Twistor property of GKZ-hypergeometric systems

1.6 MB·English

by  Takuro Mochizuki

Twistor property of GKZ-hypergeometric systems
Takuro Mochizuki
5
Abstract
1
0 Westudythemixedtwistor D-modules associated tomeromorphic functions. Inparticular, we describe
2
their push-forward and specialization under some situations. We apply the results to study the twistor

c propertyofatypeofbetterbehavedGKZ-hypergeometricsystems,andtostudytheirspecializations. Asa
e result, we obtain some isomorphisms of mixed TEP-structures in thelocal mirror symmetry.
D
Keywords: mixed twistor D-module, generalized Hodge structure, polarization, GKZ-hypergeometric system, local

1 mirrorsymmetry.
2 MSC2010: 14F1032C3832S35

]
G
1 Introduction
A
. 1.1 Mixed twistor -modules
h
D
t
a In [59], C. Simpson introduced the concept of mixed twistor structure as a generalization of mixed Hodge
m
structure. According to his principle, called Simpson’s meta theorem, most objects and theorems concerning
[ with mixed Hodge structure should have their counterparts in the context of mixed twistor structure.

The concept of mixed Hodge module due to M. Saito [53, 57] is one of the ultimate in the Hodge theory.
2
Roughly speaking, mixed Hodge modules are regular holonomic -modules equipped with mixed Hodge struc-
v
D
6 ture. One of the main results in the theory is the functoriality of mixed Hodge modules. Namely, we have
4 the standard operations on the derived category of algebraic mixed Hodge modules such as push-forward, pull
1 back, duality, inner homomorphism, tensor product, nearby cycle functor, vanishing cycle functors, which are
4
compatible with the standard operations for algebraic regular holonomic -modules.
0 D
According to Simpson’s meta theorem, we should have a twistor version of mixed Hodge modules, that is
.
1
the concept of mixed twistor -module. The concept of pure twistor -module was introduced by C. Sabbah
0 D D
[47, 50], and studied by himself and the author [35, 37]. The mixed case was studied in [40].
5
1 As in the Hodge case, we have the standard functors on the derived category of algebraic mixed twistor
: -modules,thatisoneofthe mostusefulpoints inthe theoryofmixedtwistor -modules. Anotherinteresting
v D D
point is that we have the mixed twistor -modules associated to meromorphic functions.
Xi Let f be a meromorphic function onDa complex manifold X whose poles are contained in a hypersurface H.
r Let L (f,H) be the holonomic -modules given by ( H) with the flat connection d+df. We also have the
∗ X
a D O ∗
holonomic -module L(f,H) := D(L ( f,H)), where D denotes the duality of holonomic -modules. We
! ∗
D − D
have the natural mixed twistor -modules over L (f,H) (⋆ = ,!). Applying the standard functors to such
⋆
D ∗
mixed twistor -modules, we can obtain many mixed twistor -modules. In other words, many important
D D
holonomic -modules are equipped with a natural mixed twistor structure. It would be interesting to have
D
applications of the mixed twistor structure.
In this paper, we shall explain that a type of better behaved GKZ-hypergeometric systems are naturally
equipped with the mixed twistor structure. We shall also explain an application in the study of toric local
mirror symmetry.
1.1.1 Ingredients for mixed twistor -modules
D
Originally,ingredientsfortwistor -modulesaregivenas -triplesbySabbah[47]. (See 3.1forareview.) But,
D R §
inthispaper,weconsideronlyintegrablemixedtwistor -moduleswithrealstructure. So,inthisintroduction,
D
we explain it in a slightly different way, although we use the formalism of -triples after 2.
R §
e
1
Let X be any complex manifold. Let := C X and 0 := 0 X. Let λ be the standard coordinate
X × X { }×
of C. We have the sheaf of differential operators on . Let denote the sheaf of subalgebras
X X X
D X R ⊂ D
generated by λΘ (log 0).
X
X
An -module is called strict if it is flat over . Any coheerent -module has the characteristic
X C X
R M O R M
varietyCh( )inT∗ (log 0)asinthecaseof -modules. Wesaythat isholonomicifCh( )iscontained
M X X D M M
in C Λe for a Lagrangian subvariety Λ T∗X. A real structure of striect holonomic -module is a R-
X
× ⊂ R M
perverse sheaf P on C∗ X with an isomorphism P C DR ( ).
R R C∗×X |C∗×X
× ⊗ ≃ M
Then, an integrable mixed twistor -module with real structure on X is a strict holoenomic -module
X
D R M
with a real structure and a weight filtration ( ,P ,W) satisfying some conditions. Here, W is an increasing
R
M
filtration of strict holonomic -modules with real structure. e
X
R
In this introduction,“mixed twistor -module” means “integrablemixed twistor -module with realstruc-
D D
ture”. e
1.2 Better behaved GKZ-hypergeometric systems
1.2.1 -modules
D
L. Borisov and P. Horja [5] introduced the concept of better behaved GKZ-hypergeometric system as a gener-
alization of GKZ-hypergeometric systems [15]. (See also [22].) Let us recall a special type of better behaved
GKZ-hypergeometric systems.
Let = a ,...,a Zn beafinitesubsetgeneratingZn. WehavetheconeK ( ):= m r a r
A { 1 m}⊂ R A j=1 j j j ≥
0 Rn generated by . We set K( ) := K ( ) Zn. The semigroup K( ) is the saturation of Z =
⊂ A A R A ∩ A (cid:8)P ≥(cid:12)0A
m n a n Z . We denote a =(a ,...,a ). (cid:12)
(cid:9) j=1 j j j ∈ ≥0 j j1 jn
Let Γ K( ) be any subset such that Γ+a Γ for any a . Let β Cn. The following system of
(cid:8)P ⊂ (cid:12) A (cid:9) ⊂ ∈ A ∈
differential eq(cid:12)uations GKZ( ,Γ,β) for tuples of holomorphic functions (Φc c Γ) on any open subset in Cm
A | ∈
is called the better behaved GKZ-hypergeometric system associated to ( ,Γ,β):
A
∂ Φ =Φ ( c Γ, j =1,...,m)
xj c c+aj ∀ ∈ ∀
a x ∂ +c β Φ =0 ( c Γ, i=1,...,n)
ji j xj i− i c ∀ ∈ ∀
(cid:16)Xj (cid:17)
Here, (x ,...,x ) denotes the standard coordinate system of Cm.
1 m
Remark 1.1 In [5], any tuple in a finitely generated abelian group is considered instead of a finite subset in
Zn. In that sense, the above system is a special case of better behaved GKZ-hypergeometric systems. Later,
moreover, we shall mainly consider the case β =0.
But, as in [1], we omit the existence of an element α (Zn)∨ such that α(a )=1 for any i.
i
∈
Expression as the twisted Gauss-Manin systems We can describe the corresponding -modules using
D
the twisted Gauss-Manin system. Let Tn := (C∗)n. We consider the morphism ψaff : Tn Cm given
by ψAaff(t1,...,tn) = (ta1,...,tam), where taj = ni=1taiji. Let XAaff denote the closuAre of the−im→age of ψAaff.
Let Xˇaff Xaff be the normalization. Let Dˇaff denote the complement of Tn in Xˇaff. Note that Xˇaff =
A −→ A AQ A A
SpecC[K( )] and Xaff =SpecC[Z ].
Let ΩkA (logDˇaffA) denote the s≥h0eAaf of meromorphic differential k-forms which are logarithmic along Dˇaff,
Xˇaff A A
A
studied in [3]. Let Ωk denote the sheaf of holomorphic k-forms on the normal variety Xˇaff which are 0
(Xˇaff,Dˇaff) A
A A
along Dˇaff, studied in [8]. Let q :Xˇaff Cm Xˇaff be the projection. We set
A A × −→ A
Ωk (logDˇaff Cm):=q∗Ωk (logDˇaff), Ωk :=q∗Ωk .
XˇAaff×Cm/Cm A × XˇAaff A (XˇAaff,DˇAaff)×Cm/Cm (XˇAaff,DˇAaff)
The family of Laurent polynomials mj=1xjtaj induces a meromorphic function FA on (XˇAaff,DˇAaff)×Cm.
We also have the logarithmic closed one form κ(β)= n β dt /t . We obtain the relative algebraic de Rham
P i=1 i i i
complexes:
P
•( ,β) := Ω• (logDˇaff Cm),d+dF κ(β)
C A • (XˇAaff×Cm)/Cm A × A−
(cid:16) (cid:17)
2
•( ,β) := Ω• ,d+dF κ(β)
C A ◦ (XˇAaff,DˇAaff)×Cm/Cm A−
(cid:16) (cid:17)
Let πˇ : Xˇaff Cm Cm denote the projection. Each πˇ k( ,β) is naturally a -module by
A A × −→ A∗C A ⋆ DCm
∂ (g)=∂ g+(∂ F )g. The differential d+dF κ(β) of the complexes are compatible with the actions
xj · xj xj A A−
of . So, we obtain the following -modules
Cm Cm
D D
M :=Rnπˇ •( ,β) (⋆= , ).
A,β,⋆ A∗ ⋆
C A • ◦
LetK( )◦betheintersectionofK( )andtheinteriorpartofK ( ). The -modulesM andM
R Cm A,β,• A,β,◦
A A A D
correspondto the systems GKZ( ,K( ),β) and GKZ( ,K( )◦,β) in the sense of the following lemma. (See
A A A A
Lemma E.1 and Proposition E.4 for more details.)
Proposition 1.2 LetU beanyopensubsetinCm. Wehaveanaturalbijectivecorrespondencebetweensolutions
of GKZ( ,K( ),β) (resp. GKZ( ,K( )◦,β)) and -homomorphisms M (resp. M
U A,β,• U A,β,◦
A A A A D −→ O −→
).
U
O
In this paper, we are mainly interested in the case β =0.
Expressions as the push-forward of -modules We take a toricresolutionϕ :X Xˇaff, i.e., X
D Σ1 Σ1 −→ A Σ1
be a smooth toric variety, and ϕ is a projective morphism which is Tn-equivariant. Let D := X Tn.
Σ1 Σ1 Σ1 \
We have the meromorphic algebraic function F on (X ,D ) Cm given by F . We have the algebraic
A,Σ1 Σ1 Σ1 × A
holonomic DXΣ1×Cm-modules L⋆(FA,Σ1,DΣ1 × Cm) (⋆ = ∗,!). Let πΣ1 : XΣ1 × Cm −→ Cm denote the
projection. By the results in 6.1.4 and 6.1.6, we have the following proposition, which gives expressions of
§ §
GKZ( ,K( ),0) and GKZ( ,K( )◦,0) in terms of -modules.
A A A A D
Proposition 1.3 We have natural isomorphisms
π0 L (F ,D Cm) π0 L (F ,D Cm) M ,
Σ1∗ ∗ A,Σ1 Σ1 × ≃ Σ1! ∗ A,Σ1 Σ1 × ≃ A,0,•
π0 L(F ,D Cm) π0 L(F ,D Cm) M .
Σ1∗ ! A,Σ1 Σ1 × ≃ Σ1! ! A,Σ1 Σ1 × ≃ A,0,◦
Here, π0 (⋆ = ,!) denote the 0-th cohomology of the push-forward functors of algebraic -modules with
Σ1,⋆ ∗ D
respect to π .
Σ1
Special case For any p=(p ,...,p ) Zm, we put s (p):= j p 0 and s (p):= j p 0 . We set
1 m + j − j
∈ { | ≥ } { | ≤ }
(cid:3)p = ∂xpjj − ∂x−jpj.
j∈Ys+(p) j∈Ys−(p)
We have the morphism Zm Zn given by p=(p ,...,p ) m p a . Let L denote the kernel.
−→ 1 m 7−→ j=1 j j A
For c K( ), we have the following ordinary GKZ-hypergeometric system GKZord( ,β c ) for holo-
0 ∈ A P A − 0
morphic functions Φ on any open subset of Cm:
c0
(cid:3) Φ =0 ( p L )
p c0 ∀ ∈ A
m
c β + a x ∂ Φ =0 (i=1,...,n)
0i− i ji j xj c0
(cid:16) Xj=1 (cid:17)
For any γ Cn, let I( ,γ) denote the left ideal of generated by (cid:3) (p L ) and γ + m a x ∂
∈ A DCm p ∈ A − i j=1 ji j xj
(i =1,...,n). Then, for Γ =c +Z , we have a natural isomorphism MGKZ( ,Γ,β) /I( ,β c )
0 ≥0A A ≃ DCmP A − 0
which is the -module corresponding to the system GKZord( ,β c ).
0
D A −
Suppose that K( )=Z . Then, we have a natural isomorphism
≥0
A A
M /I( ,0). (1)
A,0,• Cm
≃D A
3
Suppose moreoverK( )◦ =Z +c for an element c K( ). Then, we have a natural isomorphism
≥0 0 0
A A ∈ A
M /I( , c ). (2)
A,0,◦ Cm 0
≃D A −
For an expression c = m b a , we have the following commutative diagram:
0 i=1 i i
P a
M M
A,0,◦ A,0,•
−−−−→
≃ ≃
 b 
DCm I(yA,−c0) −−−−→ DCm Iy(A,0).
Here,theverticalmorphismsare(1)and((cid:14)2),aisthenaturalmorph(cid:14)ism,andbistheinducedbythemultiplication
of ∂bi.
xi
Q
1.2.2 Mixed twistor -modules
D
BythegeometricexpressioninProposition1.3,wenaturallyobtainmixedtwistor -modulesT overM
A,0,⋆ A,0,⋆
D
(⋆= , ). Namely,wehavethealgebraicmixedtwistor -modules (F ,D Cm)(⋆= ,!)onX Cm
• ◦ D T⋆ A,Σ1 Σ1× ∗ Σ1×
whose underlying -modules are L (F ,D Cm). We obtain the algebraic mixed twistor -modules
D ⋆ A,Σ1 Σ1 × D
T =π0 (F ,D Cm) π0 (F ,D Cm)
A,0,• Σ1∗T∗ A,Σ1 Σ1 × ≃ Σ1!T∗ A,Σ1 Σ1 ×
T =π0 (F ,D Cm) π0 (F ,D Cm)
A,0,◦ Σ1∗T! A,Σ1 Σ1 × ≃ Σ1!T! A,Σ1 Σ1 ×
Let denote the underlying -modules of T (⋆ = ,!). Let us describe them as systems of
A,0,⋆ Cm A,0,⋆
M R ∗
differential equations.
Let Γ K( ) be any subset sateisfying Γ+a Γ for any a . We consider the following system of
⊂ A ⊂ ∈ A
differentialequationsGKZRe(A,Γ,β)foratuple ΦΓ =(Φc|c∈Γ)ofholomorphicfunctions onanyopensubset
of C Cm = (λ,x ,...,x ) :
1 m
× { }
λ∂ Φ =Φ , ( c Γ, j =1,...,m)
xj c c+aj ∀ ∈
m
λ2∂ +nλ+ λx ∂ Φ =0, ( c Γ)
λ j xj c ∀ ∈
(cid:16) Xj=1 (cid:17)
m
λ(c β )+ a λx ∂ Φ =0, ( c Γ, i=1,...,n)
i− i ji j xj c ∀ ∈
(cid:16) Xj=1 (cid:17)
Wehavethe -modules GKZ( ,Γ,β)correspondingtothesystemGKZe( ,Γ,β). (See E.2.1,inparticular
R M A R A §
Lemma E.5.) According to Proposition E.6 and Remark E.7, we obtain the following proposition.
e
Proposition 1.4 GKZ( ,K( ),0) (resp. GKZ( ,K( )◦,0)) is naturally isomorphic to (resp.
A,0,•
M A A M A A M
).
A,0,◦
M
Special case For any p Zm, we set
∈
(cid:3)p := (λ∂j)pj (λ∂j)−pj.
−
j∈Ys+(p) i∈Ys−(p)
For any element c0 ∈ K(A), we consider the following system of differential equations GKZoRerd(A,β−c0)
for Φ :
c0
(cid:3) Φ =0 ( p L )
p c0 ∀ ∈ A
m
λ2∂ +nλ+ x λ∂ Φ =0
λ j xj c0
(cid:16) Xj=1 (cid:17)
4
m
λ(c β )+ a x λ∂ Φ =0 (i=1,...,n)
0i− i ji j xj c0
(cid:16) Xj=1 (cid:17)
Foranyγ Zm,let ( ,γ) denote the left idealof generatedby (cid:3) (p L ), λ2∂ +nλ+ m λx ∂ ,
∈ I A RCm p ∈ A λ j=1 j xj
and λγ + m a λx ∂ (i=1,...,n). Then, we have a natural isomorphism
− i j=1 ji j xj P
e
P GKZ( ,Γ,β) Cm/ ( ,β c0)
M A ≃R I A −
which is the -module corresponding to GKZord( ,β c ).
Cm e 0
R A −
Suppose that K( )=Z . Then, we have
≥0
A A
e
GKZ( ,K( ),0) ( ,0). (3)
Cm
M A A ≃R I A
Suppose moreoverthat K( )◦ =K( )+c for an element c (cid:14)K( ). Then, we have
A A 0 e0 ∈ A
GKZ( ,K( )◦,0) ( , c ). (4)
Cm 0
M A A ≃R I A −
For an expression c = m b a , we have the following commu(cid:14)tative diagram:
0 i=1 i i e
P
A,0,◦ A,0,•
M −−−−→ M
≃ ≃ (5)
 
Cm ( , c0) Cm ( ,0)
R IyA − −−−−→ R yI A
Here, the vertical arrows are given by ((cid:14)3), (4) and Proposition(cid:14)1.4, and the upper horizontal arrow is the
e e
morphism underlying the natural morphism of mixed twistor -modules T T , and the lower
A,0,◦ A,0,•
horizontal arrow is given by the multiplication of mi=1(λ∂xi)bi. D −→
Q
1.2.3 Relation with the reduced quantum -module of toric complete intersection
D
InspiredbytheworkofE.MannandT.Mignon[34],T.ReicheltandC.Sevenheck[45]introducedsomesystems
of differential equations to describe the reduced quantum -modules of complete intersections.
D
Let X be an n-dimensional projective toric manifold corresponding to a fan Σ. Let D (i = 1,...,m) be
i
the hypersurfaces of X corresponding to the one dimensional cones in Σ. Let K denote the canonical bundle
X
of X. Let (i = 1,...,r) be nef line bundles on X such that (K r )∨ is nef. We may assume
that = Li m β D for some β 0. We can regard ∨ asXa⊗torici=m1aLniifold. Let Zn+r be the
Lj O i=1 ji i ji ≥ Li N A ⊂
set of primitive vectors in the one dimensional cones of ∨. We have the systems of differential equations
(cid:0)P (cid:1) LLi n r
GKZoRerd(A,K(A),0) and GKZoRerd(A,K(A)◦,−c0), whereLc0 =(0,...,0,−1,...,−1)∈Zn+r. On (C∗)m+r, the
systems GKZoRerd(A,K(A),0) and GKZoRerd(A,K(A)◦,−c0) are ezqu}iv|ale{nzt to}t|he sy{stems in Definition-Lemma
6.1 of [45], i.e., we have the following commutative diagram:
( , c ) ≃ ∗ (0,0,0)
RCm+r I A − 0 |(C∗)m+r −−−−→ 0N
(cid:0) (cid:14) (cid:1)
e bb
RCm+r I(Ay,0) |(C∗)m+r −−−≃−→ ∗0M(y−r,0,0)
(cid:0) (cid:14) (cid:1)
(See [45, 6] for the -modules ∗ e(0,0,0) and ∗ (−r,0,0).) Hence, theccommutative diagram (5) implies that
§ R 0N 0M
the -modules ∗ (0,0,0) and ∗ (−r,0,0) underlie mixed twistor -modules. We also obtain that the image of
R 0N 0M D
the morphism b undeerlies a pure tbwistor -modcule, which is related with the reduced quantum -module of
D D
the ceomplete interbsection. c
Remark 1.5 In [45], they constructed -modules by using the partial Fourier-Laplace transform of GKZ-
R
hypergeometric systems and the Brieskorn lattices associated to the Hodge filtrations. (See also [44] and [49].)
They conjectured some relation of the -meodules with ∗ (0,0,0) and ∗ (−r,0,0) [45, Conjecture 6.13].
R 0N 0M
e b c
5
We review the construction in D, and we compare it with the -modules underlying our mixed twistor
§ R
-modules in Proposition D.6. (See also Proposition 6.70 and Corollary 6.71.) In this way, we can verify their
D
conjecture. In [46], they also verified their conjecture in a different waey.
1.3 Application to toric local mirror symmetry
Recall that GKZ-hypergeometric systems GKZord( ,Γ,0) and the variants GKZoerd( ,Γ,0) play important
A R A
roles in the study of mirror symmetry of toric manifolds. We shall apply the general theory of mixed twistor
-modules to the study of toric local mirror symmetry.
D
One of the important goals in the study of mirror symmetry is to obtain an isomorphism of Frobenius
manifolds associated to a mirror pair of an A-model and a B-model. Roughly, a Frobenius manifold is a
complex manifold with a holomorphic multiplication and a holomorphic inner product on the tangent bundle
satisfying some compatibility conditions. The Frobenius manifold associated to the A-model contains much
information on the genus 0 Gromov-Witten invariants. The Frobenius manifold associated to the B-model
contains much information on the generalized Hodge structure of the Landau-Ginzburg model. Hence, it is
interesting to have such an isomorphism of the Frobenius manifolds. One of the most celebrated results is due
to A. Givental who established it in the case of toric weak Fano manifolds. (See also [21, 44].)
Pursuing an analogue of such an isomorphism in the study of local mirror symmetry, Konishi and Minabe
[29, 30, 31, 32] introduced the concept of mixed Frobenius manifold as a generalization of Frobenius manifold.
Note that Frobenius manifolds do not appear at least naively in the local case. Roughly, a mixed Frobenius
manifold is a complex manifold with a holomorphic multiplication, a holomorphic filtration on the tangent
bundle, and inner products on the graded pieces with respect to the filtration, satisfying some compatibility
conditions. It looks natural to expect that mixed Frobenius manifolds appear widely.
Konishi and Minabe particularly studied the case of any toric weak Fano surface S. On the local A-side,
they constructed a mixed Frobenius manifold from the genus 0 local Gromov-Witten invariants of S. On the
local B-side, they suggestedthat the expected mixed Frobenius manifolds should be relatedwith a variationof
mixed Hodge structure associated to the corresponding Landau-Ginzburg model.
Then, it is natural to ask whether there really exists a mixed Frobenius manifold on the local B-side.
Recently, in his master thesis, Y. Shamoto proved a reconstruction theorem of mixed Frobenius manifolds as
a generalization of a reconstruction theorem of Frobenius manifolds due to C. Hertling and Y. Manin [18].
TogetherwiththedescriptionofthevariationofmixedHodgestructurein[3,30,61],heprovedtheexistenceof
mixed Frobenius manifolds associated to some toric local B-models, up to the ambiguity of the choice of inner
products on the graded pieces. It is still interesting to ask how to choose pairings on the graded pieces, and
how to obtain an isomorphism of mixed Frobenius manifolds associated to the local mirror pair.
In this paper, we study the expectation in a rough level. Instead of mixed Frobenius manifolds, we shall
study mixed TEP-structures which will be explained in the next subsection. We shall obtain an isomorphism
of mixed TEP-structures associated to a mirror pair of a local A-model and a local B-model.
1.3.1 Mixed TEP-structure
RecallthataTE-structureonacomplexmanifoldM inthesenseof[14]isalocallyfree -module with
OCλ×M V
a meromorphic flat connection : Ω1 log( 0 M) 0 M . If it is equipped with a
∇ V −→ V ⊗ Cλ×M { }× ⊗O { }×
perfect pairingP : j∗ λn suchthat (j∗P)(j∗a b)=( 1)nP(b j∗a), then ( , ,P) is called
V⊗ V −→ OCλ×M (cid:0) ⊗ (cid:1) − (cid:0) ⊗ (cid:1) V ∇
a TEP-structure or more precisely TEP(n)-structure [14] on M. Here, j : C M C M is given by
λ λ
× −→ ×
j(λ,Q)=( λ,Q). We shall often omit to denote , i.e., ( , ,P) is denoted by ( ,P).
− ∇ V ∇ V
A mixed TE-structure on a complex manifold M consists of the following:
A TE-structure on M.
• V
An increasing filtration W = W ( ) m Z on such that (i) W = 0 (m << 0) and W =
m m m
• V | ∈ V V
f
(m>>0), (ii) GrW( ) are locally free -modules, (iii) W are preserved by the connection of .
m V f (cid:0)f OCλ×M (cid:1) m f f V
f f f
If each GrW( ) is equipped with a non-degenerate pairing P : GrW( ) j∗GrW( ) λ−m such
fm V m fm V ⊗ m V −→ OCλ×M
that (GrW ,P ) is a TEP( m)-structure, then ( ,W, P ) is called a mixed TEP-structure.
m V m − V { m}
6
Forexample,agradedpolarizedvariationofmixedHodgestructurenaturallyinducesamixedTEP-structure
by the Rees construction. In that case, we can reconstruct the Hodge filtration, the weight filtration and the
polarization on the graded pieces of the original graded polarized variation of mixed Hodge structure from the
mixed TEP-structure. So, it still contains much interesting information.
1.3.2 Mixed twistor -modules with a graded polarization
D
Let ( ,P ,W) be a mixed twistor -module on a complex manifold Y. If the -module is a locally free
R Y
M D R M
-module, then ( ,W) is a mixed TE-structure by definition.
Y
O Supposethat(MM,PR,W)ispureofweightw,i.e.,GrWj =0unlessj =w. Thene,apolarizationof(M,PR,W)
is equivalentto amorphism λ−wj∗D( ), whereD is the dualityfunctor for mixedtwistor -modules.
M−→ M D
If is a locally free -module, the morphism gives a pairing : j∗ λ−w+dimY , and ( , )
Y Y
M O P M⊗ M−→ O M P
is a TEP( w+dimY)-structure.
If ( ,−P ,W) is not necessarilypure, a tuple ofpolarizations P =( m Z) on GrW( ,P ) is calleda
gradedMpolarRizationof( ,P ,W). If isalocallyfree -module,eacPhmG|rW(∈ )isalocmallyMfreeR -module.
M R M OY m M OY
We set W :=W . Then, ( ,W,P) is a mixed TEP-structure.
m−dimY m
M M M
1.3.3 fMixed TEP-structures on GKZ(f,K( ),0) and GKZ( ,K( )◦,0)
A A A A
We use the notation in 1.2. We have the open subset (Cm)reg Cm determined by the regularity condition
§ ⊂
of the Laurent polynomials F . The restriction M (⋆ = ,!) are locally free -modules. By the
A A,0,⋆|(Cm)reg
∗ O
general theory of mixed twistor -modules, are locally free -modules on C (Cm)reg. As previously
A,0,⋆
D M O ×
remarked, we obtain the mixed TE-structure ( ,W).
A,0,⋆|(Cm)reg
M
If 0 is an interior point of the convex hull of , then it turns out that we have a natural isomorphism
A
( ,P ,W) ( ,P ,W), that they are pure of weight n +m, and that they are equipped with
A,0,! R A,0,∗ R
M ≃ M
a natural polarization. In particular, the restriction to ( ,P ,W) with the polarization give a
A,0,! R |(Cm)reg
M
TEP( n)-structure. It is equivalent to the TEP-structure previously studied in the mirror symmetry [21, 44].
−
In the general case, we need to choose an additional datum for the construction of graded polarization on
( ,P ,W)and( ,P ,W). It turnsoutthat ifwearegivena pointb Zn suchthat0 isaninterior
A,0,! R A,0,∗ R
M M ∈
pointoftheconvexhullof b ,thenweobtaingradedpolarizationson( ,P ,W)and( ,P ,W)
A,0,! R A,0,∗ R
A∪{ } M M
depending on b. In particular, we obtain mixed TEP-structures ( ,W,P ) on (Cm)reg.
A,0,⋆|(Cm)reg b,⋆
M
We have the action of Tn on (C∗)m given by (s1,...,sn)(z1,...,zm) = (sa1z1,...,samzm). Let SA :=
(C∗)m/Tn. Let Sreg denote the image of (C∗)m (Cm)reg by the projection (C∗)fm S .
A ∩ −→ A
It turns out that ( ,W,P ) on (C∗)m (Cm)reg is equivariant with respect to the action. So, we
A,0,⋆ b,⋆
M ∩
obtain mixed TEP-structures ( ,W,P ) (⋆= ,!) on Sreg.
VA b,⋆ ∗ A
f
1.3.4 An isomorphism f
Let X be an n-dimensional smooth projective toric variety corresponding to a fan Σ. Let D (i = 1,...,m)
i
be the hypersurfaces of X corresponding to the one dimensional cones in Σ. Let K denote the canonical
X
bundle of X. Let (i = 1,...,r) be nef line bundles on X such that (K r )∨ is nef. We may
assumethat = Li m β D forsome β 0. LetY be the projectivecoXmp⊗letioin=1oLfi ∨ := r ∨,i.e.,
Lj O i=1 ji i ji ≥ N E i=1Li
Y =P .
Li⊕O (cid:0)P (cid:1) L
We have the degenerated quantum products (σ ) on H∗(X,C) induced by the Gromov-Witten
σ X
(cid:0)L (cid:1) • ∈ U
invariantsofY, asexplainedin 7.3.3. Here, is anappropriateopensubsetinH2(X,C). Asinthe ordinary
X
§ U
case, we have the associated TE-structure QDM X, ∨ on . Let γ H2(Y) be the first Chern class of the
X
E U ∈
tautological line bundle of Y over X. As explained in 7.3.4, we introduce a filtration W on H∗(X,C), and
f (cid:0) (cid:1)§
pairingsonGrW H∗(X,C)byusingtheactionofγonH∗(Y)andinnerproductonH∗(Y). Theconstructionwas
j
motivated by both the generaltheory of mixed twistor -modules and the constructionoffKonishiand Minabe
[31]. Thus, we obtain a mixed TEP-structure (QDM(XD, ∨),W,PE∨) on . It is equivariant with respect to
X
thetranslationactionof2π√ 1H2(X,Z)on . So,weoEbtainamixedTEUP-structure(QDM(X, ∨),W,PE∨)′
X
− U E
on /2π√ 1H2(X,Z). f
X
U −
f
7
Remark 1.6 Asexplainedin 7.3.5,thedegeneratedquantumproductsarerelatedwiththelocalGromov-Witten
§
invariants in some special cases. It was essentially given in [29], and used in [31] for the construction of the
weight filtration and the pairings.
Let ( ∨) Zn+r be the set of the primitive vectors in the one dimensionalcones in the fan corresponding
A E ⊂
to ∨. Let (X)= [ρ ] i=1,...,m Zn be the setofthe primitive vectorsin the one dimensionalconesin
i
E A { | }⊂
the fan Σ corresponding to X. In this case, ( ∨) consists of elements a (i=1,...,m+r) given as follows:
i
A E
[ρ ]+ r β n (i=1,...,m)
a := i j=1 ji j
i n (i=m+1,...,m+r)
i−m
(cid:26) P
m+i−1
Here, n = (0,...,0,1,0,...,0). We also set a := r n . Then, we have the mixed TEP-structure
i m+r+1 − i=1 i
( ,W,P ) on (Cm+r)reg. We obtain a mixed TEP-structure ( ,W,P ) on Sreg
MA(E∨) z }a|m+{r+1,∗ P VA(E∨) am+r+1,∗ A(E∨)
as the reduction.
f f
Theorem 1.7 We have the following.
An open subset U H2(X,C)/2π√ 1H2(X,Z) which contains a neighbourhood of the large radius limit
1
• ⊂ −
point.
An open subset U Sreg which contains a neighbourhood of the large radius point.
• 2 ⊂ A(E∨)
A holomorphic isomorphism ϕ:U U .
1 2
• ≃
Isomorphism of mixed TEP-structures
•
QDM(X, ∨),W,PE∨ ′ and ϕ∗ ,W,P
E |U1 VA(E∨),∗ am+r+1,∗ |U2
(cid:0) (cid:1) (cid:0) (cid:1)
up to shift of weights. f f
See Theorem 7.34 for a more refined and precise claim.
Remark 1.8 In [45], it is announced that a result in [24] implies the comparison of the TE-structures in
Theorem 1.7. In [23], the mirror theorem of Givental is generalized for non-compact or non-semipositive toric
manifolds. At this moment, it is not clear to the author if we could also deduce the comparison of the weight
filtrations from [23, 24].
1.3.5 Idea of the proof
Roughly,ourisomorphismisobtainedasthespecializationoftheisomorphismofGiventalfortheweakFanotoric
manifold Y. We naturally have H2(Y,C) = H2(X,C) Cγ. We have the quantum -module (QDM(Y), )
Y
× D P
on an appropriate open subset of H2(Y,C) 2π√ 1H2(Y,Z) associated to the Gromov-Witten invariants of
−
Y. The mixed TEP-structure QDM(X, ∨),W,PE∨ is described as the “specialization” of (QDM(Y), ).
E (cid:14) PY
(We explain the procedure “specialization” in B.1.) Let (Y) := ( ∨) a in Zn+r, which is the
m+r+1
(cid:0) § (cid:1) A A E ∪{ }
set of the primitive vectors in the one dimenfsional cones in a fan corresponding to Y. We naturally have
S = S C∗. By using the results on the specialization of mixed twistor -modules, we can describe
A(Y) A(E∨)
× D
the mixed TEP-structure ,W,P as“the specialization”of the TEP-structure( , ). We
VA(E∨)∗ am+r+1∗ VA(Y) P
have the isomorphism of Givental between the TEP-structures (QDM(Y), ) and ( , ). (See [21, 44].)
(cid:0) (cid:1) PY VA(Y) P
Hence, we can obtain the isomorphfism of mixed TEP-structures in Theorem 1.7 as the specialization of the
isomorphism of Givental.
8
1.4 The mixed twistor -modules associated to meromorphic functions
D
Motivated by the applicationsto the study of the twistor property of GKZ-hypergeometricsystems, mentioned
in the previous subsections, we shall consider the technical issues on the mixed twistor -modules. Let X be
D
a complex manifold with a hypersurface H. Let f be a meromorphic function on X whose poles are contained
in H. We havethe naturally defined integrable mixed twistor -modules with realstructure (f,H) (⋆= ,!)
⋆
over L (f,H). We also have the naturally induced polarizatioDns on GrW (f,H) which depTend on the cho∗ice
⋆ w T⋆
of an effective divisor D whose support is H. The much part of this paper is devoted to the study of such kind
ofgradedpolarizedmixedtwistor -moduleswithrealstructure. Forinstance,weconsiderthefollowingissues.
D
LetX(1) :=P1 X. WesetH(1) :=(P1 H) ( 0, X). Letι:X X(1) bethemorphisminduced
• τ× × ∪ { ∞}× −→
by 0 P1. Let f,g ( H). We have the graded polarized mixed twistor -modules with real
{ } −→ τ ∈ OX ∗ D
structure (g+τf,H(1)) on X(1) and (g,H) on X, where ⋆= ,!. Under some assumptions, we shall
⋆ ⋆
T T ∗
relateι (g,H)withthekernelandthecokernelof (g+τf,H(1)) (g+τf,H(1)). Itisnotdifficult
† ⋆ ! ∗
T T −→T
to obtain a relation in the level of mixed twistor -modules with real structures (Proposition 3.25). We
D
need more preliminaries to obtain the relation between polarizations on the graded pieces (Proposition
5.2).
Let X(1) and H(1) be as above. Let π : X(1) X be the projection. We have the graded polarized
• −→
mixedtwistor -modulesπ0 (g+τf,H(1))bytakingthepush-forwardviaπ. LetZ bethezero-setoff.
D †T⋆ f
We have the graded polarized mixed twistor -modules (g,H)[!(f) ] and (g,H)[ (f) ] on X. Under
∗ 0 ! 0
D T T ∗
some assumptions, we shall relate π0 (g +τf,H(1)) (resp. π0 (g +τf,H)) and (g,D)[!Z ] (resp.
†T∗ †T! T∗ f
(g,D)[ Z ].) Again, it is not difficult to obtain a relation in the level of mixed twistor -modules with
! f
T ∗ D
realstructure(Proposition3.22). Weneedmorepreliminariestoobtaintherelationbetweenpolarizations
on the graded pieces (Proposition 5.7).
We also need the compatibility of various standard functors for -modules and flat bundles, which are
D
explained in the appendix sections A and B.
§ §
Acknowledgement This study grows out of my attempt to understand the works of H. Iritani [20, 21], Y.
Konishi, S. Minabe [30, 31, 32] and T. Reichelt, C. Sevenheck [44, 45]. I am grateful to them for helpful
discussions on various occasions. In particular,Reichelt and Sevenheck explained their notation, and remarked
metheissueofthenormalityconditioninthestudyofGKZ-hypergeometricsystemsandtoricvarieties. Ithank
C. Sabbah for his kindness and for discussions on many occasions. I thank Y. Shamoto for some discussion. I
thank G. Wilkinfor his interestto this study andhis kindness. I amgratefulto C.Hertling and U.Walther for
some discussions. I thank A. Ishii, A. Moriwaki, M.-H. Saito, Y. Tsuchimoto, T. Xue and K. Vilonen for their
kindness and supports.
This workispartiallysupportedbythe Grant-in-AidforScientific Research(C)(No. 22540078),the Grant-
in-AidforScientificResearch(C)(No. 15K04843),theGrant-in-AidforScientific Research(A)(No. 22244003)
and the Grant-in-Aid for Scientific Research (S) (No. 24224001), Japan Society for the Promotion of Science.
This research was partially completed while the author was visiting the Institute for Mathematical Sciences,
National University of Singapore in 2014. This research was partially completed while the author stayed in
the MathematischesForschungsinstitutOberwolfach,partiallysupportedbythe SimonsFoundationandbythe
Mathematisches Forschungsinstitut Oberwolfach.
2 -modules associated to meromorphic functions
D
2.1 Purity condition
2.1.1 -modules associated to meromorphic functions
D
LetX be acomplexmanifoldwithahypersurfaceD. Let ( D)be the sheafofmeromorphicfunctions onX
X
O ∗
whose poles are contained in D. Let D denote the duality functor on the category of holonomic -modules on
D
X. For any coherent -module M, we set M( D) := ( D) M and M(!D) := D D(M)( D) . The
DX ∗ OX ∗ ⊗OX ∗
(cid:0) (cid:1)
9
restriction of M(⋆D) (⋆ = ,!) to X D is naturally isomorphic to M . We have the canonical morphism
|X\D
∗ \
M(!D) M( D) whose restriction to X D is the identity.
−→ ∗ \
Let f be a meromorphic function on (X,D), i.e., a section of ( D). Let (f) and (f) denote the
X 0 ∞
O ∗
effective divisors obtained as the zeroes and the poles of f. The supports of the divisors are denoted by (f)
0
| |
and (f) .
∞
| |
We obtain the meromorphic flat bundle L (f,D) := ( D)v with v = vdf. We naturally regard it
∗ X
O ∗ ∇
as a -module. We set L(f,D) := L (f,D)(!D) = D L ( f,D) . The image of the canonical morphism
X ! ∗ ∗
D −
L(f,D) L (f,D) is independent of the choice of D such that (f) D, and denoted by L(f). We
! ∗ ∞
−→ (cid:0) (cid:1)| | ⊂
have natural isomorphisms L (f,D) L(f)(⋆D) for ⋆ =!, . Indeed, L(f) L (f,D) naturally induces
⋆ ∗
≃ ∗ −→
L(f)( D) L (f,D) which is clearly an isomorphism. By using the duality, we obtain L(f,D) L(f)(!D).
∗ !
∗ −→ ≃
When D = (f) , we set L (f):=L (f, (f) ).
∞ ⋆ ⋆ ∞
| | | |
2.1.2 Purity
We continue to use the notation in 2.1.1. We introduce a condition.
§
Definition 2.1 We say that a meromorphic function f on X is pure at P X if the canonical morphism
∈
L(f) L (f) is an isomorphism on a neighbourhood of P. We say that f is pure if it is pure at any point
! ∗
−→
of X.
Because D L (f) L( f) and D L(f) L ( f), we have the following easy lemmas.
∗ ! ! ∗
≃ − ≃ −
Lemma 2.2 L(cid:0)et f be(cid:1)a meromorphic fu(cid:0)nction(cid:1)on X.
f is pure if and only if f is pure.
• −
f is pure if and only if the morphisms L(f) L (f) and L( f) L ( f) are epimorphisms.
! ∗ ! ∗
• −→ − −→ −
Let ϕ : Y X be a proper morphism of complex manifolds such that ϕ induces an isomorphism Y
−→ \
ϕ−1(D) X D.
≃ \
Lemma 2.3 Let f be a meromorphic function on (X,D). Suppose that (ϕ∗f) =ϕ−1(D) and that ϕ∗(f) is
∞
pure. Then, f is also pure, and it satisfies (f) =D.
| ∞| (cid:12) (cid:12)
(cid:12) (cid:12)
Proof We set f := ϕ∗(f) and D := ϕ−1(D). Because D = ϕ (f ) (f) D, we have (f) =
Y Y Y ∞ ∞ ∞
| | ⊂ | | ⊂ | |
D. We have ϕi L (f ,D ) = 0 (i = 0), and ϕ0 L (f ,D ) ( D) ϕ0 L (f ,D ) . Hence, we have
+ ∗ Y Y 6 + ∗ Y Y (cid:0)∗ ≃ (cid:1)+ ∗ Y Y
ϕ0 L (f ,D ) L (f,D). By the duality, we have ϕ0 L(f ,D ) L(f,D). Then, from the purity of
+ ∗ Y Y ≃ ∗ (cid:0) + ! Y(cid:1) Y ≃ ! (cid:0) (cid:1)
f , we obtain that L(f,D) L (f,D) is an isomorphism.
Y (cid:0) (cid:1) ! −→ ∗ (cid:0) (cid:1)
2.1.3 Vanishing of cohomology for pure functions
Thepurityconditionsometimesimpliesthevanishingofthecohomology. Wementionatypicalcase. LetX bea
complexmanifoldwithahypersurfaceD. Letf beameromorphicfunctionon(X,D)suchthat(i) (f) =D,
∞
| |
(ii) f is pure. Let F :X S be a proper morphism of complex manifolds.
−→
Proposition 2.4 Suppose that RiF Ωj ( D) = 0 for (i,j) Z2 such that i+j > dimX. Then, we
∗ X ⊗O ∗ ∈ ≥0
have FkL(f)=0 (k =0).
† 6 (cid:0) (cid:1)
Proof BecauseDL(f) L( f)andbecause f isalsopure,itisenoughtoprovethatFkL(f)=0fork>0.
≃ − − †
We have only to prove the claim locally aroundany point of S. We set ω :=ΩdimX and ω :=ΩdimS. Recall
X X S S
thatF L(f)isobtainedasRF ω F−1 ω−1 L L(f) . Wehavethestandardfreeresolution
† ∗ X⊗F−1OS DS⊗ S ⊗DX
DX ⊗Ω•X[dimX] of the right D(cid:16)X(cid:2)-modules ωX. B(cid:0)y the assu(cid:1)m(cid:3)ption, if i(cid:17)+j >dimX, we have
RjF Ωi (F−1 ω ) L(f) RjF Ωi L(f) ( ω )=0.
∗ X ⊗F−1OS DS ⊗ S ⊗OX ≃ ∗ X ⊗ ⊗OS DS ⊗ S
(cid:16)(cid:2) (cid:3) (cid:17) (cid:0) (cid:1)
Hence, we have the desired vanishing FkL(f)=0 for k >0.
†
10