J , V b c ; K × V b c ; L ). Actually, let us denote by γ the gradienttree obtained as the composition of γ with the projection P → P . We see that γ is thegradient tree defining the product m ( V a b ; I , V b c ; J ). Similarly, we consider γ . Then, wehave A ( γ ) = A ( γ ) + A ( γ ), and we see that this is compatible with the product( e a b ; I ⊗ e a b ; J ) · ( e b c ; K ⊗ e b c ; L ) = ( e a b ; I · e b c ; K ) ⊗ ( e a b ; J · e b c ; L )= e − A ( γ ) e a c ; I + K ⊗ e − A ( γ ) e a c ; J + L . This completes the proof pf Theorem 5.10. (cid:3)
Proof of Proposition 5.13
Each γ is obtained from the pair ( γ , γ ) in the proof ofTheorem 5.10. In particular, the image γ ( T ) of a trivalent tree T by γ is obtainedas ( γ , γ )( T ) ⊂ P × P . By Proposition 5.8, γ ( T ) ⊂ ∂ ( P ) if V a b = P or V b c = P ,i.e., if a , b , c do not satisfy a = b = c . Similarly, γ ( T ) ⊂ ∂ ( P ) unless a = b = c .On the other hand, when at least either L ( a ,a ) = L ( b ,b ) or L ( b ,b ) = L ( c ,c ) is satisfied,at least one of the inequalities a < b , b < c , a < b , b < c is satisfied. Thus, at least neither a = b = c nor a = b = c is satisfied. This means that at least either γ ( T ) ⊂ ∂ ( P ) or γ ( T ) ⊂ ∂ ( P ) holds, which implies that γ ( T ) ⊂ ∂ ( P ). (cid:3) Lastly, we show more explicitly the gradient trees γ defining the products m ( V a b ; I × V a b ; J , V b c ; K × V b c ; L ) . A different point from the previous subsection is that V a b ; I × V a b ; J , V b c ; K × V b c ; L and V a c ; I + K × V a c ; J + L may not be points even if V a b ; I × V a b ; J = P and V b c ; K × V b c ; L = P .There are 4 × a < b < c , a < b < c .(1) only one of the equations a = b , b = c , a = b , b = c is satisfied.(2) a = b < c and a = b < c , or a < b = c and a < b = c .(2’) a = b < c and a < b = c , or a < b = c and a = b < c .(2”) a = b = c and a < b < c , or a < b < c and a = b = c .(3) three of the equations a = b , b = c , a = b , b = c are satisfied and theremaining one is the inequality.(4) a = b = c and a = b = c .The case (4) corresponds to P · P = P in Mo ( P ). The cases (3) and (2) correspond to theproducts P · V = V or V · P = P for some V . The case (2”) corresponds to the product( P × V ) · ( P × W ) = P × ( V · W ) or ( V × P ) · ( W × P ) = ( V · W ) × P , where V · W isa product in Mo ( P ) or Mo ( P ). Thus, the argument reduces to the one in the previoussubsection. In the case (0), all V a b ; I × V a b ; J , V b c ; K × V b c ; L and V a c ; I + K × V a c ; J + K consist of points. Thus, the situation is just the product of the ones in Lemma 5.6.Now we discuss more carefully the cases (1) and (2’). In the case (1), if a = b , then V a b ; I × V a b ; J = P × { v a b ; J } which is not a point. In this case, a gradient tree γ ∈ GT ( P × { v a b ; J } , { v a c ; K } ×{ v b c ; L } ; { v a c ; K }×{ v a c ; J + L } ) belongs to GT (( v a c ; K , v a b ; J ) , ( v a c ; K , v b c ; L ); ( v a c ; K , v a c ; J + L ))and the image γ ( T ) is a straight segment connecting ( v a c ; K , v a b ; J ) and ( v a c ; K , v b c ; L )on which ( v a c ; K , v a c ; J + L ) sits. Similarly, in the case (2’), if a = b and b = c , thenwe have V a b ; I × V a b ; J = P × { v a b ; J } , V b c ; K × V b c ; L = { v b c ; K } × P . A gradient tree γ ∈ GT ( P × { v a b ; J } , { v b c ; K } × P ; { v b c ; K } × { v a b ; J } ) belongs to GT (( v b c ; K , v a b ; J ) , ( v b c ; K , v a b ; J ); ( v b c ; K , v a b ; J )), and then the image γ ( T ) is just thepoint ( v b c ; K , v a b ; J ) which is the intersection P × { v a b ; J } ∩ { v b c ; K } × P . References [1] M. Abouzaid. Morse homology, tropical geometry, and homological mirror symmetry for toric vari-eties.
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