STUDIES OF CLOSED/OPEN MIRROR SYMMETRY FOR QUINTIC THREE-FOLDS - PDF document

STUDIES OF CLOSED/OPEN MIRROR SYMMETRY FOR QUINTIC THREE-FOLDS THROUGH LOG MIXED HODGE THEORY

0. Introduction 1. Log mixed Hodge theory 1.1. Category B (log) 1.2. Ringed space ( S log , O log S ) 1.3. Toric variety 1.4. Graded polarized LMH 1.5. Nilpotent orbit and period map 1.6. Moduli of LMH of specified type 3. Quintic threefolds 3.2. Quintic threefold and its mirror 3.3. Picard-Fuchs equation on B-model of mirror V ◦ 3.4. A-model of quintic V 3.5. Z -structure 3.6. Correspondence table 3.9. Proof of (4) in Introduction 4. Proof of (6) in Introduction 2

0. Introduction Fundamental Diagram For classifying space D of MHS of specified type, D SL(2) , val ֒ → D BS , val     � � − D ♯ D Σ , val ← − − − Σ , val − − − − → D SL(2) D BS     � � D ♯ D Σ ← − − − − Σ Hope to understand Hodge theoretic aspect of MS by this. 3

Mirror symmetry for quintic 3-folds Mirror symmetry for A-model of quintic 3-fold V and B-model of its mirror V ◦ was predicted in [CDGP91], and proved in following (1)–(3), which are equivalent. Every statement is near large radius point q 0 of complexified K¨ ahler moduli KM ( V ) and maximally unipotent monodromy point p 0 of complex moduli M ( V ◦ ). t := y 1 /y 0 , u := t/ 2 πi and q := e t = e 2 πiu from 3.3 below and respective ones in 3.4 below. 4

Φ V GW ( t ) = Φ V ◦ (1) ( Potential. [LLuY97]) GM ( t ). (2) ( Solutions. [Gi96], [Gi97p]) H 2 ) H 3 1 + tH + d Φ td Φ ( ( ) J V := 5 H 5 + dt − 2Φ dt 5 I V := 5 H ( y 0 + y 1 H + y 2 H 2 + y 3 H 3 ) Then, y 0 J V = I V . (3) ( Variation of Hodge structure. [Morrison97]) ∼ ← ( p 0 ∈ M ( V ◦ )) by canonical coordinate ( q 0 ∈ KM ( V )) ∼ ← M ( V ◦ ) to q = exp(2 πiu ), lifts over the punctured KM ( V ) ( H V , S, ∇ middle , H V ∼ Z , F ; ˜ ← ( H V ◦ , Q, ∇ GM , H V ◦ Z , F ; 1 , [pt]) Ω , g 0 ) . 5

Our (4) below is equivalent to (1)–(3). (4) ( Log period map ) σ : monodromy cone transformed by a level structure into End of reference fiber of local system for A- and B- models. Then, we have diagram of horizontal log period maps ∼ ← ( p 0 ∈ M ( V ◦ )) ( q 0 ∈ KM ( V )) ↘ ↙ ([ σ, exp( σ C ) F 0 ] ∈ Γ( σ ) gp \ D σ ) with extensions of specified sections in (3), where ( σ, exp( σ C ) F 0 ) is nilpotent orbit and Γ( σ ) gp \ D σ is fine moduli of LH of specified type. 6

Open mirror symmetry for quintic 3-folds (5) ( Inhomogenous solutions , [Walcher07], [PSW08p], [MW09]) L : Picard-Fuchs differential operator for quintic mirror. T A = u ( 1 1 n d q d/ 2 ) ∑ 2 ± 4 + . 2 π 2 d odd ∫ C + T B = Ω , { C ± , line } = { x 1 + x 2 = x 3 + x 4 = 0 } ∩ X ψ . C − √ z ) 15 ( z = 1 L ( y 0 ( z ) T A ( z )) = L ( T B ( z ))(= ψ 5 ) . 16 π 2 7

In a neighborhood of MUM point p 0 , we have the following (6). (6) ( Computations of admissible normal function and domainwall tension on MUM point ) H Q := H V ◦ Q , T := T B L Q : translation of local system Q ⊕ H Q by T e 0 in E xt 1 ( Q , H Q ) eron model for admissible normal function over T e 0 , whose J L Q : N´ weak fan is constructed in [KNU13p, N´ eron models for admissible normal functions] transl pol H O / ( F 2 + H Q ) S := ( z 1 / 2 -plane) − ≅ ( F 2 ) ∗ / H Q − − − → J L Q ≅   � H O / ( F 1 + H Q ) ≅ ( F 3 ) ∗ / H Q ¯ J L Q ≅ 8

To state following assertions, we use e 0 , e 1 which are part of basis of H O respecting Deligne decomposition at p 0 (see 6 (2B)). (6.1) T e 0 as truncated normal function S → ¯ J 1 ,L Q . (6.2) Truncated normal function in (6.1) uniquely lifts to admissible normal function S → J 1 ,L Q . (6.3) Followings are mirror: 0 → H 4 ( V, Z ) → H 4 ( V − Lg ) → H 2 ( Lg ) → 0 2 ) → 1 0 → Z e 1 (gr M 2 Z e 1 (gr M 2 ) → (2-torsion) → 0 Here Lg is real Lagrangian, and M = M ( N, W ) around MUM point p 0 . (6.4) (5) tells that inverse of admissible normal function in (6.2) from its image is given by 16 π 2 / 15 times L applying to extension of L Q . 9

1. Log mixed Hodge theory 1.1. Category B (log) S : subset of analytic space Z . Strong topology of S in Z is strongest one among topologies on S s.t. for ∀ analytic space A and ∀ morphism f : A → Z with f ( A ) ⊂ S , f : A → S is continuous. Log structure on local ringed space S is sheaf of monoids M ∼ on S and homomorphisim α : M → O S s.t. α − 1 O × → O × S . S fs means finitely generated, integral and saturated. 10

Analytic space is call log smooth if, locally, it is isomorphic to open set of toric variety. Log manifold is log local ringed space over C which has open covering ( U λ ) λ satisfying: For each λ , there exist log smooth fs log analytic space Z λ , finite subset I λ of global log differential 1-forms Γ( Z λ , ω 1 Z λ ), and isomorphism of log local ringed spaces over C between U λ and open subset in strong topology of S λ := { z ∈ Z λ | image of I λ in stalk ω 1 z is zero } in Z λ . 11

1.2. Ringed space ( S log , O log S ) S ∈ B (log). S log := { ( s, h ) | s ∈ S , h : M gp → S 1 hom. s.t. h ( u ) = u/ | u | ( u ∈ O × S,s ) } s endowed with weakest topology s.t. followings are continuous. (1) τ : S log → S, ( s, h ) �→ s . (2) For ∀ open U ⊂ S and ∀ f ∈ Γ( U, M gp ), τ − 1 ( U ) → S 1 , ( s, h ) �→ h ( f s ). τ is proper, surjective with τ − 1 ( s ) = ( S 1 ) r ( s ) , r ( s ) := rank( M gp / O × S ) s varies with s ∈ S . Define L on S log as fiber product: exp → τ − 1 ( M gp ) L − − − − ∋ ( f at ( s, h ))       � � � exp → Cont( ∗ , S 1 ) Cont( ∗ , i R ) − − − − ∋ h ( f ) 12

ι : τ − 1 ( O S ) → L is induced from exp → τ − 1 ( O × τ − 1 ( O S ) S ) ⊂ τ − 1 ( M gp ) f ∈ − − − −       � � � exp ( f − ¯ Cont( ∗ , S 1 ) f ) / 2 ∈ Cont( ∗ , i R ) − − − − → Define τ − 1 ( O S ) ⊗ Sym Z ( L ) O log := ( f ⊗ 1 − 1 ⊗ ι ( f ) | f ∈ τ − 1 ( O S )) . S Thus τ : ( S log , O log S ) → ( S, O S ) as ringed spaces over C . For s ∈ S and t ∈ τ − 1 ( s ) ⊂ S log , let t j ∈ L t (1 ≤ j ≤ r ( s )) s.t. images in ( M gp / O × S ) s of exp( t j ) form a basis. Then, O log S,t = O S,s [ t j (1 ≤ j ≤ r ( s )] is polynomial ring. 13

1.3. Toric variety σ : nilpotent cone in g R , i.e., sharp cone generated by finite number of mutually commutative nilpotent elements. Γ : subgroup of G Z , and Γ( σ ) := Γ ∩ exp( σ ). Assume σ is generated over R ≥ 0 by log Γ( σ ). P ( σ ) := Γ( σ ) ∨ = Hom(Γ( σ ) , N ). toric σ := Hom( P ( σ ) , C mult ) ⊃ torus σ := Hom( P ( σ ) gp , C × ), 0 → Z → C → C × → 1 induces e 0 → Hom( P ( σ ) gp , Z ) → Hom( P ( σ ) gp , C ) → Hom( P ( σ ) gp , C × ) → 1, − where e ( z ⊗ log γ ) := e 2 πiz ⊗ γ ( z ∈ C , γ ∈ Γ( σ ) gp = Hom( P ( σ ) gp , Z )). ρ ≺ σ induces surjection P ( ρ ) ← P ( σ ) hence open toric ρ ֒ → toric σ . 0 ρ ∈ toric ρ is P ( ρ ) → C mult ; 1 �→ 1, other elements of P ( ρ ) �→ 0. 0 ρ ∈ toric ρ ⊂ toric σ by above open immersion. Then, as set, toric σ = { e ( z )0 ρ | ρ ≺ σ, z ∈ σ C / ( ρ C + log Γ( σ ) gp ) } . 14

For S := toric σ , polar coordinate R ≥ 0 × S 1 → R ≥ 0 · S 1 = C induces τ : S log = Hom( P ( σ ) , R mult ≥ 0 ) × Hom( P ( σ ) , S 1 ) = { ( e ( iy )0 ρ , e ( x )) | ρ ≺ σ, x ∈ σ R / ( ρ R + log Γ( σ ) gp ) , y ∈ σ R /ρ R } → S = Hom( P ( σ ) , C mult ) , τ ( e ( iy )0 ρ , e ( x )) = e ( x + iy )0 ρ . By 0 → ρ R / log Γ( ρ ) gp → σ R / log Γ( σ ) gp → σ R / ( ρ R + log Γ( σ ) gp ) → 0, τ − 1 ( e ( a + ib )0 ρ ) = { ( e ( ib )0 ρ , e ( a + x )) | x ∈ ρ R / log Γ( ρ ) gp } ≅ ( S 1 ) r , as set, where r := rank ρ varies with ρ ≺ σ . H σ = ( H σ, Z , W, ( 〈 , 〉 w ) w ) : canonical local system on S log by representation π 1 ( S log ) = Γ( σ ) gp ⊂ G Z = Aut( H 0 , W, ( 〈 , 〉 w ) w ). 15

1.4. Graded polarized LMH S ∈ B (log). Pre-graded polarized log mixed Hodge structure on S is H = ( H Z , W, ( 〈 , 〉 w ) w , H O ) consisting of H Z : local system of Z -free modules of finite rank on S log , W : increasing filtration W of H Q := Q ⊗ H Z , 〈 , 〉 w : nondegenerate ( − 1) w -symmetric Q -bilinear form on gr W w , H O : locally free O S -module on S satisfying: ∃ O log ⊗ Z H Z ≅ O log ⊗ O S H O ( log Riemann-Hilbert correspondence ), S S ∃ FH O : decreasing filt. of H O s.t. F p H O , H O /F p H O locally free. Put F p := O log ⊗ O S F p H O . Then τ ∗ F p = F p H O . S 〈 F p (gr W w ) , F q (gr W w ) 〉 w = 0 ( p + q > w ). 16