arXiv:1303.2560v2 [math.AG] 25 Oct 2014
PERIOD INTEGRALS AND THE RIEMANN-HILBERT CORRESPONDENCE
AN HUANG, BONG H. LIAN, AND XINWEN ZHU
Abstract. A tautological system, introduced in [17][18], arises as a regular holonomic system of partial differential equations that governs the period integrals of a family of complete intersections in a complex manifoldX, equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [4], and was verified for the case of projective homogeneous space under an assumption.
In this paper, we prove this conjecture in full generality. By means of the Riemann- Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action.
Contents
1. Introduction 1
2. CY hyperplane sections 6
3. !-fibers ofτ 12
4. The geometry 13
5. A formula forτ: CY case 14
6. General type hyperplane sections 18
7. A formula forτ: general type case 21
8. Projective homogeneous spaces 23
9. Rank 1 points of 1-step flags 25
10. Rank 1 points ofr-step flags 27
Appendix A. Theory ofD-modules 36
References 43
1. Introduction
Let Gbe a connected algebraic group over a field k of characteristic zero. Let X be a projectiveG-variety and letL be a very ampleG-linearized invertible sheaf overX which gives rise to aG-equivariant embedding
X →P(V),
where V = Γ(X,L)∨. Let r = dimV. We assume that the action of G on X is locally effective, i.e. ker(G → Aut(X)) is finite. Let Gm be the multiplicative group acting on V by homotheties. Let ˆG = G×Gm, whose Lie algebra is ˆg= g⊕ke, where e acts on V by identity. We denote by Z : ˆG → GL(V) the corresponding group representation,
1
and Z : ˆg→ End(V) the corresponding Lie algebra representation. Note that under our assumptions,Z: ˆg→End(V) is injective.
Let ˆı: ˆX ⊂V be the cone ofX, defined by the idealI( ˆX). Letβ : ˆg→kbe a Lie algebra homomorphism. Then atautological systemas defined in [17][18] is the cyclicD-module on V∨
τ(G, X,L, β) =DV∨/DV∨J( ˆX) +DV∨(Z(ξ) +β(ξ), ξ∈ˆg), where
J( ˆX) ={Db|D∈I( ˆX)}
is the ideal of the commutative subalgebraC[∂]⊂DV∨ obtained by the Fourier transform of I( ˆX) (see §A for the review of the Fourier transform and in particular (A.6) for the notation).
Given a basis {ai} of V, we have Z(ξ) = P
ijξijai∂aj, where (ξij) is the matrix rep- resenting ξ in the basis. Since the ai are also linear coordinates on V∨, we can view Z(ξ) ∈ Derk[V∨] ⊂ DV∨. In particular, the identity operator Z(e) ∈ EndV becomes the Euler vector field onV∨.
We recall the main motivation for studying tautological systems. LetX′ be a compact complex manifold (not necessarily algebraic), such that the complete linear system of anti- canonical divisors inX′is base point free. Letπ:Y →B:= Γ(X′, ωX−1′)sm be the family of smooth CY hyperplane sections Ya ⊂X′, and let Htopbe the Hodge bundle over B whose fiber at a∈ B is the line Γ(Ya, ωYa) ⊂Hn−1(Ya), where n = dimX′. In [18], the period integrals of this family are constructed by giving a canonical trivialization of Htop. Let Π = Π(X′) be the period sheaf of this family, i.e. the locally constant sheaf generated by the period integrals (Definition 1.1 [18].)
LetV = Γ(X′, ωX−1′)∨,X be the image of the natural mapX′ →P(V), and L=OX(1).
LetGbe a connected algebraic group acting onX.
Theorem 1.1. The period integrals of the familyπ:Y →B are solutions to τ=τ(G, X,L, β0),
whereβ0 is the Lie algebra homomorphism which vanishes ongandβ0(e) = 1.
This was proved in [17] forX′ a partial flag variety, and in full generality in [18], where the result was also generalized to hyperplane sections of general type.
We note that when X′ is a projective homogeneous manifold of a semisimple group G, in which case we have X = X′, τ is amenable to explicit descriptions. For example,
one description says that the tautological system can be generated by the vector fields corresponding to the linearGaction onV∨, and a twisted Euler vector field, together with a set of quadratic differential operators corresponding to the defining relations ofX inP(V) under the Pl¨ucker embedding. The case where X is a Grassmannian has been worked out in detail [17]. Furthermore, when the middle primitive cohomologyHn(X)prim = 0, it is also known that the systemτ is complete, i.e. the solution sheaf coincides with the period sheaf [4].
We now return to a general tautological systemτ. Applying an argument of [13], we find that ifGacts onX by finitely many orbits, and if the character D-module on ˆG
Lβ:=DGˆ/DGˆ(ξ+β(ξ), ξ ∈ˆg)
on ˆG is regular singular, then τ is regular holonomic. See [17] Theorem 3.4(1). In this case, if X = ⊔rl=1Xl is the decomposition into G-orbits, then the singular locus of τ is contained in∪rl=1Xl∨. HereXl∨⊂V∨is the conical variety whose projectivizationP(Xl∨) is the projective dual to the Zariski closure ofXlinX. From now on we assume thatGacts onX by finitely many orbits, and Lβ is regular singular. Note that the latter assumption is always satisfied whenGis reductive.
Let us now turn to the main problem studied in this paper. In the well-known applications of variation of Hodge structures in mirror symmetry, it is important to decide which solutions of our differential system come from period integrals. By Theorem 1.1, the period sheaf is a subsheaf of the solution sheaf of a tautological system. Thus an important problem is to decide when the two sheaves actually coincide, i.e. whenτis complete. Ifτ is not complete, how much larger is the solution sheaf relative to the period sheaf? From Hodge theory, we know that (see Proposition 6.3 [4]) the rank of the period sheaf is given by the dimension of the middle vanishing cohomology of the smooth hypersurface Ya. Therefore, to answer those questions, it is clearly desirable to know precisely the holonomic rank ofτ. For a brief overview of known results on these questions in a number of special cases, see Introduction in [4].
Conjecture 1.2. (Holonomic rank conjecture) Let X be ann-dimensional projective ho- mogeneous space of a semisimple groupG. The solution rank ofτ=τ(G, X, ωX−1, β0) at the pointa∈V∨ is given by dimHn(X−Ya).
In [4], the following is proved,
Theorem 1.3. Assume that the natural map
g⊗Γ(X, ω−rX )→Γ(X, TX⊗ωX−r) is surjective for eachr≥0. Then conjecture 1.2 holds.
In this paper, we prove this in full generality.
Theorem 1.4. Conjecture 1.2 holds.
This will be proved in §2. There are at least two immediate applications of this result.
First we can now compute the solution rank forτ forgenerica∈V∨. Corollary 1.5. The solution rank ofτ at a smooth hyperplane sectiona is
dimHn(X)prim+ dimHn−1(Ya)−dimHn+1(X).
where the first term is the middle primitive cohomology ofX =G/P withn= dimX.
The last two terms of the rank above can be computed readily in terms of the semisimple groupGand the parabolic subgroupP by the Lefschetz hyperplane and the Riemann-Roch theorems. (See Example 2.4 [17]). A second application of Theorem 1.4 is to find certain exceptional pointsain V∨ where the solution sheaf ofτ degenerates “maximally”.
Definition 1.1. A nonzero section a ∈ V∨ = Γ(X, ωX−1) is called a rank 1 point if the solution rank ofτ at ais 1. In other words, HomDV∨(τ,OV∨,a)≃C.
Corollary 1.6. Any projective homogeneous variety admits a rank 1 point.
We will construct these rank 1 points in two explicit but different ways. The first, which works forG=SLl, is a recursive procedure that produces such a rank 1 point by assembling rank 1 points from lower step flag varieties, starting from Grassmannians, and by repeatedly applying Theorem 1.4. The second way, which works for any semisimple groupG, is by using a well-known stratification of the flag varietyG/Bto produce an open stratum inX =G/P with a one dimensional middle degree cohomology. The complement of this stratum is an anticanonical divisor, hence a rank 1 point ofX by Theorem 1.4.
The geometric formula in Conjecture 1.2 appears to go well beyond the context of ho- mogeneous spaces. Theorem 1.4 can be seen as a special case of the following much more general theorem. Consider a smooth projectiveG-varietyX withL=ωX−1 very ample. Set τ =τ(G, X,L, β) and V∨ = Γ(X,L). We introduce some more notations. Let L∨ be the total space ofLand ˚L∨ be the complement of the zero section. Let
ev:V∨×X ։L∨, (a, x)7→a(x) be the evaluation map, andL⊥ := ker(ev). Finally let
π∨:U :=V∨×X−L⊥→V∨.
Note that this is the complement of the universal family of hyperplane sections L⊥ ։ V∨, (a, x)7→a. Put
DX,β = (DX⊗kβ)⊗Uˆgk,
wherekβis the 1-dimensional ˆg-module given by the characterβ (see§A for the notations).
We now state a main result of this paper.
Theorem 1.7. For β(e) = 1, there is a canonical isomorphism τ≃H0π+∨(OV∨⊠DX,β)|U.
Corollary 1.8. SupposeGacts onX by finitely many orbits, andk=C. Then the solution rank ofτ ata∈V∨ is given by dimHcn(Ua,Sol(DX,β)|Ua), whereUa=X−Ya.
More generally, we have
Theorem 1.9. For β(e)∈/Z≤0, andL=ω−1X , there is a canonical isomorphism τ≃H0π∨+ev!(D˚L∨,β)[1−r].
In addition to proving Conjecture 1.2 as a special case, Theorem 1.7 can also be used to derive the well-known formula for the solution rank of a GKZ system [8] at generic pointa.
But since Corollary 1.8 holds for arbitrarya∈V∨, it holds in particular foracorresponding the union of allT-invariant divisors inX (which is anticanonical). In this case, Theorem 1.7 implies thatais a rank 1 point – a result of [10] based on Gr¨obner basis theory but motivated by applications to mirror symmetry. Thus Theorem 1.7 and Corollary 1.8 interpolate a result of [8] and [10] by unifying the rank formula at generic point and at those exceptional rank 1 points, and at the same time, generalize them to an arbitraryG-variety.
Theorems 1.9 and 2.1 are clearly motivated by period integral problems in Calabi-Yau geometry. Equally important parallel problems for manifolds of general type have also been systematically studied [18][7]. In this paper, we develop the general type analogues of those two main theorems. Roughly speaking,ωX−1is replaced by an arbitrary very ample invertible sheafL onX, and τ by a larger differential system defined on Γ(X,L)∨×Γ(X,L ⊗ωX)∨. This class of systems arise naturally from period integrals of general type hypersurfaces in X. The precise statements will be formulated and proved in§6 and§7.1.
We now outline the paper. In§2, we prove Theorem 1.7 and a number of its consequences, including Theorem 1.4. We also describe explicitly the “cycle-to-period” mapHn(X−Ya)→ HomDV∨(τ,OV∨,a) as a result of Theorem 1.7, and use it to answer a question recently communicated to us by S. Bloch. While §2 deals only with the case β(e) = 1, we remove this assumption in §§3-5. In §3, we study the !-fibers of τ, and describe some vanishing
results at the special point a = 0. We describe the geometric set up in §4 for proving Theorem 1.9. The key step of the proof, involving an exact sequence forτ, is done in§5.
In§6 and §7.1, we prove the general type analogues of Theorems 1.9 and 2.1. Finally, we apply our results to construct rank 1 points for partial flag varieties in the caseG=SLlin
§§9-10, and for general semisimple groups in §8. The appendix §A collects some standard facts on D-modules.
Acknowledgements. S. Bloch has independently noticed the essential role of the Riemann- Hilbert correspondence in connecting the de Rham cohomology and solution sheaf of a tautological system. We thank him for kindly sharing his observation with us. We also thank T. Lam for helpful communications. A.H. would like to thank S.-T. Yau for advice and continuing support, especially for providing valuable resources to facilitate his research.
B.H.L. is partially supported by NSF FRG grant DMS 1159049. X.Z. is supported by NSF grant DMS-1313894 and DMS-1303296 and the AMS Centennial Fellowship.
2. CY hyperplane sections
We begin with Theorem 1.7 : X is aG-variety withL=ωX−1 very ample, and β(e) = 1.
This is in fact a special case of the more general Theorem 1.9 and therefore can be also obtained by the methods introduced in later sections. However, we decide to deal with this case first for several reasons. On the one hand, the proof given here is different from the later method and is more direct. On the other hand, the subcase whenβ(g) = 0, i.e. β=β0, which is important to mirror symmetry, is already covered by Theorem 1.7.
Letn= dimX. LetU =V∨×X−V(f), whereV(f) =L⊥ is the universal hyperplane section, so thatUa=X−V(fa) whereV(fa) =Ya, the zero locus of the sectionfa≡a∈V∨. Letπ∨:U →V∨denote the projection. The restriction ofβtogis still denoted byβ when no confusion arises. Put DX,β = (DX⊗kβ)⊗Ugk. Note that if G acts onX by finitely many orbits, then DX,β is (G, β)-equivariant holonomic D-module on X (see Lemma A.5 and A.6), and therefore
N := (OV∨⊠DX,β)|U is a holonomic D-module onU.
Theorem 2.1. Assume thatβ(e) = 1. Then there is a canonical isomorphismτ ≃H0π+∨N. Corollary 2.2. If β(g) = 0. There is a canonical surjective map
τ→H0π∨+OU.
Proof. Note that there is always a surjective mapDX,0 =DX/DXg→DX/DXTX =OX. The corollary follows from the fact thatπ+∨ is right exact asπ∨:U →V∨ is affine.
We turn to the solution sheaf of τ via the Riemann-Hilbert correspondence. Assume G acts onX by finitely many orbits. Let us writeF = Sol(DX,β). This is a perverse sheaf on X.
Corollary 2.3. Let k = C and a ∈ V∨. Then the solution rank of τ at a is given by dimHc0(Ua,F|Ua).
Proof. Denote Sol(N) = G. According to the Riemann-Hilbert correspondence, Sol(τ) =
pR0π∨! G, where pR0π!∨ denotes the 0th perverse cohomology ofπ∨! . Then the non-derived solution sheaf clSol(τ) = HomDV∨(τ,OV∨) is given by H−r(pR0π∨! G), the (−r)th (stan- dard) sheaf cohomology ofpR0π!∨G. However, as Rπ∨! G lives in positive perverse degrees, H−r(pR0π∨! G) =H−rRπ!∨G=R−rπ!∨G. AsG=C[r]⊠F|U, the claim follows.
Remark 2.1. We will give more explicit descriptions of the perverse sheaf F in various situations later on. For example, in the case X is a homogenous G-variety and β(g) = 0, thenF =C[n].
Now we prove Theorem 2.1. We will assumeβ(g) = 0 to simplify notations.
Let us write
(2.1) R:=DV∨/DV∨J( ˆX),
which is a left DV∨-module. Observe that for any ξ ∈ ˆg, DVI( ˆX)Z∨(ξ) ⊂ DVI( ˆX), so DV∨J( ˆX)Z(ξ) ⊂ DV∨J( ˆX). Therefore, DV∨J( ˆX) can be regarded as a right ˆg-module, on which ξ ∈ ˆg acts via the right multiplication by Z(ξ). Accordingly, R is also a right ˆ
g-module. In addition, by definition we have
(2.2) τ= (R ⊗kβ)⊗ˆgk,
wherekβ is the 1-dimensional representation of ˆgdefined byβ.
We now convertRto a left ˆg-module (cf. [4,§2].) Let{ai}be a basis ofV and{a∗i} the dual basis. Observe that asOV∨-modules, one can write
R ≃ OV∨⊗S, where
(2.3) S=k[∂ai]/J( ˆX)≃ OV/I( ˆX)
is identified with the homogeneous coordinate ring of ˆX, andOV∨ acts on the first factor1. If we convert the right action of ˆgonRdescribed above to a left actionα, thenαwill be the sum of the following two actions: the first is the action of ˆgon the second factor through
1TheDV∨-module structure onRis given as follows:∂ai acts onOV∨⊗Sas∂ai⊗1 + 1⊗a∗i.
the dual representationZ∨: ˆg→EndV∨→EndS, which is denoted by α1; to describe the second actionα2, observe that the natural multiplication map
(V ⊗V∨)⊗(OV∨⊗S)→(OV∨⊗S),
inducesV ⊗V∨→EndRandα2is viaZ∨: ˆg→V⊗V∨→End(R). Explicitly, if we write a⊗b∈ OV∨⊗S, then
(2.4) α1(ξ)(a⊗b) =a⊗Z∨(ξ)(b).
Let’s writeZ∨(ξ) =−P
ijξijai⊗a∗j. Then
(2.5) α2(ξ)(a⊗b) =−X
ij
ξijaai⊗ba∗j.
Letf =Pai⊗a∗i ∈ R, which can be regarded as the universal section of the line bundle OV∨⊠L overV∨×X. Recall that U=V∨×X−L⊥. Then
OU = (OV∨⊗S( ˆX))(f)
is the homogeneous localization ofRwith respect tof, where the degree of a⊗b∈ OV∨⊗ S( ˆX) is the degree of b in the graded ring S( ˆX). As L−1 =ωX, we can regard f−1 as a rational section ofOV∨⊠ωX, regular onU. ThenOUf−1 can be identified with the regular sections ofOV∨⊠ωX overU. In other words,
(2.6) OUf−1≃ωU/V∨ = (OV∨⊠ωX)|U.
Therefore, it is equipped with a (DV∨ ⊠DopX)|U module structure (see [5, VI, §3] for the definition of rightDX module structure onωX). Asgmaps to the vector fields onX,OUf−1 is aDV∨×g-module. We will describe this structure more explicitly. First, we describe the DV∨-module structure. Letθ be a vector field onV∨, andξ∈g. It is enough to describe θ(f−1) and (f−1)ξ. Let us writeZ∨(ξ) =−P
ijξijai⊗a∗j as before.
Lemma 2.4. We have
θ(f−1) =− P
iθ(ai)⊗a∗i
f2 ∈ OUf−1, and
(f−1)ξ=− P
ijξijai⊗a∗j
f2 ∈ OUf−1.
Proof. Letv ∈V∨, regarded as a section ofL. Thenv−1 is a rational section of ωX, and ω= 1⊗v−1is a rational section ofOV∨⊠ωX, obtained by pullback of a rational section of ωX. Note thatg= (1⊗v)/f ∈ OU, and we can write f−1=g(1⊗v−1). By definition, for a vector fieldθonV∨,θ(ω) = 0, and forξ∈g,ωξ =−1⊗Lieξv−1, where Lieξ :ωX→ωX
is the Lie derivative (see [5, VI,§3] for the definition of right D-module structures onωX).
Therefore
θ(f−1) =θ(g)ω, (f−1)ξ= (gξ)ω−g(1⊗Lieξv−1).
Note that
θ(g) =θ( 1 Pai⊗av∗i
) =−
Pθ(ai)⊗av∗i
(Pai⊗av∗i)2 =−g2X
θ(ai)⊗a∗i v =−g
Pθ(ai)⊗a∗i
f .
Therefore, the first equation holds. On the other hand (g)ξ= (1⊗v
f )ξ=−1⊗Z∨(ξ)(v)
f −(1⊗v)Pξijai⊗a∗j
f2 .
To prove the second, we need to understand Lieξv−1. We consider a more general situation.
Let X be a Fano variety. Assume that L = ωX−1 is very ample, and X → P(V) be the closed embedding where V = Γ(X,L)∨. Then g = Γ(X, TX) is a Lie algebra and L is naturally g-linearized. Therefore, V∨ = Γ(X,L) is a natural g-module with action Z∨ : g → End(V∨). As Z∨(ξ) = −P
ijξijai⊗a∗j, we have Z∨(ξ)(v) =−P
ijξijai(v)a∗j forv ∈V∨. On the other hand, recall that gacts onωX by Lie derivatives. Note that for v∈V∨,v−1 can be regarded as a rational section ofωX.
Lemma 2.5. Let ξ∈g,06=v∈V∨. Then
Lieξv−1=−Z∨(ξ)(v) v v−1.
Proof. Consider the 1-parameter subgroupgt=exp(tξ). Then d
dtgt∗(v−1) =−(v◦gt)−2d
dt(v◦gt).
Now sett= 0.
Now Lemma 2.4 follows.
Note that explicitly, theDV∨×g-module structure onOUf−1can be described as follows.
Letθ=∂ai be a vector field onV∨andξ=−Pξijai⊗a∗j ∈g,m=fl+11 (a⊗b)∈ OUf−1, wherea∈ OV∨ andb∈S is homogeneous of degreek, then
∂ai(m) = ∂ai(a)⊗b
fl+1 + (−1)l+1(l+ 1)a⊗ba∗i fl+2 , (m)ξ= 1
fl+1(a⊗Z∨(ξ)(b))−l+ 1 fl+2(X
ij
ξijaai⊗ba∗j).
We extend this to a ˆg-module by requiring thateacts by zero onOUf−1.
Now, we have the following technical lemma. Recall thatβ(e) = 1.
Lemma 2.6. The map φ:R ⊗kβ→ OUf−1 given by φ(a⊗b) = (−1)ll!
fl+1 a⊗b
is aDV∨׈g-module homomorphism. In addition, it induces an isomorphism τ= (R ⊗β)⊗ˆgk≃(OUf−1)⊗ˆgk= (OUf−1)⊗gk.
Proof. A direct calculation shows thatφis aDV∨׈g-module homomorphism. Namely, we know that∂ai acts onRby∂ai⊗1 + 1⊗a∗i. Therefore,
φ(∂ai(a⊗b)) =φ(∂ai(a)⊗b+a⊗ba∗i) =(−1)ll!
fl+1 (∂ai(a)⊗b) +(−1)l+1(l+ 1)!
fl+2 (a⊗ba∗i), which is the same as∂aiφ(a⊗b). Theg-equivariance can be checked similarly.
Clearlyφis surjective, with the kernel spanned by (l+1)a⊗b+f(a⊗b) forbhomogeneous of degreel. But (a⊗b)α(e) = (l+ 1)a⊗b+f(a⊗b). The lemma is proved.
To apply this lemma, recall the definition ofπ+∨ forπ∨:U →V∨a smooth morphism of algebraic varieties. Asπ∨ is an affine morphism,
π∨+N = Ω•U/V∨⊗ N[dimX].
In particular,
H0π∨+N = coker((OV∨⊠ΩdimX X−1⊗DX⊗gk)|U →(OV∨⊠ωX⊗DX⊗gk)|U).
As coker(ΩdimX X−1 ⊗DX → ωX ⊗DX) = ωX as right DX-modules, H0π∨+N is exactly (OV∨⊠ωX)|U⊗gk≃τ. This completes the proof of Theorem 2.1.
We continue to let X be a general smooth projective G-variety, and let β(e) = 1. We further assume thatk=Candβ(g) = 0, and consider some consequences of Theorem 2.1.
By taking the solution sheaves on both sides in Corollary 2.2, we get an injective map (2.7) Hn(X−V(fa))≃Hom(H0π+∨OU,OV∨,a)→Hom(τ,OV∨,a),
where the first isomorphism follows from the same argument as in Corollary 2.3 and the Poincare duality. This gives an explicit lower bound for the solution rank ofτ at any point a. For applications, we need to give a more geometric and explicit description of this map.
Note that we can interpret 1/f as a family (parametrized byV∨) of meromorphic top forms on X, whose fiber overa∈V∨ has poles alongV(fa). We denote this family of top forms onX by Ωa. These forms can also be given as follows.
Consider the principal Gm-bundle π∨ : ˚L∨ → X (with right action). Then there is a natural one-to-one correspondence between sections of L and Gm-equivariant morphism f : ˚L∨ →k, i.e. f(m·h−1) =hf(m). We shall write fa the function that represents the sectiona. Letw= (w1, .., wn) be local coordinates onX, andzwbe the coordinate induced on the fibers ofL∨. Putω=dzw∧dw1∧ · · · ∧dwn. Then it can be shown thatω defines a global non-vanishing form onL∨. (See [18, Prop. 6.1].) Letx0 be the vector field generated by 1∈k=Lie(Gm). Then Ω :=ix0ω is aG-invariantGm-horizontal form of degree dimX on ˚L∨. Moreover, since
Ωa := Ω fa
is G×Gm-invariant, it defines a family of meromorphic top form on X with pole along V(fa) [18, Thm. 6.3]. Then the isomorphism in Lemma 2.6 sends the generator “1” ofτ to Ωa. Consider the “cycle-to-period” map defined in [18]
Hn(X−V(fa))→Hom(τ,OV∨,a), γ7→
Z
γ
Ωa.
Corollary 2.7. The cycle-to-period map Hn(X−V(fa))→Hom(τ,OV∨,a), γ7→ hγ,Ω
fai= Z
γ
Ω fa
, is injective.
The rest of the section will not be used in the sequel. We note that the argument of Corollary 2.3 has the following interesting topological consequence, which answers a question S. Bloch communicated to us. LetX ⊂PN be ann-dimensional smooth projective variety.
LetV(f)→V∨= Γ(X,L) be the universal family of hyperplane sections of X.
Corollary 2.8. Let a∈V∨. Then for a′ close to a, the map Hn(X−V(fa))→Hn(X − V(fa′))induced by parallel transport is injective.
Proof. As argued in Corollary 2.3, Hn(X −V(fa)) can be identified with the stalk of the classical solutions of some regular holonomic system onV∨. Since any analytic solution to a regular holonomic system ataextends to some neighborhood ofa, the map between stalks of the classical solution sheaf of this regular holonomic system given by analytic continuation
is injective.
Our result sheds new light on the well-studied toric case, i.e. the original GKZ A- hypergeometric differential equations. We assume thatX is a toric variety, with the action of the torusG=T. Then ˆG=T ×Gm. Then Theorem 2.1 takes a particular easy form in the following situation.
Corollary 2.9. [10] If β = β0, and X is smooth toric variety, G = T is the algebraic torus ofX, andYa is the anticanonical divisor ofX given by the union ofG-invariant toric divisors in X, thena is a rank 1 point.
Proof. Note that in this caseDX,β|X−Ya ≃ OX−Ya. Therefore, Hom(τ,OV∨,a)≃Hn(Tn),
which is one-dimensional.
3. !-fibers of τ
In the following three sections, we considerτ whenβ(e) is not necessarily 1. Here we will give a formula of the !-fibers ofτata∈V∨. Fora∈V∨, letia :{a} →V∨ be the inclusion and for simplicity, let us write
τa! =i!aτ.
This is a complex of vector spaces and our goal is to give an expression of this complex.
By (2.2) we have
τa! =ka⊗LOV∨ ((R ⊗β)⊗ˆgk)[−dimV],
where ka =OV∨/ma is the residual field ata, andma is the maximal ideal ofOV∨ corre- sponding toa.
The advantage of this expression of τa! is that we can first calculate ka ⊗LOV∨ R as a (complex of) right ˆg-modules, and then taking the Lie algebra coinvariants. Namely, we have the Koszul resolution ofka, which gives the complex that calculatesτa!
(3.1) τa! = (^
V ⊗ OV∨⊗S)⊗ˆg(−β).
whereV ⊗ OV∨→ OV∨ is given byv⊗17→v−v(a). In general, this complex is difficult to compute. However, whena= 0, this is more tractable, as we shall see.
First, for a general pointa∈V∨we can express the degreer-term as
(3.2) Hrτa! ≃H0(ˆg, S⊗β),
where the action of ˆgonS will be the sum of two actions (induced by the actionsα1 and α2 of g on OV∨⊗S, as described in (2.4) and (2.5)). Concretely, the first action is via Z∨ : ˆg→EndV∨→EndS, and the second is via the ξ(b) = −P
ξijai(a)ba∗j forb ∈S. If a 6= 0, S is not a finite dimensional ˆg-module and this Lie algebra coinvariant is difficult to compute. On the other hand, if a= 0, the second action vanishes andS decomposes as finite dimensional representations of ˆg.
Lemma 3.1. Assume that β(e)6∈Z≤0. ThenHrτ0! = 0.
Proof. The homothetyGmacts onS by nonnegative weights. Therefore, ifβ(e)6∈Z≤0, the
coinvariant ofS⊗β with respect to thisGmis zero.
From now on, we assume thatβ(e)6∈Z≤0.
Let us calculateHr−1τ0!. We have
(V ∧V)⊗(R ⊗β) −−−−→m2 V ⊗(R ⊗β) −−−−→m1 R ⊗β
y y y
(V ∧V)⊗(R ⊗β)⊗ˆgk −−−−→d2 V ⊗(R ⊗β)⊗ˆgk −−−−→d1 (R ⊗β)⊗ˆgk.
Then
Hr−1τ0! =m−11 ((R ⊗β)ˆg)/(Imm2+V ⊗(R ⊗β)ˆg)
As the Koszul complex is acyclic away from degree zero, we can rewrite the above as Hr−1τ0! = (R ⊗β)ˆg∩Imm1/(Imm1)ˆg.
Consider
0→(R ⊗β)ˆg∩Imm1/(Imm1)ˆg→(R ⊗β)ˆg/(Imm1)ˆg→(R ⊗β)ˆg/(R ⊗β)ˆg∩Imm1→0.
Note that 0 =Hrτ0! implies that (R ⊗β)ˆg+ Imm1=R ⊗β. Therefore, (R ⊗β)ˆg/(R ⊗β)ˆg∩Imm1=R ⊗β/Imm1. We therefore can write
Hr−1τ0! = ker((R ⊗β)ˆg/(Imm1)ˆg→ R ⊗β/Imm1).
Therefore, there is a surjective map
H1(ˆg, S⊗β)→Hr−1τ0!,
where ˆgacts onS viaZ. (SoS are direct sums of finite dimensional representations of ˆg.) Lemma 3.2. For β(e)6∈Z≤0, we have H1(ˆg, S⊗β) = 0. Therefore, Hr−1τ0! = 0.
Proof. Consider the ˆgcoinvariants functor as the composition ofgcoinvariants functor, and theCcoinvariants functor. TheE2terms of the Grothendieck spectral sequence contributing toH1(ˆg, S⊗β) areH1(C, H0(g, S⊗β)) andH0(C, H1(g, S⊗β)). AsS⊗β breaks as direct sums according to weights as ag-module, andCacts on each given weight piece as the weight plus β(e), it is clear that under the above assumption on β(e), both H1(C, H0(g, S⊗β))
andH0(C, H1(g, S⊗β)) are zero.
4. The geometry
LetX be a smooth projective variety andLa very ample line bundle which givesX → P(V), whereV∨= Γ(X,L). Let ˆı: ˆX →V be the closed embedding of the cone ofX into V. LetLbe the totally space ofL∨. Then
iL:L→X×V
is a rank one subbundle of the trivial vector bundle over X with fiber V. The following diagram is commutative
L −−−−→iL X×V
π
y yπ Xˆ −−−−→ˆı V
and the left vertical arrow realizesLas the blow-up of ˆX at the origin. We denote the open immersion
j˚L: ˚L=L−X →L, whereX is regarded as the zero section ofL.
LetL∨ be the dual ofL, i.e., the total space ofL, andj˚L∨ : ˚L∨→L∨be the open subset away from the zero section. The the dual ofiLis the evaluation map
ev:X×V∨→L∨ which sends (x, a) to a(x)∈L∨.
LetiL⊥ :L⊥ →X×V∨ be the orthogonal complement ofL inX×V∨, i.e. the kernel ofev. The projection
L⊥i→L⊥ X×V∨π
∨
→V∨
realizesL⊥ as the universal family of hyperplane sections ofX. We still denote this projec- tion by π∨. Let jU : U =X×V∨−L⊥ →X×V∨ be the complement. Fora∈V∨, the fiberUa ofU →V∨ overaisX−V(fa), wherefa is the section ofLgiven byaandV(fa) is its divisor. Note that the following diagram is Cartesian.
(4.1)
U −−−−→jU X×V∨
ev
y yev
˚L∨ −−−−→j˚L∨ L∨.
5. A formula for τ: CY case
We will complete the proof of Theorem 1.9 in this section.
Leti0:{0} →V be the inclusion of the origin, andj0 : ˚V →V be the open embedding of the complement. Let ˚X= ˆX− {0}. The open inclusion ˚X →Xˆ is still denoted byj0and the closed inclusion ˚X → ˚V is denoted by ˚ı. By specializing (A.4), we have the following important sequence for ˆτ =Four(τ)
(5.1) 0→i0,+H−1i+0τˆ→H0j0,!(ˆτ|˚V)→τˆ→i0,+H0i+0τˆ→0.
First we make a simplification of this sequence.
Lemma 5.1. For β(e)6∈Z≤0,i0,+H0i+0τˆ= 0.
Proof. Assume thatH0i+0ˆτ=kℓ, so thati0,+H0i+0τˆ=δ0ℓ. I.e. there is a surjective map of D-modules ˆτ→δ0ℓ onV. Taking the Fourier transform, we therefore have a surjective map τ → OVℓ∨. Taking the right exact functor Hri!0, i.e., the rth cohomology of the !-fibers at 0∈V∨, we have a surjective mapHrτ0! →kℓ. By Lemma 3.1,ℓ= 0.
As a result, under our assumption
(5.2) 0→i0,+H−1i+0τˆ→H0j0,!(ˆτ|V˚)→τˆ→0.
Let d = dimkH−1i+0ˆτ. Then i0,+H−1i+0τˆ = δ0d. Taking the Fourier transform of this sequence, we therefore obtain
(5.3) 0→ OVd∨ → Four(H0j0,!(ˆτ|V˚))→τ →0.
We next understandFour(H0j0,!(ˆτ|V˚)). Clearly, ˆτ is set-theoretically supported on ˆX.
Note that the Fourier transform ofZ(ξ) +β(ξ) isZ∨(ξ) +β′(ξ), where β′(ξ) =β(ξ)−trZ(ξ).
We have the following lemma.
Lemma 5.2. For L=ωX−1, we have ˆτ|X˚=DX,β˚ ′, where DX,β˚ ′ is the D-module onX˚ as introduced in Lemma A.6.
Proof. Recall that ˆτ is defined as ˆ
τ=DV/DVI( ˆX) +DV(Z∨(ξ) +β′(ξ), ξ∈ˆg).
Let R′ :=DV/DVI( ˆX). Then similar to (2.2), ˆτ = (R′⊗kβ′)⊗ˆgk. Consider the closed embedding ˚X → ˚V. Then as explained in §A, there is a D˚V ×DX˚-bimodule, DV˚←X˚ = DV˚/D˚VI( ˚X)⊗ωX/˚ ˚V. As the relative canonical sheafωX/˚ V˚ is trivial, R′|V˚ =D˚V←X˚. In particular, R′ also admits a right DX˚ structure, and it is clear that the right action of ˆg on R′ is induced from the map ˆg → DX˚. Therefore, R′|X˚ = DX˚ as DX˚-bimodules, and ˆ
τ|X˚=DX,β˚ ′, as claimed.
Note that ˚L≃X˚and therefore ˆτ|X˚≃DX,β˚ ′ can be regarded as a D-module on ˚L, which is naturally (Gm, β′(e))-equivariant. Then
(5.4) H0j0,!(ˆτ|˚V) =H0π!(iL,!j˚L,!DX,β˚ ′).
According to Lemma A.14,
(5.5) Four(H0π!(iL,!j˚L,!DX,β˚ ′)) =H0π∨! FourX(iL,!j˚L,!DX,β˚ ′).
By (A.7),
(5.6) FourX(iL,!j˚L,!DX,β˚ ′) =ev!FourX(j˚L,!DX,β˚ ′)[1−r].
Lemma 5.3. There is a canonical isomorphism
FourX(j˚L,!DX,β˚ ′)≃j˚L∨,+D˚L∨,β.
Proof. Instead of the original formula, we can proveFourX(j˚L∨,+D˚L∨,β)≃j˚L,!DX,β˚ ′. First note that the +-restriction ofFourX(j˚L∨,+D˚L∨,β) alongX →Lis zero. In fact, we have the following more general fact. We keep the notationsL,L∨, j˚Letc. We consider theGm-action onLby homothethies.
Lemma 5.4. Let M be aGm-monodromic holonomic D-module on˚L (see §A for the ter- minology.) Then the+-fiber of FourX(j˚L,!M)along X→L∨ is zero.
Proof. One can check this pointwise on X and by the base change of Fourier transform (Lemma A.14), one can assumeX is a point. Then it follows from Example A.12.
Therefore, by (A.4), it is enough to showFourX(j˚L∨,+D˚L∨,β)|˚L =D˚L,β′. By definition, we can write
FourX(j˚L∨,+D˚L∨,β)|˚L=pL,+(ex⊗D˚L×X˚L∨→˚L∨⊗g(k−β))[1].
In other words,FourX(j˚L∨,+D˚L∨,β)|˚L is calculated as the cokernel of the map ex⊗D˚L×X˚L∨→˚L∨⊗g(k−β)→∇ Ω˚L∨/X⊗O˚L∨ ex⊗D˚L×X˚L∨→˚L∨⊗g(k−β).
Note thatM =D˚L×X˚L∨→˚L∨⊗g(k−β) is a cyclicD˚L×X˚L∨-module, with a canonical generator
“1”. For a local sectionD∈D˚L×X˚L∨, let [D] =D“1” denote the corresponding local section ofM. Note that Ω˚L/Xandexare canonically trivialized asO-modules. Indeed, by definition, the underlyingO-module ofexis the structure sheaf. On the other hand, if locally onX, we choosesa section ofL, regarded as a coordinate function onL, andt the dual coordinate onL∨. Then the 1-formdt/t is independent of the choice and defines the trivialization of Ω˚L∨/X. Therefore, the underlying O-modules of both terms in this complex are M. Then the D-module structure is given as follows: forD∈D˚L×X˚L∨,
t∂t([D]) = [t∂tD] + [tsD], s∂s([D]) = [s∂sD] + [tsD].
As a result,FourX(j˚L∨,+D˚L∨,β)|˚L=M/(t∂t+ts)M.
More explicitly, as O-modules, ex :=m!ex is canonically trivialized, as was said above.
Letf ∈Γ(OV×V∨), then unravelling the definitions, we have the following action of ∂t on the elementf⊗m−1(1)∈ex:
(5.7) ∂t(f⊗m−1(1)) =∂tf ⊗m−1(1) +f ∂t(st)⊗m−1(∂st1) = (∂t+s)f⊗m−1(1) Note that∂st = 1 inex. Therefore, for 1⊗1⊗[D]∈ex⊗M, we have
(5.8) ∇(1⊗1⊗[D]) =dt⊗∂t(1⊗[D]) =dt⊗s⊗[D] +dt⊗1⊗[∂tD] = dt
t ⊗1⊗[tsD+t∂tD]
So one gets the above identity for the Fourier transform.
To proceed, we first consider N = D˚L×X˚L∨→˚L∨/(t∂t+ts)D˚L×X˚L∨→˚L∨, which is a D- module on ˚L. We define a D-module homomorphism D˚L → N, D 7→ D“1”, which we claim is an isomorphism. Indeed, we can assume thatXis affine and the line bundleL→X is trivial. Then it is a direct calculation.
Finally, note that both N and D˚L are right ˆg-modules. The ˆg-module structure on N comes from ˆg → D˚L∨ acting on D˚L×X˚L∨→˚L∨ from the right, and the ˆg-module structure on D˚L comes from ˆg → D˚L∨ acting itself from the right. Under the above isomorphism,
D˚L=N⊗kβ′−β. Now Lemma 5.3 follows.
Lemma 5.5. Assume that β(e)6∈Z≤0. We haved= dimHr−1τ0! = 0.
Proof. The second statement follows from Lemma 3.2. We need to establish the first equality.
For simplicity, let us denoteN :=ev!(D˚L∨,−β)[1−r]. This is a plain D-module onU.
Takingi!0 of (5.2), it is enough to show that
Hri!0H0π∨+N =Hr−1i!0H0π+∨N = 0.
Consider the distinguished triangle
i!0H≤−1π+∨N →i!0π+∨N →i!0H0π∨+N →. The long exact sequence associated to this triangle is
Hr−1i!0π∨+N →Hr−1i!0H0π∨+N →Hri!0H≤−1π+∨N →Hri!0π+∨N →Hri!0H0π∨+N →0.
Note thatUdoes not intersect withX×{0} ⊂X×V∨. Therefore,i!0π+∨N = 0. This implies that Hri!0H0π+∨N = 0, and Hr−1i!0H0π+∨N = Hri!0H≤−1π∨+N. But H≤−1π∨+N sits in cohomological degree≤ −1 andi!0has cohomological amplituder,Hri!0H≤−1π∨+N = 0.
We can now complete the proof of Theorem 1.9.
Proof. Combining (5.4)-(5.6) and Lemma 5.3, we can rewrite (5.2) as (5.9) 0→ OVd∨ →H0π∨+ev!(D˚L∨,β)[1−r]→τ →0.
Theorem 1.9 follows immediately from Lemma 5.5 and the sequence (5.9).
Remark 5.1. Note that explicitly,
ev!(D˚L∨,β)[1−r] =DU/DUTU/˚L∨+DU(ξ+β(ξ), ξ∈ˆg),
whereTU/˚L∨ is the relative tangent sheaf, and ˆgacts onX×V∨ diagonally. In the special caseβ(e) = 1, it reduces toN = (OV∨⊠DX,β)|U as in Theorem 2.1.
6. General type hyperplane sections
LetX be a projectiveG-variety,La very ampleG-linearized invertible sheaf overX, and X →P(V)
the associatedG-equivariant embedding, where V = Γ(X,L)∨. Put W = Γ(X,L ⊗ωX)∨, r= dimV, and s= dimW.
For simplicity, we assume thatL ⊗ωX is base point free. (ThatW 6= 0 actually suffices for the following results.) Thus, we have a morphismX→P(V)×P(W). Let
I ⊂k[V ×W]
be the bihomogeneous ideal defining the image, and letId be the subspace ofI consisting of the degW =delements.
LetG2
mbe the multiplicative group acting onV ×W by homotheties. Let ˆG=G×G2
m, whose Lie algebra is ˆg=g⊕keV ⊕keW, whereeV, eW act respectively on V, W by their identities. We denote byZV : ˆG→GL(V) andZW : ˆG→GL(W) the corresponding group representations, and ZV : ˆg→End(V),ZW : ˆg→End(W) the corresponding Lie algebra representations. In particular, ZV(eV), ZW(eW) are the respective Euler vector fields on V, W. As before, we denote the Fourier transform by b:DV∨×W∨ →DV×W.
Let ˆı: ˆX ⊂V be the cone ofX, defined by the idealI( ˆX). Letβ : ˆg→kbe a Lie algebra homomorphism. We extend the definition of atautological systemgiven in§1 as follows [18].
Definition 6.1. LetτV W =τV W(G, X,L, β) be the cyclicD-module onV∨×W∨given by DV∨×W∨/DV∨×W∨J +DV∨×W∨JW +DV∨×W∨(ZV(ξ) +ZW(ξ) +β(ξ), ξ∈g)ˆ where
J =Ib, JW = \ Sym2W∨.
Note that when β(eW) = 0, we have τV W = τ ⊠OW∨ where τ = τ(G, X,L, β) is as defined in§1. (See first paragraph of §7.)
To apply Definition 6.1 to the geometric problem at hand, we first prove
