Picard-Fuchs Systems Arising from Toric and Flag Varieties

Picard-Fuchs Systems Arising From Toric and Flag Varieties

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A dissertation presented

Chenglong Yu

The Department of Mathematics

in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics

Harvard University Cambridge, Massachusetts

Picard-Fuchs systems arising from toric and ﬂag varieties

Abstract

This thesis studies the Picard-Fuchs systems for families arising as vector bundles zero loci in toric or partial flag varieties, including Riemann-Hilbert type theorems and arithmetic properties of the differential systems. The theory of tautological systems is proposed by Lian, Song and Yau in [51, 52]. These Picard-Fuchs type differential systems arise from the variation of Hodge structures of complete intersections in variety X with large symmetries, generalizing Gel’fand-Kapranov- Zelevinski systems for toric varieties [25]. The form of tautological systems makes it natural to introduces the powerful tool of D-modules into the study of hypersurface family in X. Following this direction, Riemann-Hilbert type theorems are obtained by Bloch, Huang, Lian, Srinivas, Yau and Zhu in [6, 34, 35], including solution rank formula, completeness and geometric interpretation of solution sheaves. In the first part of this thesis, we generalize these results in two aspects, one is the construction of tautological systems for vector bundles, the other is Riemann-Hilbert type theorems in this case. In the second part of the thesis, we examine an explicit description of Jacobian rings for homogenous vector bundles. This can be viewed as a description of the graded quotients of tautological systems with respect to the natural filtered D-module structure. We consider a set of cohomological vanishing conditions that imply such a description, and we verify these

iii conditions for some new cases. We also observe that the method can be directly extended to log homogeneous varieties. We apply the Jacobian ring to study the null varieties of period integrals and their derivatives, generalizing a result in [16] for projective spaces. As an additional application, we prove the Hodge conjecture for very generic hypersurfaces in certain generalized flag varieties. The last part is devoted to the arithmetric properties of fundamental periods near large complex structure limit. Motivated by the work of Candelas, de la Ossa and Rodriguez- Villegas [14], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [67] on higher Hasse-Witt matrices for toric hypersurfaces following Katz’s method of local expansion [38, 39]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz’s congruence relations in terms of expansion coefficients [39] to Dwork’s congruence relations [21] about periods.

iv Contents

1 Introduction 1 1.1 Tautological systems for homogeneous vector bundles ...... 3 1.2 Jacobian ring for homogenous bundles ...... 4 1.3 Hasse-Witt matrices, unit roots and period integrals ...... 5 1.4 Notations ...... 8

2 Tautological systems for homogenous vector bundles 8 2.1 Period integrals for zero loci of vector bundles ...... 8 2.1.1 Calabi-Yau bundles and adjunction formulas ...... 10 2.1.2 Tautological systems ...... 13 2.1.3 Examples ...... 18 2.2 Global residue on P(E∨)...... 19 2.3 Solution rank ...... 20 2.3.1 Lie algebra homology description ...... 20 2.3.2 Perverse sheaves description ...... 21 2.3.3 Irreducible homogeneous vector bundles ...... 23 2.3.4 Complete intersections ...... 24 2.4 Chain integrals in log homogenous varieties ...... 25

3 Jaobian ring for homogenous vector bundles 26 3.1 Line bundles ...... 26 3.2 Vanishing conditions ...... 33 3.3 An application to diﬀerential zeros of period integrals ...... 38 3.4 Zero loci of vector bundle sections ...... 41

v 3.5 Hypersurfaces in log homogenous varieties ...... 46 3.6 Hodge conjecture for very generic hypersurfaces ...... 48

4 Arithmetic properties of period integrals 49 4.1 Introduction ...... 49 4.1.1 Notations ...... 50 4.1.2 Statement of the theorem ...... 52 4.2 Local expansions and Hasse-Witt matrices ...... 54 4.3 Generalized ﬂag vareities ...... 62 4.4 Complete intersections ...... 68 4.5 Frobenius matrices of toric hypersurfaces ...... 70 4.6 Unit root of toric Calabi-Yau hypersurfaces and periods ...... 79 4.7 Frobenius matrices for Calabi-Yau hypersurfaces ...... 81 4.7.1 Unit root of Calabi-Yau hypersurfaces in G/P ...... 82

vi Acknowledgements

First and foremost I would like to thank my advisor Professor Shing-Tung Yau, for his careful guidance, inspiring discussions, constant support and encouragement. Without his generosity in sharing ideas and his invaluable support I would not be able to ﬁnish my thesis work. I would like to thank Professor Bong Lian for sharing his ideas and discussions of all kinds of interesting topics. I thank Professor An Huang for many discussions on our collaborations. I thank Professor Wilfried Schmid for his advises and encouragement. I thank Professor Joe Harris for serving on my thesis committee. I also thank Professor Matt Kerr, Professor Colleen Robles, Professor Mao Sheng and Professor Charles Doran for their interests in my thesis work and helpful conversations. I would also like to thank Professor Gerard van der Geer for his guidance during my last year in Tsinghua and the invitation to Amsterdam, which was a memorable experience for me. I also want to thank Dr. Yu Zhou for encouraging me to switch from chemistry to mathematics. Yau’s students seminar opened my eyes to many interesting topics and has been a great source for me to learn mathematics. I owe a lot to the participants, my colleagues and friends Peter Smillie, Jie Zhou, Teng Fei, Yu-Wei Fan, Jonathan Zhu, Baosen Wu, Tristan Collins, Yi Xie, Takahashi Ryosuke, Dingxin Zhang, Zijian Yao, Meng Guo, Boyu Zhang, Ziliang Che, Yusheng Luo, Eduard Duryev, Koji Shimizu, Zhiwei Zheng for helpful and enjoyable discussions on mathematics. Most computation for the last part of this thesis was done in Xiaolin Shi and Yixiang Mao’s living room. I appreciate their warm and heartfelt hospitality. My thanks also go to the stuﬀ members in the department and Bok center. I really

vii appreciate the help from Jameel Al-Aidroos, Janet Chen, Robin Gottlieb, Yu-Wen Hsu, Oliver Knill, Pamela Pollock and Kate Penner on teaching calculus. I beneﬁted a lot from the support of the teaching team at Harvard. Without the apprentice and coaching system, I would not overcome the language problems and have the conﬁdence to stand in front of my students. I am also grateful for the help from Maureen Armstrong, Pam Brentana, Diana Chen, Susan Gilbert, Larissa Kennedy, Anna Kreslavskaya, Sarah LaBauve, Gaby Leon-Guerrero and Irene Minder. Last but not the least I would like to thank my family for their love. Without their understanding and unconditional support, I would not have the courage to pursue my dream and academic career. I also want to thank Wenyu for all the great memories together. This thesis is dedicated to them.

viii 1 Introduction

Period integrals connect Hodge theory, number theory, mirror symmetry, and many other important areas of math. The study of periods has a long history dating back to Euler, Legendre, Gauss, Abel, Jacobi and Picard in the form of special functions, which are periods of curves. Period integrals first appears in the form of hypergeometric functions. The name hypergeometric functions first appeared in the book Arithmetica Infinitorum (1655) by John Wallis. Legendre first made the connection to geometry of the hypergeometric functions through the theory of elliptic integrals. On the other hand, Euler studied the Euler-Gauss hypergeometric equations. In special case, it is the Picard-Fuchs equation for the Legendre family of elliptic curves. Euler obtained the power series solutions to the Euler-Gauss equations and also wrote then in terms of period integrals. He introduced the important idea that instead of considering at one elliptic integral alone, one should look at the family of the integrals and consider the corresponding differential equations. This is the first example of variation of Hodge structures and Picard-Fuchs systems. The theory of deformations of higher dimensional complex varieties was pioneered by Kodaira and Spencer [42, 41, 43]. The modern study of variation of Hodge structures starts from the work of Griffiths [29], and continues by Deligne [17, 18, 19] and Schmid [60]. In particular, the periods of hypersurfaces in Pn is studied in [29] from the point of view of variation of Hodge structures. Thanks to the relations between pole order filtrations and Hodge filtrations, there is a reduction of pole method to compute the Picard-Fuchs systems for hypersurfaces in Pn. Two natural generalizations of Pn are toric varieties and homogenous Fano varieties. For toric varieties, Gel’fand, Kapranov and Zelevinski [25] found a hypergeometric system that annihilates periods of toric hypersurfaces. More recently, Lian, Song, and Yau

1 [51, 52] established a holonomic differential system called the tautological system for varieties with large symmetry, which generalizes Gel’fand-Kapranov-Zelevinski systems. The machinery of D-module is introduced in tautological systems and opens new ways to study hypersurfaces in homogenous Fano varieties. For instance, in [6, 34], Huang, Lian, Yau, together with their collaborators, obtained general solution rank formulas, which recovered the formula for Gel’fand-Kapranov-Zelevinski system and also proved the completeness of tautological systems for homogenous Fano varieties. More interestingly, new potential can- didates for large complex structure limites appeared naturally as the projected Richardson varieties. For Gel’fand-Kapranov-Zelevinski systems, there are also new results inspired by this approach. For example, in [35], the solutions of Gel’fand-Kapranov-Zelevinski systems are realized as chain integrals or semiperiods. There are already many interesting phenomenon in toric hypersurfaces, especially Calabi- Yau hypersurfaces, in which Gel’fand-Kapranov-Zelevinski system plays an important role. For instance, the famous mirror symmetry between genus zero Gromov-Witten invariants and periods of mirror toric Fano varieties in the sense of Batyrev [4], was proved inde- pendently by Givental [26, 27] and Lian-Liu-Yau [48, 49, 50]. On the B-side of mirror symmetry, the solutions to Gel’fand-Kapranov-Zelevinski systems are explicitly written down as B-series [30] and have interesting number theoretical properties. Similar problems for periods of Calabi-Yau hypersurfaces in homogenous Fano varieties are still open and need further exploration. The theory of tautological systems behaves very nicely for homogenous varieties. For instance, the solutions can all be realized as period integrals along homology classes of the complement of the hypersurfaces. On the other hand the solutions to Gel’fand-Kapranov- Zelevinski systems correspond to period integrals, or Euler integrals on complement inside the affine tori. In some sense, the solutions forget the compactifications of algebraic tori.

2 A conjecture proposed by Hosono-Lian-Yau [30] sets an algorithm to exclude those extra solutions and characterizes periods from B-series, which is called hyperplane conjecture. In the case of projective spaces, the conjecture is proved recently by Lian-Zhu [53]. They arrived at the solution because Pn is both toric and homogenous and the solutions to tautological systems are period integrals. The conjecture in general case is still open. The thesis is mainly about the theory of tautological systems in more general settings and arithmetic properties of period integrals. The main results are contained in [32, 33, 31].

1.1 Tautological systems for homogeneous vector bundles

In Section 2, we extend the theory of tautological systems to vector bundles and study the related Riemann-Hilbert problems. In the construction of tautological systems for complete intersections [51, 52], a key ingredient is a global residue formula on principal bundle. In the principal bundle setting, there is a holomorphic n-form Ω such that for any section

0 −1 Ω f ∈ H (X,KX ), the residue form along Yf = {f = 0} can be written as Res( f ). It is related to the Calabi-Yau structure deﬁned in [52]. See Deﬁnition 2.1 for details. For

Pn P i ˆ instance, the n-form used in is Ω = i(−1) xidx0 ∧ · · · dxi · · · ∧ dxn under homogenous Ω coordinates. For any homogenous polynomial f of degree n+1, the n-form f is well-deﬁned n on P with simple poles along Yf . For vector bundles, we have similar constructions in principal bundle associated with the vector bundle. Straightforward calculations conﬁrm that the following symmetry operators and geometric operators are in the Picard-Fuchs system. See Theorem 2.7. Using Cayley’s trick, we can relate the systems to hypersurfaces in the projectivations of the vector bundle. This enables us to apply similar D-module machinary as [34] to study solution ranks for the new system. Even for complete intersections, the result seems to be

3 new. In this case, the theory is like a mixture of toric and homogenous varieties.

1.2 Jacobian ring for homogenous bundles

In Section 3, we study the Jacobian ring description of variation of Hodge structures arising from the family appeared in Section 2. In particular, this allows us to have an algorithm to compute the Hodge numbers for vector bundle zero loci in flag varieties. In [29], Griffiths introduced a Jacobian ring description for Hodge structures of hypersurfaces in a projective space. This was later generalized to hypersurfaces of sufficiently high degree by Green [28]. Green’s description involves the first prolongation bundle. For complete intersections or more general vector bundle sections, this was also studied by Flenner in connection to the local Torelli problem [23] using spectral sequence argument and Koszul resolution. See [66] or [44] for details of Jacobian ring in terms of the first prolongation bundle for complete intersections or vector bundle sections. In [5], Batyrev and Cox obtained similar descriptions for toric hypersurfaces. It is generalized to complete intersections in toric varieties by Mavlyutov [55]. The corresponding Jacobian ring is explicit in the sense that it only depends on the combinatorial data associated to a toric variety. We study the Jacobian rings for varieties given by zero loci of vector bundles sections from the point of tautological systems. In the case of homogenous vector bundle on flag varieties, the Jacobian ring can be explicitly given in terms of representations of G. See Theorems 4.6 and 3.18. Some of these descriptions may be well-known to experts. For example, in the case of hypersurfaces in irreducible Hermitian symmetric spaces, the description was given by Saito in [59]. We also show similar results for hypersurfaces in log homogenous varieties following Batyrev’s work on toric hypersurfaces [4]. In section 3.3, we apply our results to study tautological systems associated to the period integrals of

4 those families. In Section 3.6, we prove Hodge conjecture for very generic hypersurfaces in certain generalized ﬂag varieties.

1.3 Hasse-Witt matrices, unit roots and period integrals

Section 4 is devoted to some arithmetic properties of period integrals of hypersurfaces in both toric and ﬂag varieties. In particular, we proved a conjecture made by Vlasenko [67] on an algorithm to compute the unit root part of the F -crystal associated to toric hypersurfaces. The relations among Hasse-Witt matrices, unit roots of zeta-functions and period integrals were pioneered in Dwork’s work on the variation of zeta-functions of hypersurfaces [20], [21], [22]. Some well-known examples are Legendre family

y2 = x(x − 1)(x − λ), (1.1) see Example 8 in [37]; and the Dwork family

n+1 n+1 X0 + ··· Xn − (n + 1)tX0 ··· Xn = 0, (1.2) see 2.3.7.18 and 2.3.8 in [36] and also [69]. There is a canonical choice of holomorphic

n n-forms ωλ for these Calabi-Yau families since they are hypersurfaces in P . These families

−(n+1) both have maximal unipotent monodromy at λ = t = 0. The period integral Iγ of ωλ over the invariant cycle γ near λ = 0 is the unique holomorphic solutions to the corresponding Picard-Fuchs equation. On the other hand, these families are deﬁned over Z. We can consider the p-reductions of these families and the Hasse-Witt matrices associated to ωλ. According to a theorem of Igusa-Manin-Katz, they are solutions to Picard-Fuchs

5 equations mod p. We ﬁrst state the relations for Dwork family, see [69]. The period is given by hypergeometric series

  1 , 2 , ··· , n  n+1 n+1 n+1  Iγ = F (λ) = nFn−1  ; λ (1.3) 1, 1, ··· , 1

∞ 1 2 n X n+1 n+1 ··· n+1 = r r r λr. (r!)n r=0

The Hasse-Witt matrix H-Wp is given by the truncation of F (λ)

p−1 1 2 n X n+1 n+1 ··· n+1 H-W (λ) = (p−1)F (λ) = r r r λr. (1.4) p (r!)n r=0

Here (k)F (λ) means the truncation of F (λ) with terms of λ of degree less or equal than k. Let F (λ) g(λ) = ∈ Z [[λ]] (1.5) F (λp) p

Some congruences relations for the coeﬃcients of g implies that g can be viewed an element in lim Z [λ, ((p−1)F (λ))−1]/psZ [λ, ((p−1)F (λ))−1] and it satisﬁes Dwork congruences ←−s→∞ p p

(ps−1) (F (λ)) s g(λ) ≡ s−1 mod p . (1.6) (p −1)(F )(λp)

Especially it is related to Hasse-Witt matrix by

F (λ) ≡ (p−1)F (λ) mod p. (1.7) F (λp)

r n+1 Let q = p and t ∈ Fq. Assume p - n + 1 and t 6= 0, 1 H-Wp(λ) 6= 0. Then there exists exactly one p-adic unit root in the factor of zeta function of Dwork family corresponding

6 to Frobenius action on middle crystalline cohomology. It is given by

g(λˆ)g(λˆp) ··· g(λˆpr−1 ) (1.8) with λˆ being the Teichmüllerlifting under λ → λp. We generalize the above relation to hypersurfaces in toric varieties and partial flag varieties. We first prove the mod p results. Complete intersections are treated in Section 4.4. The key algorithm of Hasse-Witt matrix is a generalization of the result on hypersurfaces in Pn. See Katz’s algorithm 2.7 in [36]. For general hypersurfaces in a Fano variety X, we use the Cartier operator on ambient space to localize the calculation in terms of local expansion similar to [39]. When X is toric variety, the algorithm depends on the toric data associated to X. The algorithm implies generic invertibility of Hasse-Witt matrices for toric hypersurfaces, generalizing Adolphson and Sperber’s result for Pn in [2], see remark 4.16 and corollary 4.17. For generalized flag varieties, Bott-Samelson desingularization is used to reduce the calculation to a similar situation as toric varieties. The affine charts on Bott-Samelson varieties also give an explicit algorithm to calculate the power series expansions of period integrals of hypersurfaces in G/P . The second part of the section applies Katz’s local expansion method [38, 39] to prove a conjecture in [67]. The crystalline cohomology of the hypersurface family has an F -crystal structure. When the Hasse-Witt matrix is invertible, there exists a unit root part of the F -crystal. We consider the p-adic approximation of the Frobenius matrix on the unit root part. Especially, the Hasse-Witt matrix is the Frobenius matrix mod p. In [39], Katz gives a p-adic approximation of the Frobenius matrix in terms of the local expansions of top forms on a formal chart along a section of the family. In [67], Vlasenko constructed a sequence of matrices related to a Laurent polynomial f and proved congruence relations similar to

7 Katz’s algorithm in [39]. The p-adic limit is conjectured to be the Frobenius matrix for hypersurfaces in Pn when f is a homogenous polynomial. According to Corollary 4.12 in section 4.2, the ﬁrst matrix α1 mod p appeared in [67] is the Hasse-Witt matrix for toric hypersurfaces . So it is natural to generalize Vlasenko’s conjecture to toric hypersurfaces. We give a proof of the conjecture in section 4.5.

1.4 Notations

We ﬁrst ﬁx the following notations in the following discussions.

1. Let G be a complex Lie group and g = Lie(G)

2. Let Xn be a smooth projective variety together with action of G.

3. Let Er be a G-equivariant vector bundle on X with rank r.

∨ 0 ∨ 4. Assume V = H (X,E) has basis a1, ··· , aN and dual basis ai .

∨ 5. Let f ∈ V be an section and Yf the zero locus of f. We further assume Yf is smooth with codimension r. See the discussion about smoothness in section 3.4.

2 Tautological systems for homogenous vector bun-

dles

2.1 Period integrals for zero loci of vector bundles

Let Xn be a smooth n-dimensional Fano variety and E be a vector bundle of rank r over X. Denote the dual space of global sections by V = H0(X,E)∨. Assume that any generic

∨ section s ∈ V deﬁnes a nonsingular subvariety Ys = {s = 0} in X with codimension r.

8 (When E is very ample, the zero locus of a generic section is either empty or smooth due to a Bertini-type theorem for vector bundles proved by Cayley’s trick. For example, see Lemma 1.6 in [65]. When it is empty, we can consider the quotient bundle of E by the trivial line bundle.) The dimension of Ys is denoted by d = n − r. Consider the family of nonsingular varieties formed by zero loci of sections in V ∨, denoted by π : Y → B = V ∨ − D, where ∼ −1 D is the discriminant locus. If we further assume det E = KX , the adjunction formula implies that ∼ ∼ KYs = KX ⊗ det E|Ys = OYs . (2.1)

∼ A section s of KX ⊗ det E = OX gives a family of holomorphic top forms Ωs on Ys

corresponding to constant section 1 of OYs , also called the residue of s. We want to consider the period integral Z Ωs, γ ∈ Hn(Ys) (2.2) γ

If E splits as a direct sum of line bundles, the residue map is defined on line bundles and generalized to E by induction. In the nonsplitting case, we can apply the residue formula in the splitting case locally and glue it together to get a global residue formula. The first isomorphism in (2.1) is induced by writing s as line bundle sections and independent on the decomposition. This direct formula turns out to be hard for computations. Following the idea in [51],[52], we use the Calabi-Yau bundle structures to lift the bundles and sections to the principal bundles. Similar computations give us the differential relations from the symmetry of the bundles and geometric constrains from the defining ideal of P(E∨) the projectivation of E∨ in P(V ).

9 2.1.1 Calabi-Yau bundles and adjunction formulas

Motivated by the residue formulas for projective spaces and toric varieties, the notion of Calabi-Yau bundles is introduced in [52] and used to write down an adjunction formula on principal bundles. The canonical sections of holomorphic top forms used in period integral are given by this construction. First we recall the deﬁnition of Calabi-Yau bundles in [52] and adapted it to the local complete intersections.

Deﬁnition 2.1 (Calabi-Yau bundle). Denote H and G to be complex Lie groups. Let p: P → X be a principal H-bundle with G-equivariant action. A Calabi-Yau bundle structure on (X,H) says that the canonical bundle of X is the associated line bundle with character χ: H → C∗. The following short exact sequence

∗ 0 → Ker p∗ → TP → p TX → 0 (2.3) induces an isomorphism ∼ ∗ ∨ KP = p KX ⊗ det(P ×ad(H) h ). (2.4)

∼ Fixing an isomorphism P ×H Cχ = KX , the isomorphism (2.4) implies that KP is a trivial bundle on P and has a section ν which is the tensor product of nonzero elements in Cχ and det h∨. This holomorphic top form satisﬁes that

∗ −1 h (ν) = χ(h)χh (h)ν, (2.5)

where χh is the character of H on det h by adjoint action. The tuple (P, H, ν, χ) satisfying (2.5) is called a Calabi-Yau bundle.

∼ Conversely, any section ν satisfying (2.5) determines an isomorphism P ×H Cχ = KX .

10 Since the only line bundle automorphism of KX → X ﬁxing X is rescaling when X is compact, such ν is determined up to rescaling (Theorem 3.12 in [52]). So the equivariant action of G on P → X changes ν according to a character β−1 of G. We say the Calabi-Yau bundle is (G, β−1)-equivariant.

Example 2.2. Let X be CPd+1 and P be Cd+2\{0} with nature actions of G = GL(d+2, C)

∗ −1 and H = C . The volume form is ν = dx0 ∧ · · · ∧ dxd+1. The character β = det g for any g ∈ G.

−1 When E is the line bundle KX , the following is the residue formula for Calabi-Yau bundles:

Theorem 2.3 ([52], Theorem 4.1). If (P, H, ν, χ) is a Calabi-Yau bundle over a Fano

d manifold X, the middle dimensional variation of Hodge structure R π∗(C) associated with the family π : Y → B of Calabi-Yau hypersurfaces has a canonical section of the form

ι ··· ι ν ω = Res ξ1 ξm . (2.6) f

Here ξ1, ··· , ξm are independent vector ﬁelds generating the distribution of H-action on P , C C ∼ −1 and f : B × P → is the function representing the universal section of P ×H χ−1 = KX .

When E is a direct sum of line bundles associated to characters of H, the residue formula is similar to (2.6) by induction.

Deﬁnition 2.4. When E is a vector bundle associated with representation ρ: H → GL(W ), we can also construct a residue formula as follows. Under the assumption that

E = P ×H W , a sections of E is a (H, ρ)-equivariant map f : P → W , i.e. f(p·h) = h·f(p).

1 r Choose a basis e1, ··· , er for W , then f = f e1 + ··· + f er. Assume f deﬁnes a smooth

11 Calabi-Yau subvariety Yf with codimsion r in X. We have the following residue formula:

ν ω = ι ··· ι Res (2.7) ξ1 ξm f 1 ··· f r

The residue deﬁnes a holomorphic top form on the zero locus of f i = 0 on P , which is the restriction of the principal bundle on the Calabi-Yau subvariety. After contracting with

ξ1, ··· , ξm, the holomorphic d-form w is invariant under the action of H and vanishes for the vertical distribution, hence it deﬁnes a d-form on Yf .

The vector bundle E can be associated with diﬀerent principal bundles. The residue construction is canonical in the following sense.

Proposition 2.5. For diﬀerent choice of principal bundles P or basis e1, ··· , er, the residue form ω is unique up to a scalar which is independent of f.

Proof. Firstly, for diﬀerent choices of basis of W , the functions fi are changed by a linear transformation. Hence the denominator of the residue formula is changed by the determinant of the linear transformation along the common zero locus of fi. So the residue is changed by a scalar. Secondly, we prove the independence on the choice of principal bundles. Let P 0 be the frame bundle of E. We only need to compare any P with P 0. The frame bundle can be

0 0 constructed by P = P ×H GL(W ). So we have a quotient map from P × GL(W ) to P . Especially we have a principal bundle map

c: P → P 0, (2.8) which is equivariant under the actions of H and GL(W ) on both sides related by ρ: H →

12 GL(W ). Then we have a morphism between the exact sequence

∗ 0 Ker p∗ TP p TX 0 (2.9)

0 0 0∗ 0 Ker p ∗ TP p TX 0

∗ ∼ Notice that one exact sequence gives an isomorphism p KX = KP ⊗ det(P ×ad(H) h) and

induces the form ν and ιξ1 ··· ιξm ν. So we have

∗ 0 ι ··· ι ν = c (ι 0 ··· ι 0 ν ). (2.10) ξ1 ξm ξ1 ξr

Furthermore, the functions fi deﬁned on P are also pull back of the corresponding functions on P 0. So we have the same residue formula.

With the canonical choice of ω, the period integral for the family is deﬁned to be

Z Πγ = ω. (2.11) γ

Here γ is a local horizontal section of the d-th homology group of the family. The period integrals are local holomorphic functions on the base B and generates a subsheaf of OB called period sheaf.

2.1.2 Tautological systems

In order to study the period sheaf, we look for diﬀerential operators which annihilate the period integrals. In [51] and [52], tautological systems are the introduced and the solution sheaves contain the period sheaves. When H is the complex torus and X toric variety, tautological systems are the Gel’fand-Kapranov-Zelevinski systems and extended Gel’fand-

13 Kapranov-Zelevinski systems. When X is homogenous variety, tautological systems provide new interesting D-modules. The notion of tautological system also provides convenient ways to apply D-module theory to study the solution sheaves and period sheaves. The regularity and holonomicity are discussed in [51], [52]. The Riemann-Hilbert problems and geometric realizations are discussed in [6], [34], [35]. The diﬀerential operators in tautological systems come from two sources: one from symmetry group G called symmetry operators and the other from the deﬁning ideal of

−1 X in the linear system |KX | called geometric constrains. In this section we have similar

∨ ∨ ∨ constructions. First we ﬁx a basis a1, ··· , am of V and dual basis a1 , ··· , am of V . Viewing

∨ ∨ ∨ ai as coordinates on V , the universal section of E is denoted by f = a1a1 + ··· + amam.

∨ According to the discussion in last section, the section ai corresponds to a map fi : P → W

1 r and a tuple of functions fi = (fi , ··· , fi ). Then the residue formula has the following form:

ν ω = ιξ1 ··· ιξm Res P 1 P r . (2.12) ( i aifi ) ··· ( i aifi )

Considering the action of G on V , we have a Lie algebra representation

Z : g → End V (2.13)

P ∨ ∨ For any x ∈ g, we denote Z(x) = i,j xjiajai and the dual representation Z (x) = P ∨ i,j −xijai aj. From Proposition 2.5, the uniqueness of residue formula, we know that the G action changes period integral according to a character of G. So the ﬁrst order diﬀerential op-

P ∂ erators xijai + β(x) annihilate the period integral. More speciﬁcally, consider the ij ∂aj action of the one parameter group exp(tx) acting on the period integral. From Cartan’s

14 formula,

ν ν Lxιξ1 ··· ιξm P 1 P r = d(ιxιξ1 ··· ιξm P 1 P r ). (2.14) ( i aifi ) ··· ( i aifi ) ( i aifi ) ··· ( i aifi )

So we have Z ν ιξ1 ··· ιξm Res(Lx P 1 P r ) = 0. (2.15) γ ( i aifi ) ··· ( i aifi ) The Lie derivative is

ν Lx P 1 P r (2.16) ( i aifi ) ··· ( i aifi ) P k ν X i,j aixijfj ν = −β(x) + (2.17) P 1 P r P 1 P k 2 P r ( aif ) ··· ( aif ) ( a f ) ··· ( a f ) ··· ( a f ) i i i i k i i i i i i i i i X ∂ ν = (−β(x) − x a ) . (2.18) ij i ∂a (P a f 1) ··· (P a f r) ij j i i i i i i

So we have X ∂ (β(x) + x a )Π = 0. (2.19) ij i ∂a γ ij j The geometric constrains arise from the following observation. Consider the ﬁrst order diﬀerential operators:

k ∂ ν X fj ν = − ; (2.20) P 1 P r P 1 P k 2 P r ∂aj ( aif ) ··· ( aif ) ( a f ) ··· ( a f ) ··· ( a f ) i i i i k i i i i i i i i i the second order diﬀerential operators:

2 ∂ ν X a b 1 r = Pabfl fj ν (2.21) ∂al∂aj f ··· f a,b

Here Pab are rational functions of f1, ··· , fr, not depending on j, l. By induction, we will

15 have similar formula for higher order diﬀerential operators ∂i1,··· ,is and the P coeﬃcients are independent of the multi-index i1, ··· , is.Notice that we can switch the order of l and

∨ ∨ j. So we have Pab = Pba. On the other hand, consider the product of al and aj in H0(X, Sym2 E). The symmetric product Sym2 E is associated with the symmetric product

∨ ∨ 2 of the representation of ρ. Hence al · ak can be viewed as a map P → Sym W

X a X a X a a 2 X a b a b ( fl ea) · ( fj eb) = fl fj ea + (fl fj + fj fl )eaeb. (2.22) a b a a

Consider the elements in the kernel of the map

H0(X,E) ⊗ H0(X,E) → H0(X, Sym2 E). (2.23)

∨ 2 The Fourier transform of these elements annihilate period integral. For example if (a1 ) −

∨ ∨ a2 a3 = 0, then

2 2 ∂ ∂ ν X a 2 a a X a b a b a b ( 2 − ) 1 r = ( Paa((f1 ) − f2 f3 ) + Pab(2f1 f1 + f2 f3 + f3 f2 ))ν = 0. ∂a1 ∂a2∂a3 f ··· f a a

a ∨ 2 ∨ ∨ This is because the terms of fi are the coeﬃcients of (a1 ) − a2 a3 written under the basis

2 ea, eaeb. In order to describe the geometric origin of the diﬀerential operators above, we need the well-known facts relating the vector bundle E and hyperplane line bundle O(1) on P(E∨).

Proposition 2.6. Assume E → X is a holomorphic vector bundle on complex manifold X and O(1) → P(E∨) is the hyperplane bundle on the projectivation of E∨. There is a

16 canonical ring isomorphism

0 k ∼ 0 ∨ ⊕k H (X, Sym (E)) = ⊕kH (P(E ), O(k)). (2.25)

Proof. It follows from Leray-Hirsch spectral sequence computing H0(P(E∨), O(k)).

The above identiﬁcation of V ∨ with H0(P(E∨), O(1)) gives a map P(E∨) → P(V ) by O(1) when |O(1)| is base-point free. Consider the ideal determined by image of this map I(P(E∨),V ), which is the kernel of the map

k ∨ 0 k ⊕k Sym (V ) → ⊕kH (X, Sym (E)) (2.26)

With the discussion above, we collect all the diﬀerential operators in the following theorem.

Theorem 2.7. The period integral Πγ satisﬁes the following system of diﬀerential equations:

∨ Q(∂a)Πγ = 0 (Q ∈ I(P(E ),V )) (2.27)

(Zx + β(x))Πγ = 0 (x ∈ g) (2.28) X ∂ ( a + r)Π = 0. (2.29) i ∂a γ i i

The last operator is called Euler operator and comes from ω being homogenous of degree

−r with respect to ai. We can also view Euler operator as symmetry operator. Consider the frame bundle of E with structure group H = GL(r). It has a symmetry G = C∗ acting as the center of H. The symmetry operator of G is the Euler operator. We call the diﬀerential system in Theorem 2.7 tautological system for (X,E,H,G). It’s

17 the same as the cyclic D-module τ(G, P(E∨), O(−1), βˆ) deﬁned in [51] [52] by

∨ ˆ τ = DV ∨ /DV ∨ (J(P(E )) + Z(x) + β(x), x ∈ gˆ). (2.30)

∨ ∨ ˆ ∗ Here J(P(E )) = {Q(∂a)|Q ∈ I(P(E ))}, G = G × C with Lie algebra gˆ = g ⊕ Ce and βˆ = (β, r). We can apply the holonomicity criterion for tautological system in [51],[52].

Theorem 2.8. If the induced action of G on P(E∨) has ﬁnite orbits, the corresponding tautological system τ is regular holonomic.

2.1.3 Examples

r Example 2.9 (Complete intersections). When E = ⊕1Li is a direct sum of very ample line bundles, the above system recovers the tautological system for complete intersections in [52]. This case is equivalent to say that the structure group of E is reduced to the complex torus (C∗)r. So we have symmetry group (C∗)r acting on the ﬁbers of E. This

ˆ 0 ∨ gives the usual Euler operators in [52]. Let Xi be the cone of X inside Vi = H (X,Li) P ∨ r under the linear system of Li. The cone of (E ) inside V = ⊕i=1Vi is ﬁbered product ˆ Xi over X. So the geometric constrains are the same as [52]. Assume X is a G-variety

∨ consisting of ﬁnite G-orbits and Li are G-homogenous bundles. Then P(E ) admits an action of G˜ = G × (C∗)r−1 with ﬁnite orbits. This is the same holonomicity criterion as [52] for complete intersections.

Example 2.10 (Homogeneous varieties). Let G be a semisimple complex Lie group and X = G/P is a generalized ﬂag variety quotient by a parabolic subgroup P . This forms a principal P -bundle over X. We assume E to be a homogenous vector bundle from a representation of P and the action of G on P(E∨) is transitive. Then the projectivation

18 of P(E∨) is also a generalized flag variety for a parabolic subgroup P 0 ⊂ P . Hence the G-action on P(E∨) is transitive. If O(1) is very ample on P(E∨), the defining ideal of P(E∨) in P(V ) is given by the Kostant-Lichtenstein quadratic relations. Furthermore, any character of G is trivial, hence β is zero. So the differential system is regular holonomic and explicitly given in this case.

2.2 Global residue on P(E∨)

The residue formula in Deﬁnition 2.4 is motivated by residue formulas for hypersurfaces. It is locally the same as complete intersections. In this section, we introduce another approach more directly related to the geometry of P(E∨) by Cayley’s trick. Let P = P (E∨) be the projectivation of E∨ and O(1) be the hyperplane section bundle on P. The projection map is denoted by π : P → X. From now on, we assume E is ample and by deﬁnition, is equivalent to O(1) being ample. We collect the propositions relating the geometry of X and P in the following. Proposition 2.11 and 2.13 are from [66],[44] and [55]. Proposition 3.15 is from Corollary 4.9 in [44].

Proposition 2.11. 1. There is a natural isomorphism H0(X,E) ∼= H0(P, O(1)). The corresponding section in H0(P, O(1)) is also denoted by f.

2. Let Y˜ be the zero locus of f in P. Then Y˜ is smooth if and only if Y is smooth with codimension r or empty.

∼ ∗ 3. There is an natural isomorphism KP = π (KX ⊗ det E) ⊗ O(−r)

From now on, we assume Y is smooth with codimesion r ≥ 2.

n−r n−r Deﬁnition 2.12. The variable cohomology Hvar (Y ) is deﬁned to be cokernel of H (X) → Hn−r(Y ).

19 Proposition 2.13. There is an isomorphism

n+r−1 P ˜ ∼ n+r−2 ˜ ∼ n−r H ( − Y ) = Hvar (Y )(−1) = Hvar (Y )(−r) (2.31)

∼ −1 We now consider the Calabi-Yau case, equivalently det E = KX , for simplicity. Then we have the vanishings in the Hodge ﬁltration F n+r−1−k = 0 for k < r−1 and isomorphisms

0 0 n n+r−1 ˜ n−r n−r H (P, OP) → H (P,KP ⊗ O(r)) → F H (P − Yf ) → F H (Yf ). (2.32)

Proposition 2.14. The constant function 1 is sent to holomorphic top form ωf on Yf via this sequence of isomorphisms. Then ωf is the same as ω in Deﬁnition 2.4.

Consider the principle bundle adjunction formula for base space P. Let (P, H, ν, χ) is

P 0 P Ω a Calabi-Yau bundle over . The image of 1 in H ( ,KP ⊗ O(r)) has the form f r on principle bundle P . If we write f as universal section, then similar calculation can recover the diﬀerential operators in Theorem 2.7.

2.3 Solution rank

Now we discuss the solution rank for the system. There are two versions of solution rank formula for hypersurfaces. One is in terms of Lie algebra homology, see [6]. One is in terms of perverse sheaves on X, see [34]. Here we have similar description for zero loci of vector bundle sections.

2.3.1 Lie algebra homology description

We ﬁx some notations. Let R = C[V ]/I(P(E∨)) be the coordinate ring of P. Let Z : gˆ → End(V ) be the extended representation by e acting as identity. We extend the character

20 β : gˆ → C by assigning β(e) = r.

f ∼ Deﬁnition 2.15. We deﬁne DV ∨ -module structure on R[a]e = R[a1, ··· aN ] as follows.

The functions ai acts as left multiplication on R[a1, ··· aN ]. The action of ∂ai on R[a1, ··· aN ]

∨ is ∂ai + ai .

Then we have the following DV ∨ -module isomorphism.

Theorem 2.16. There is a canonical isomorphism of DV ∨ -module

τ ∼= R[a]ef /Z∨(gˆ)R[a]ef (2.33)

This leads to the Lie algebra homology description of (classical) solution sheaf

Corollary 2.17. If the action of G on P(E∨) has ﬁnitely many orbits, then the stalk of the solution sheaf at b ∈ V ∨ is

∼ f(b) ∨ f(b) ∼ f(b) sol(τ) = HomD(Re /Z (gˆ)Re , Ob) = H0(gˆ, Re ) (2.34)

2.3.2 Perverse sheaves description

We follow the notations in [34].

1. Let L∨ be the total space of O(1) and ˚L∨ the complement of the zero section.

2. Let ev : V ∨ × P → L∨, (a, x) 7→ a(x) be the evaluation map.

3. Assume L⊥ = ker(ev) and U = V ∨ × P − L⊥. Let π : U → V ∨. Notice that U is the complement of the zero locus of the universal section.

4. Let DP,β = (DP ⊗ kβ) ⊗Ug k, where kβ is the 1-dimensional g-module with character β and k is the trivial g-character.

21 5. Let N = OV ∨ DP,β.

We have the following theorem. See Theorem 2.1 in [34].

Theorem 2.18. There is a canonical isomorphism

∼ 0 ∨ τ = H π+(N |U ) (2.35)

Proof. The proof follows the same arguments of Theorem 2.1 in [34]. The only diﬀerence

∨ −r is that the universal section f = ai ⊗ ai deﬁnes a trivialization of OV ∨ O(1). So f

−1 instead of f deﬁnes a nonzero section of OV ∨ ωP. Hence we have isomorphism

−r ∼ OU f = ωU/V ∨ . (2.36)

Let R := OV ∨ /OV ∨ J(P) be a DV ∨ × gˆ-module. Then from Theorem 2.16, we have

∼ τ = (R ⊗ kβ) ⊗ˆg k. (2.37)

The DV ∨ × gˆ-morphism in the technical Lemma 2.6 is now changed to

−r φ: R ⊗ kβ → OU f (2.38) by setting (−1)l(l + r)! φ(a ⊗ b) = a ⊗ b (2.39) f l+r

Here we identify R with OV ∨ ⊗ S and S is the graded coordinate ring of (P, O(1)). The

22 element b ∈ S has degree l. Since β(e) = r, we have an isomorphism induced by φ

∼ ∼ −r τ = (R ⊗ kβ) ⊗ˆg k = (OU f ) ⊗g k. (2.40)

A direct corollary is the following

Corollary 2.19. If β(g) = 0, there is a canonical surjective map

0 ∨ τ → H π+OU . (2.41)

In terms of period integral, we have an injective map

Hn+r−1(X − Yb) → Hom(τ, OV ∨,b) (2.42) given by Z Ω γ 7→ r (2.43) γ f

We have similar solution rank formula. We assume G-action on P has ﬁnitely many

an an P orbits. Let F = Sol(DP,β) = RHomDan (DP,β, OP ) be a perverse sheaf on .

∨ 0 Corollary 2.20. Let b ∈ V . Then the solution rank of τ at b is dim Hc (Ub, F|Ub ).

Now we apply the solution rank formulas to diﬀerent cases.

2.3.3 Irreducible homogeneous vector bundles

In this subsection, we assume X is homogeneous G-variety and the lifted G-action on P is also transitive. In other words, we have X = G/P and P = G/P 0 with P/P 0 ∼= Pr−1. Then

23 we have the following corollary

Corollary 2.21. If β(g) = 0, then the solution rank of τ at point b ∈ V ∨ is given by

n−r dim Hvar (Yb).

Proof. The solution sheaf in this case is F ∼= C[n + r − 1]. So the solution rank is

0 C n−r dim Hc (Ub, [n + r − 1]) = dim Hn+r−1(Ub) = dim Hvar (Yb) (2.44)

Example 2.22. Let X = G(k, l) be Grassmannian and F be the tautological bundle of

∼ ∨ l−1 ∼ −1 rank k. Then E = F ⊗ O( k ) is an ample vector bundle with det E = KX . The corresponding P is homogenous under the action of SL(l + 1)

2.3.4 Complete intersections

We ﬁrst ﬁx some assumptions.

1. Let E split as direct sum of homogenous G-line bundles L1, ··· ,Lr.

2. Let G˜ = G × (C∗)r−1 acting on P as Example 2.9.

3. Let G-action of X have ﬁnitely many orbits. Then G˜-action on P has ﬁnitely many orbits.

4. We further assume β(g) = 0. This implies β(g˜) = 0. In this case, there are always some invariant divisor on P.

r−1 Let [t1, ··· , tr] be the local homogenous coordinates on P in P -direction. Each ti comes ∨ P from the coordinate on Li . Then ti = 0 deﬁnes globally a divisor on . We denote it by

24 ˚ ˚ Di. The complement of ∪iDi is denoted by P. Let j : P → P be the open embedding. We treat the following two special cases with X being homogenous or toric. Let X = G/P be a homogenous G-variety. Then we have the isomorphism

∼ ! Proposition 2.23. DP,β = j!j DP,β

Proof. The proof follows from the proof in [35]. The key observation is that Lemma 3.2 and Corollary 3.3 only uses the condition that toric divisors have normal crossing singularities and the ambient space is a log homogenous variety with respect to these divisors. See the section about chain integral in log homogenous varieties.

So we have the following description of solution rank

Theorem 2.24. The solution rank at point b ∈ V ∨ is given by

˜ ˜ Hn+r−1(P − Yb, (P − Yb) ∩ (∪iDi)) (2.45)

This theorem is not satisfying because the ﬁnal cohomology is not directly related to

X. Let Y1, ··· ,Yr be the zero locus of the Li component of section sb. From the geometric realization of some solutions as period integral as rational forms along the cycles in the complement of Y1 ∪ · · · ∪ Yr, we have the following conjecture:

Conjecture 2.25. There is a natural isomorphism of solution sheave as period integrals

∼ HomDV ∨ (τV , O)|b = Hn(X − (Y1 ∪ · · · ∪ Yr)) (2.46)

2.4 Chain integrals in log homogenous varieties

Let X be a complex variety with normal crossing divisor D. The log D tangent bundle

TX (− log D) is a subsheaf of TX deﬁned as follows. If x1, ··· , xn is the local coordinate of

25 X and D is the hyperplanes deﬁned by z1 = 0, ··· , zr = 0, then the generating sections of TX (− log D) are z1∂1, ··· , zr∂r, ∂zr+1, ··· , zn. Then we say X is log homogenous if

0 TX (− log D) is globally generated. Let g be H (X,TX (− log D) and G be the corresponding Lie group. Then X is an G-variety. Let L be a G-equivariant line bundle deﬁned on X. We assume L is very ample and L+KX is base point free. Then the period integrals of sections

∨ 0 ∨ 0 in W = H (X,L + KX )on hypersurface families cut out by V = H (X,L) satisfies the tautological systems τVW with character β = 0. See [52] and [34]. Then τ is holonomic because G-action on X is stratified by D with finite orbits. See [8] for discussion of log homogenous varieties.

We consider the solution rank of τVW in this case. Following the same proof in [35], we have the following description of solution rank of τVW .

Theorem 2.26. There is an natural isomorphism

∼ HomDV ∨×W ∨ (τVW , O)|(a, b) = Hn(X − Ya, (X − Ya) ∩ (∪D)) (2.47) given by period integral.

3 Jaobian ring for homogenous vector bundles

In this section, we examine an explicit description of Jacobian rings for homogenous vector bundles.

3.1 Line bundles

In this subsection, we consider the case that E is an line bundle L. The Hodge structure of Yf is determined by ambient space X by Lefschetz hyperplane theorem except the

26 n−1 middle dimension. Let Uf = X − Yf and Hvan (Y ) is the kernel of Gysin morphism Hn−1(Y ) → Hn+1(X). Hodge structures of Y and U are related by the Gysin sequence

n n n−1 0 → Hprim(X) → H (U) → Hvar (Y )(−1) → 0 (3.1)

n n−2 n Here Hprim(X) is the corker of the Gysin morphism H (Y ) → H (X). We will consider X to be a Fano variety. So Hn(X, O) = 0 and F 0Hn(U) = F 1Hn(U).

0 k Deﬁnition 3.1. Let R be the graded ring R = ⊕k≥0H (X,L ). The generalized Jacobian ideal J is the graded ideal generated by f, LZ f for Z ∈ g. Here the Lie derivative LZ f is

0 0 k+1 from the natural g-action on H (X,L). Then M = ⊕k≥0H (X,KX ⊗ L ) is a graded R-module.

This deﬁnition gives Green’s Jacobian ring [28] under suitable vanishing conditions, see Proposition 3.9. We have the following proposition.

Proposition 3.2. There is an natural morphism from

(M/JM)k → F n−kHn(U)/F n−k+1Hn(U).

It is compatible with multiplication map H0(X,L) × (M/JM)k → (M/JM)k+1 and the Higgs ﬁeld from Gauss-Manin connection

This is a direct result from Griﬃths’ study of rational forms [29].

Proposition 3.3. There is a natural map

0 n n−k n αk : H (X, ΩX ((k + 1)Y )) → F H (U) (3.2)

0 n−1 with dH (X, ΩX (kY )) contained in the kernel.

27 Proof of Proposition 3.2. See [68], Theorem 6.5 for the construction of αk. Letα ¯k be the induced map to the quotient F n−kHn(U)/F n−k+1Hn(U). Now we prove that JM k−1 ⊂

k kerα ¯k. The elements of M can be realized as rational forms on X as follows. Consider a

G-equivariant principal bundle P over X such that L and KX are associated bundles. Then 0 ˜ 0 k f ∈ H (X,L) and P ∈ H (X,KX ⊗ L ) are viewed as functions on P. There exists an

ΩP˜ n-form Ω deﬁned on P such that f k+1 is the pull-back of a rational n-form on X. See [52] for the construction of Ω from principal bundle version of adjunction formula. We identify Ωn−1

0 k 0 k ∼ with TX ⊗ KX and consider the morphism g ⊗ H (X,KX ⊗ L ) → H (X,TX ⊗ L ⊗ KX ) =

0 n−1 H (X, ΩX (kL)) given by X X ΩTi Z ⊗ T 7→ ι (3.3) i i Zi f k i i

0 k Here Zi is a basis of g, Ti ∈ H (X,KX ⊗ L ) and ι is the contraction between vector ﬁelds with forms. Let γ ∈ H0(X, Ωn−1(kY )) be such a diﬀerential form, then

X ΩTi X ΩTi dγ = d ι = L (3.4) Zi f k Zi f k i i by Cartan’s formula. Assume P˜ satisﬁes

Ω(P˜ − fQ) = dγ (3.5) f k+1 for some Q ∈ H0(X, Ωn(kY )). Then we have

˜ Ω(P − fQ) X ΩTi = L (3.6) f k+1 Zi f k X (β(Zi)Ti + LZ Ti)f − kTiLZ f = Ω( i i ) (3.7) f k+1 i

Here the G-action lifts to P and the corresponding character on the n-form Ω is denoted

28 ˜ P P ˜ by β. So P = (Q + i β(Zi)Ti + LZi Ti)f − k i TiLZi f. Since such P ∈ kerα ¯k and Q can

0 k k−1 be any section in H (X,KX ⊗ L ), we have JM ⊂ kerα ¯k. Hence this induces a map (M/JM)k → F n−kHn(U)/F n−k+1Hn(U).

ΩP The Gauss-Manin connection on the universal family is given by diﬀerentiating f k+1 .

Now we discuss the case X being homogenous and suﬃcient conditions for the above map being isomorphism. Let X = G/P be a generalized ﬂag variety. We consider the following two vanishing conditions for the line bundle L.

p q l H (X, ΩX ⊗ L ) = 0 with p > 0, q ≥ 0, l ≥ 1 (3.8)

1 k H (X, (G ×adP p) ⊗ L ⊗ KX ) = 0 for k ≥ 1 (3.9)

We postpone the discussion about the conditions to next section.

Theorem 3.4. Let 0 ≤ k ≤ n − 1 and assume L satiesﬁes conditions (3.9) and (3.8) for (p, q, l) in the following range {1 ≤ p ≤ k, q = n − p, l = k − p + 1} ∪ {1 ≤ p ≤ k − 1, q = n − p − 1, l = k − p} ∪ {1 ≤ p ≤ k − 1, q = n − p, l = k − p}, then the map

(M/JM)k → F n−kHn(U)/F n−k+1Hn(U) is an isomorphism.

Proof. The proof follows the argument in Theorem 6.5 [68]. Under condition (3.8) with {1 ≤ p ≤ k, q = n − p, l = k − p + 1} ∪ {1 ≤ p ≤ k − 1, q = n − p − 1, l = k − p}, the map

0 n 0 n−1 n−k n αk : H (X, ΩX ((k + 1)Y ))/dH (X, ΩX (kY )) → F H (U) (3.10)

29 is an isomorphism for 1 ≤ k ≤ n − 1. Consider the exact sequence

0 → p → g → g/p → 0. (3.11)

According to condition (3.9), the short exact sequence of vector bundles induces a surjective

0 k 0 n−1 0 n−1 morphism g ⊗ H (X,KX ⊗ L ) → H (X, ΩX (kL)). So any γ ∈ H (X, ΩX (kL)) has

0 n n−k+1 n the form in (3.4). The map αk−1 : H (X, ΩX (kY )) → F H (U) is surjective under condition (3.8) for 1 ≤ p ≤ k − 1, q = n − p, l = k − p. So the kernel ofα ¯k is given by JM k−1.

0 Remark 3.5. If L and KX are multiples of the same ample line bundle L , then R and M can be embedded in the coordinate ring R0 = ⊕Hk(X, (L0)k). We can deﬁne the Jacobian ideal

0 0 k J to be the ideal generated by f, LZ f in R . Then the degree-k summand (M/JM) is the corresponding summand in R0/J. When X = Pn, we can take L0 to be O(1) bundle. Let L = O(d). The vanishing conditions are satisﬁed except kd = n for condition 3.9. When kd 6= n, this is the same Jabocian ring description for Hodge structures of hypersurfaces in

∂f projective spaces. More specifically, the Jacobian ideal defined here are generated by xj ∂xi if we view f as polynomial of homogenous coordinates [x0, ··· , xn]. The usual Jacobian ideal are generated by ∂f . When kd 6= n, the corresponding degree part in the usual ∂xi P ∂f Jacobian ring are quotients of elements in the form gi with gi homogenous with i ∂xi ∂f degree greater than 0, which are the same degree part of ideal generated by xj . ∂xi Remark 3.6. For hypersurfaces in irreducible Hermitian symmetric spaces, the Jacobian ring defined here is already given by Saito in [59]. See Lemma 4.1.12 [59]. The vanishing conditions required here is slightly weaker than the ones used in [59]. This is because the approach in Theorem 6.5 [68] used the exactness of log de-Rham complex in degree ≥ 2

k n−k,c with smooth Y . So the cohomology of closed log forms H (X, ΩX (log Y )) computes the

30 hypercohomology of

n−k n−k+1 0 → ΩX (log Y ) → ΩX (log Y ) → · · · .

n−k,c ∼ n−k,c On the other hand, ΩX (log Y ) = ΩX (Y ) has a resolution

n−k n−k+1 0 → ΩX (Y ) → ΩX (2Y ) → · · · .

The proof in [59] used the exact sequence

n−k n−k d n−k+1 n−k+1 d 0 → ΩX (log Y ) → ΩX (Y ) −→ ΩX (2Y )/ΩX (Y ) −→

d n n ··· −→ ΩX ((k + 1)Y )/ΩX (kY ) → 0

The vanishing

k−l n−k+l n−k+l H (X, ΩX ((l + 1)Y )/ΩX (lY )) = 0 and

k−l−1 n−k+l n−k+l H (X, ΩX ((l + 1)Y )/ΩX (lY )) = 0 requires more vanishing conditions (3.8). See Section 3.5 for the application of this exact sequence in the log homogenous case, which follows the same argument for hypersurfaces in algebraic tori [4].

0 1 Remark 3.7. The Kodaira-Spencer map H (X,L) → H (Y,TY ) has kernel equal to J if

1 −1 2 −1 H (X,TX ⊗ L ) = 0, and is surjective if H (X,TX ⊗ L ) = 0. In this case, the multiplication map H0(X,L)/J × (M/JM)k → (M/JM)k+1 gives the Higgs ﬁeld in universal deformation family of Y . For example this holds for Grassmannians G(a, b) with b ≥ 5 with any ample line bundle L.

31 n Remark 3.8. In order to get similar description as P and coordinate ring C[a1, ··· an+1], we consider M to be the total space of H-principal bundle over X with G-equivariant action. The character χ: H → C∗ is associated with the line bundle L. In many cases, the total space M is embedded in affine space M¯ as Zariski open set and global sections of structure sheaf of M is extended to M¯ . Assume that the G-action also extends to M¯ . For example, when X = G(a, b) is the Grassmannian, we can take M to be the Stiefel manifold and M¯ the affine space Aab. The coordinate ring R is identified with C[M¯ ]H,χ, which is the functions that are equivariant under the H action by characters mχ. The basis can be given by standard monomials. Then the sections f and LZ fare identified with elements in C[M¯ ] similar as the Pn case.

Now we discuss the relation to Green’s Jacobian ring [28]. See Saito’s identification of two definitions for Hermitian symmetric spaces, Lemma 3.2.3 [59]. First we recall Green’s definition of Jacobian ring. Let ΣL be the first prolongation bundle. It is the bundle of first order differential operators on L. The differentiation of f gives a section df ∈

0 ∗ 0 k 0 k+1 H (X, ΣL ⊗ L). This induces a map H (X,L ⊗ KX ⊗ ΣL) → H (X,L ⊗ KX ). The

Jacobian ring Rk is the cokernel of this map.

k ∼ Proposition 3.9. If L is ample and satisﬁes condition 3.9, then (M/JM) = Rk.

Proof. The proof is the same as Lemma 3.2.3 [59]. Consider the exact sequence

0 → OX → ΣL → TX → 0

k twisted by L ⊗ KX . We have

0 k 0 k 0 k 0 → H (X,L ⊗ KX ) → H (X,L ⊗ KX ⊗ ΣL) → H (X,L ⊗ KX ⊗ TX ) → 0.

32 0 k 0 k Since g ⊗ H (X,L ⊗ KX ) → H (X,L ⊗ KX ⊗ TX ) is surjective, similar calculation as

0 k 0 k+1 Proposition 3.3 shows that the image of H (X,L ⊗ KX ⊗ ΣL) → H (X,L ⊗ KX ) is JM k−1.

3.2 Vanishing conditions

In this section we discuss the vanishing conditions, especially when Y is Calabi-Yau. First we discuss condition (3.9) when L = −KX .

0 For l = 1, the vanishing condition (3.9) is equivalent to H (X,TX ) = g. When G is simple, there are three series of exceptional cases that this condition fails. See [3], Chapter 3.3, Theorem 2. For l ≥ 2, we prove that vanishing condition (3.9) holds for any X = G/P with

L = −KX .

i k Proposition 3.10. If L = −KX , then we have H (X, (G×adP p)⊗L ) = 0 for i ≥ 1, k ≥ 1

Proof. Without loss of generality, we assume G is simple. We ﬁrst ﬁx some notations. Let

Φ+ be the set of positive roots for g and B− ⊂ P be the corresponding negative Borel subgroup. Let p = t ⊕ L g , where Φ is the set of positive roots generated α∈Φ−∪ΦP α P ∨ by {α1, ··· , αl}\{αj1 , ··· αjm }. The pairing on t induced by the Killing form is denoted (α,β) by (, ). The pairing h, i is defined by hα, βi = 2 (β,β) . Consider the Jordan-Hölder p- representation filtration of p = V0 ⊃ V1 ⊃ · · · ⊃ Vt with irreducible factors Wi = Vi/Vi+1.

The corresponding associated bundles over X are also denoted by Vi and Wi. Then the

ss highest weight of Wi as maximal semisimple Lie subalgebra p ⊂ p, denoted by λi, is either 0 or in Φ ∪ Φ . The weight for L is 2ρ = P α. − P P α∈Φ+\ΦP

Now we prove all the higher cohomology groups of Wi vanish. The complement of

{αj1 , ··· , αjm } in the Dynkin diagram is decomposed as connected components D1,D2, ··· ,Da.

33 ss The semisimple part p is the direct sum of Lie subalgebras corresponding to Dj. Then

ss ss λi restricted to the Cartan of p is dominant weight for p . So hλi, βi ≥ 0 for all β ∈ ΦP . The paring used here is induced by the Killing form on pss. Since Killing form is the unique bilinear pairing invariant under adjoint action up to rescaling, we use the same notation.

We claim hλi + 2kρP + ρ, βi ≥ 0 for any β ∈ Φ+. For simple root β, we have hρP , βi ≥ 0

and hρ, βi = 1. If β ∈ {α1, ··· , αl}\{αj1 , ··· αjm }, then hλi, βi ≥ 0. If β = αjb , then hλi, βi ≥ −3. In this case, we have hρP , αjb i ≥ 1 since h2ρ − 2ρP , αjb i ≤ 0. Here 2ρ − 2ρP is the sum of positive roots in Dj, hence has nonpositive product with αjb .

The above method also proves the vanishing condition (3.9) for irreducible Hermitian symmetric spaces.

Proposition 3.11. Then condition 3.9 holds for any ample line bundle on irreducible Her- mitian symmetric space X that is not isomorphic to projective spaces. Here G is Aut(X).

Proof. Let X = G(a, l + 1) be a Grassmannian not isomorphic to Pl. Let O(1) be the

k positive generator of Picard group of X. Then L ⊗ KX = O(t) for some integer t. The tangent bundle TX is associated bundle of g/p and g/p is irreducible pss representation with highest weight being the highest long root. We use the previous notations and ΦP is generated by {α1, ··· , αl}\{αj}. Let ωi be the ith fundamental weight. Then the highest long root is ω1 + ωl. The line bundle O(1) is the associated line bundle with weight ωj with

0 k 0 k j 6= 1, l. If t ≥ 0, then H (X,L ⊗ KX ) 6= 0 and H (X,TX ⊗ L ⊗ KX ) is an irreducible

0 k 0 k g-module. So the map g ⊗ H (X,L ⊗ KX ) → H (X,TX ⊗ L ⊗ KX ) is surjective. If t < 0,

0 k then H (X,TX ⊗ L ⊗ KX ) = 0 since ω1 + ωl − tωj is not dominant weight. Let X be any other irreducible Hermitian symmetric space G/P . Let P be maximal parabolic subgroup P with αj removed. The highest long root λ = aiωi. If X is not isomorphic to projective spaces, then aj = 0. The argument for Grassmannian applies to this case.

34 Now we discuss vanishing condition (3.8). When X is Hermitian symmetric, the homogeneous bundle is decomposed as direct sum of irreducible vector bundles and the corresponding highest weights are given by the criterion of Kostant [45]. In [62], Snow has found some suﬃcient conditions. See the Proposition in Section 1 of [62]. Especially for Grassmannian X = G(a, l + 1), vanishing condition 3.8 holds for L = O(t) with t ≥ l. When t < l, there are some cases not satisfying the vanishing condition. For instance, the

p n−p a2−a cohomology H (X, Ω ⊗ O(2)) 6= 0 when X = G(a, 2a) and p = 2 . See [61] Theorem 3.2 and 3.3.

Example 3.12. Here we show some examples for which vanishing condition (3.8) holds. The upshot is that the homogenous bundles involved are not direct sum of irreducible vector bundles and the irreducible factors in Jordan-H¨olderﬁltration admits nontrivial higher cohomology.

n −1 Let X = G/B with g = sll+1 and L = KX with l = 1, 2, 3, 4. Then we have p ∼ n−p i j Ω ⊗ L = ∧ TX . When l = 1, 2, we have H (X, ∧ TX ) = 0, i ≥ 0 from Borel-Weil-Bott.

i 2 ∼ For l = 3, the only remaining case is H (X, ∧ TX ) = 0 for i ≥ 1. Since TX = G×B (g/b), it

3 2 has a natural ﬁltration given by the heights of positive roots V0 = g /b− ⊃ V1 = g /b− ⊃

1 V2 = g /b− ⊃ 0 with factors

∼ M Wi = Vi/Vi+1 = Lα (3.12) ht(α)=3−i

2 2 The induces a ﬁltration ∧ V0 = V0 ∧ V1 ⊃ · · · ∧ V2 with successive quotients W01 =

2 2 W0 ⊗ W1,W02 = W0 ⊗ W2,W11 = ∧ W1,W12 = W1 ⊗ W2,W22 = ∧ W2.

35 Then we have the following spectral sequence from the ﬁltration

0 210 ⊕ 012 0 0 0 0

−1 0 103 ⊕ 301 ⊕ 020 0 0 0

−2 0 0 020 020 ⊕ 020 0

−3 0 0 0 101 ⊕ 101 101

−4 0 0 0 0 0

0 1 2 3 4 (3.13)

p,q p+q The terms E1 in the spectral sequence are H (Wp) with Wij reindexed and m1m2m3 P stands for irreducible g-module with highest weight i miωi, where ωi is the i-th funda- 4,−3 mental weight. Now we study the diﬀerential 101 ⊕ 101 → 101. The E1 component is

1 3,−3 0 H (W4) with W4 generated by v1 = eα1 ∧ eα3 and E1 is H (W3) with W3 generated by v2 = eα1+α2 ∧ eα3 , v3 = eα2+α3 ∧ eα1 . Notice that σα2 (α2 + α3 + ρ) − ρ = α1 + α2 + α3 = 101.

Denote the minimal parabolic subgroup generated by α2 by Pα2 . Consider the projection

∗ map πα2 : G/B → G/Pα2 and the spectral sequence converging to R πα2 V0 induced by the same ﬁltration. The ﬁbers for the bundles in the spectral sequence are determined by the action of the Borel of sl2 generated by α2. The short exact sequence is given by

0 → W4 → V3 → W3 → 0 (3.14)

36 with action of sl2 on

fα2 v1 = 0, fα2 v2 = v1, fα2 v3 = −v1 (3.15)

hα2 v1 = −2v1, hα2 v2 = 0, hα2 v3 = 0 (3.16)

˜ If we consider the subspace V3 generated by v1, v2(or v3), then the restriction of 3.14 on P1 = Pα2 /B is isomorphic to

0 → O(−1) → C2 → O(1) → 0 (3.17)

1 3 0 1 2 0 3 ˜ 1 ∼ P C ∼ ˜ ∼ twisted by O(−1). Hence we have R πα2 (V )|P = H ( , O(−1))⊗ = 0 and R πα2 (V ) =

0 1 0. The boundary map R πα2 (W3) → R πα2 (W4) is isomorphism on each component. From the following commutative diagram

0 0 0 1 H (G/Pα2 ,R πα2 (W3)) H (G/Pα2 ,R πα2 (W4)) (3.18)

0 1 H (G/B, W3) H (G/B, W4)

3,−3 4,−3 We conclude that d1 : E1 → E1 is isomorphism restricted to each component. 2,−2 3,−2 The same argument shows that d1 : E1 → E1 is isomorphism when projected to 3,−2 each components of E1 . But the second diﬀerential is diﬃcult to calculate this way. There is another approach for calculating the cohomology for homogenous bundles. See proposition 2.8 in [46]. It combines the BGG resolution of Verma modules and Bott’s theorem on cohomology of homogenous vector bundles. The highest weight λ part of the

37 cohomology of vector bundle G ×B E is given by the following sequence

0 → E[λ] → ⊕w∈W,l(w)=1E[wλ] → · · · → E[w0λ] → 0 (3.19)

Apply this sequence to our situation, we only need to check λ = (020) and E = ∧2(g/b).

∨ (λ,αi )+1 The ﬁrst diﬀerential is realized as multiplication by fi . In this case, we have

∗ d1(eα1+α2+α3 ∧ eα2 ) = f1(eα1+α2+α3 ∧ eα2 ) f3(eα1+α2+α3 ∧ eα2 ) = eα2+α3 ∧ eα2 eα1+α2 ∧ eα2

∗ and d1(eα1+α2 ∧eα2+α3 ) = f1(eα1+α2 ∧eα2+α3 )f3(eα1+α2 ∧eα2+α3 ) = eα2 ∧eα2+α3 eα1+α2 ∧eα2 .

∗ The choices of plus or minus sign are the same in two expressions. Hence d1 is surjective. We

1 2 i j have H (X, ∧ TX) = 0. When l = 4, the same calculation shows vanishing H (X, ∧ TX ) = 0, i > 0.

3.3 An application to diﬀerential zeros of period integrals

Let X be an n-dimensional smooth projective variety such that its anti-canonical line

−1 bundle L := KX is very ample. Let G be a connected algebraic group acting on X. We assume that G acts with ﬁnitely many orbits. We shall regard the basis elements ai of

∨ ∨ ∨ V = Γ(X,L) as linear coordinates on V . Let B := Γ(X,L)sm ⊂ V be the space of smooth sections.

top Let π : Y → B be the family of smooth CY hyperplane sections Ya ⊂ X, and let H

n−1 be the Hodge bundle over B whose ﬁber at a ∈ B is the line Γ(Ya, ωYa ) ⊂ H (Ya). In [52] the period integrals of this family are constructed by giving a canonical trivialization of Htop. Let Π be the period sheaf of this family, i.e. the locally constant sheaf generated by the period integrals. [16] initiated a study of the zero loci of derivatives of period integrals under this canon-

38 ical trivilization. Namely, For any δ ∈ DV ∨ , denote

N (δ) = {b ∈ B | δs(b) = 0, ∀periods s}. (3.20)

In [16], it has been shown that N (δ) is algebraic, and in the case when X = Pn, an explicit equation for N (δ) was given – see Thm 7.2 in [16]. This explicit equation in particular gives a natural stratiﬁcation of the zero loci. In [16], based on the theory of tautological systems applied to general homogeneous varieties G/P , Thm 7.2 is a direct consequence of Lemma 7.1, whose proof relies on the algebraic and geometric rank formulae (Thm 2.9 in [6], and Thm 1.4 in [34]) for tautological systems that both apply to general G/P , together with the Jacobian ring description of the Hodge structure of hypersurfaces in Pn. Now, the generalization of the latter to certain cases of homogeneous varieties directly provides a generalization of Thm 7.2 to those cases, where the statement stays the same word by word. In the following, we state and prove the generalization of Lemma 7.1, therefore establishing the generalization of Thm 7.2 as mentioned. We will follow notations

f(b) in [34][6]. In particular, H0(gˆ, Re ) denotes a particular Lie algebra coinvariant space, which appears as the right hand side of the algebraic rank formula for tautological systems – see section 2 of [6].

Lemma 3.13 (Degree bound lemma). Let X be a homogeneous variety G/P of dimension n, where Theorem 4.6 holds, let gˆ = Lie(G) ⊕ C. Let Zi be a basis of gˆ. Suppose f(b) is

f(b) f(b) nonsingular. For h ∈ R, he ≡ 0 in H0(gˆ, Re ) iﬀ

f(b) X f(b) he = Zi(rie )

for some ri ∈ R, and deg ri ≤ deg h − 1, ∀i.

39 Proof. The ‘if’ direction is obvious. For the ‘only if’ direction, recall the homogeneous

Jacobian ideal J :=< Zif(b)|Zi ∈ gˆ > of R. Let Bk denote a C-basis for the degree k part of R/J. First, since f(b) is homogeneous of degree 1, the degree 0 part of R/J is nonzero, and is spanned by 1. For any h ∈ R, consider expanding the highest degree component of h, which we denote by h0, in degree = deg h part of R/J in terms of the chosen basis: P i.e. by deﬁnition, there exist elements si ∈ R, such that h0 − siZi(f(b)) can be written as a linear combination of the chosen basis elements in degree = deg h. Obviously, we can require that deg si ≤ deg h − 1 for each i by dropping all higher degree components of each of these ri, if there are any. Working degree by degree, it is clear that we can choose ri ∈ R

f(b) P f(b) P P with deg ri ≤ deg h−1, ∀i, such that he = Zi(rie )+ ckBk, where ckBk denote

f(b) a linear combination of elements of the Bk with all k ≤ deg h. Therefore, H0(gˆ, Re ) is spanned by Bk.

−1 On the other hand, by Theorem (4.6) applied to the Calabi-Yau case L = KX , and

n taking the sum over k, one has dimC R/J = H (X − V (f(b))). Combining the algebraic and geometric rank formula for tautological systems applied to G/P [6][34], we have in

n f(b) this case, h (X − V (f(b))) = dim H0(gˆ, Re ). Therefore, the collection of Bk consists of

f(b) f(b) linearly independent elements, and he = 0 in H0(gˆ, Re ) iﬀ all coeﬃcients ck = 0.

Remark 3.14. Here our grading convention on R is such that f(b) is of degree 1. 1 ∈ C acts on V ∨ by identity. In the algebraic rank formula for tautological systems, there is a twist of the action of the Euler operator by a constant, however this twist does not aﬀect the above proof, since the degree 0 part of R/J is of dimension 1, and is spanned by 1.

40 3.4 Zero loci of vector bundle sections

In this section we describe Hodge structure of zero loci of vector bundle sections on homogenous variety X = G/P . Again, we use the following notations.

1. Let G be a semisimple algebraic group with parabolic subgroup P and X = G/P be the corresponding ﬂag variety.

2. Let ρ: P → GL(r) be a semisimple representation of P . The associated homogenous

r vector bundle is E = G ×P C .

3. Consider the space of global sections V ∨ = H0(X,E) and f ∈ V ∨. The zero locus is

deﬁned by Yf = {f = 0}.

There are two diﬀerent typical examples, one is that E is the direct sum of line bundles

E = L1 ⊕ L2 · · · ⊕ Lr, the other is that ρ is an irreducible representation of P . The idea follows the Cayley trick in the description of complete intersections. See Konno’s paper [44] for Green’s Jacobian ring and application to Torelli theorem for complete intersections. For complete intersections in toric varieties, see [55]. See Section 2.2 for the collections of results and notations needed in this section.

n−r In order to describe the Hodge structure on Hvar (Y ), we consider the following vanishing conditions.

Hp(X, Ωq−a ⊗ ∧aE⊗Sl−aE) = 0 for p > 0, q ≥ 0 and 0 ≤ a ≤ l − 1 X (3.21) p q−a a+1 l−a−1 or H (X, ΩX ⊗ ∧ E⊗S E) = 0 for p > 0, q ≥ 0 and 0 ≤ a ≤ l − 1

1 k H (X, (G ×adP p) ⊗ S (E) ⊗ det E ⊗ KX ) = 0 for k ≥ r (3.22)

Proposition 3.15. If E satisﬁes the vanishing condition 3.21, then P satisﬁes the vanishing

41 condition

p q H (P, ΩP ⊗ O(l)) = 0 for p > 0, q ≥ 0 and l ≥ 1 (3.23)

Proof. Consider the exact sequence

∗ 0 → ker π∗ → TP → π TX → 0 (3.24)

∨ q We denote ker π∗ by Tv and (ker π∗) by Ωv for simplicity. The bundle ∧ ΩP admits a

a ∼ a q−a ∗ p a ﬁltration with graded pieces Gr = ∧ Ωv ⊗ ∧ π ΩX . It is suﬃcient to prove H (P, Gr ⊗ O(l)) = 0. Consider the Leray spectral sequence

b,c b c a b+c P a E2 = H (X,R π∗(Gr )) → H ( , Gr ).

According to projection formula, we have

c a ∼ c a q−a R π∗(Gr ) = R π∗(Ωv ⊗ O(l)) ⊗ ∧ ΩX (3.25)

c a c P ∨ a The ﬁber of R π∗(∧ Ωv⊗O(l)) over a point x is canonically isomorphic to H ( (E ), ΩP ∨ ⊗ x (Ex ) O(l)). If c > 0 or l ≤ a, this is zero according to Bott vanishing theorem. If c = 0, l ≥ a+1,

0 P ∨ a then H ( (E ), ΩP ∨ ⊗ O(l)) is an irreducible representation of GL(Ex). Claim this is an x (Ex ) a l−a irreducible factor of ∧ (Ex) ⊗ S (Ex) as GL(Ex)-representation. The proof of the claim is in the following lemma.

Lemma 3.16 (Bott). Let W be r-dimensional vector space. Let P(W ∨) be projective space consisting of the lines in the dual vector space. Assume 1 ≤ a ≤ l − 1 and Ωa ⊗ O(l) has a natrual GL(W )-action induced by the tautological action on O(1). Then H0(P(W ∨), Ωa ⊗ O(l)) is an irreducible factor of ∧a(W ) ⊗ Sl−a(W ) or ∧a+1(W ) ⊗ Sl−a−1(W ) as GL(W )-

42 representation.

Proof. The Euler sequence

0 → Ω1 → W ⊗ O(−1) → C → 0 (3.26) is an exact sequence of homogenous GL(W )-bundle. It induces an exact sequence

0 → Ωa → ∧aW ⊗ O(−a) → Ωa−1 → 0 (3.27)

There is an exact sequence of GL(W )-representations

0 → H0(P(W ∨), Ωa ⊗ O(l)) → ∧a(W ) ⊗ Sl−a(W ) → H0(P(W ∨), Ωa−1 ⊗ O(l)) → 0 (3.28)

Hence it is an irreducible summand of ∧a(W ) ⊗ Sl−a(W ).

Deﬁnition 3.17. Let Sk(E) be the symmetric product of E. Then the coordinate ring of

0 k 0 k+1−r P is R = ⊕k≥0H (X,S (E)) graded by k. Let M = ⊕k≥r−1H (X,S (E)⊗det E ⊗KX ) be a graded R-module with gradings k. The Jacobian ideal J is the ideal in R generated

0 ∨ k+1−r by f and LZ f, Z ∈ g. Denote Nk = H (X,E ⊗ S (E) ⊗ det E ⊗ KX ). There is a map

∨ from Nk to Mk deﬁned by pairing the E component with f.

Theorem 3.18. If E satisﬁes the vanishing conditions (3.22), (3.21) for p, q, l in the range {1 ≤ p ≤ k +r −1, q = n+r −1−p, l = k +r −p}∪{1 ≤ p ≤ k +r −2, q = n+r −p−2, l = k + r − 1 − p} ∪ {1 ≤ p ≤ k + r − 2, q = n + r − 1 − p, l = k + r − 1 − p} and (3.21) with p = 1, q − a = n, a = r − 2, l − a = k − r + 2, then we have the following description of the

43 Hodge structure of Y

n−r−k,k ∼ k+r−1 k+r−1 k+r−2 Hvar (Y ) = M /N f + JM (3.29)

0 Proof. According to Proposition 3.15, we have a surjective map H (P,KP ⊗ O(k + 1)) →

n+r−1−k n+r−1 ˜ 0 n−r−2 F H (P − Y ) with kernel equal to dH (P, ΩP ⊗ O(k)). Using Leray spectral

0 k sequence, we have H (P,KP ⊗O(k +1)) is nonzero only if k ≥ r −1 and isomorphic to M .

0 n−r−2 n−r−2 ∼ Now we describe H (P, ΩP ⊗ O(k)). There is a natural isomorphism ΩP ⊗ O(k) =

TP ⊗ KP ⊗ O(k). So we consider the exact sequence

∗ 0 → Tv → TP → π TX → 0 (3.30)

twisted by KP ⊗ O(k). Here Tv = ker π∗. This induces a long exact sequence

0 0 0 → H (P,Tv ⊗ KP ⊗ O(k)) → H (P,TP ⊗ KP ⊗ O(k)) →

0 ∗ 1 H (P, π TX ⊗ KP ⊗ O(k)) → H (P,Tv ⊗ KP ⊗ O(k))

1 First we claim H (P,Tv ⊗ KP ⊗ O(k)) = 0. There is an exact sequence

1 0 1 0 → H (X,R π∗(Tv ⊗ O(k − r)) ⊗ KX ⊗ det E → H (P,Tv ⊗ KP ⊗ O(k)) →

0 1 H (X,R π∗(Tv ⊗ O(k − r)) ⊗ KX ⊗ det E)

1 r−1 We have R π∗(Tv ⊗O(k−r)) = 0 according to Bott’s vanishing theorem on P . According

0 to Lemma 3.16, the bundle R π∗(Tv ⊗ O(k − r)) ⊗ KX ⊗ det E is a direct summand of

r−2 k−r+2 1 0 ∧ E ⊗ S E ⊗ KX . So H (X,R π∗(Tv ⊗ O(k − r)) ⊗ KX ⊗ det E) also vanishes due to condition 3.21.

44 0 Now we describe H (P,Tv ⊗ KP ⊗ O(k)). Consider the Euler sequence

∗ ∨ 0 → C → π E ⊗ O(1) → Tv → 0 (3.31)

twisted by KP ⊗ O(k). Then we have an surjective map

k 0 N → H (P,Tv ⊗ KP ⊗ O(k)) (3.32)

1 0 n+r−2 k k−1 since H (P,KP⊗O(k)) = 0. The image under the map d: H (P, Ω (kY˜ )) → M /M f is described as follows. The paring of E∨⊗E → C induces a pairing H0(X,E∨⊗Sk+1−r(E)⊗

0 0 k+1−r 0 det E ⊗ KX ) ⊗ H (X,E) → H (X,S (E) ⊗ det E ⊗ KX ). Since f ∈ H (X,E), this gives a map Nk → Mk by pairing with f.

0 ∗ Next we describe H (P, π TX ⊗ KP ⊗ O(k)). It is zero when k = r − 1 and isomorphic

0 k−r to H (X,TX ⊗ S (E) ⊗ KX ⊗ det E)) when k ≥ r. Consider the exact sequence

0 → p → g → g/p → 0 (3.33)

k−r twisted by S (E) ⊗ KX ⊗ det E. Under condition 3.22, we have a surjective map

0 k−r 0 k−r g ⊗ H (X,S (E) ⊗ KX ⊗ det E) → H (X,TX ⊗ S (E) ⊗ KX ⊗ det E)) (3.34)

0 k−r k k−1 The image of g ⊗ H (X,S (E) ⊗ KX ⊗ det E) under d to M /M is described as Lie derivative of g on M k. So F n+r−k−1Hn+r−1(X − Y )/F n+r−kHn+r−1(X − Y ) ∼= M k/N kf + JM k−1.

Remark 3.19. We have a natural pairing E ⊗ Sk−rE → Sk−r+1E. This induces a map

0 Mk−1 → Nk and commute with the pairing with H (X,E). So the Jacobian ideal J in

45 0 Deﬁnition 3.17 can be replaced by J generated by LZ f for Z ∈ g.

3.5 Hypersurfaces in log homogenous varieties

Let Xn be a smooth projective variety with simple normal crossing divisor D. The log tangent bundle TX (− log D) is a subsheaf of TX defined as follows. If z1, ··· , zn is the local coordinate of X and D is the hyperplanes defined by z1 = 0, ··· , zr = 0, then the generating sections of TX (− log D) are z1∂1, ··· , zr∂r, ∂zr+1, ··· , ∂zn. We say X is log homogenous if TX (− log D) is globally generated and log parallelizable if TX (− log D) is trivial. Toric varieties and flag varieties are examples of log homogenous varieties. See

0 [9, 10] for discussion of log homogenous varieties. Let g be H (X,TX (− log D)) and G be a corresponding Lie group making X as an G-variety. Let X0 = X − D be the open G-orbit. Let L be a G-equivariant line bundle on X. The section f ∈ H0(X,L) deﬁnes a hypersurface Y . We say it is nondegenerate if Y + D is still simple normal crossing. Let

Y0 = Y − D. In this section we discuss similar Jacobian ring description of Hodge groups of X0 − Y0. It is a straightforward result following Batyrev’s work on the mixed hodge structures of aﬃne hypersurfaces in algebraic tori [4].

0 k The same as deﬁnition 3.1. Let R be the graded ring R = ⊕k≥0H (X,L ). The generalized Jacobian ideal J is the graded ideal generated by f, LZ f for Z ∈ g. Then

0 k+1 M = ⊕k≥0H (X,KX (D) ⊗ L ) is a graded R-module. Let ΩX,D = ΩX (log D). Deﬁne

RX to be the kernel of action map OX ⊗ g → TX (− log D) following the notation in [10]. We consider the following two vanishing conditions.

p q l H (X, ΩX,D ⊗ L ) = 0 for p > 0, q ≥ 0, l ≥ 1 (3.35)

1 k H (X,RX ⊗ KX (D) ⊗ L ) = 0 for k ≥ 1 (3.36)

46 Theorem 3.20. Under conditions (3.36) and (3.35) for {p = k−l+1, q = n−k+l−1, 1 ≤ l ≤ k} ∪ {p = k − l + 1, q = n − k + l, 0 ≤ l ≤ k} ∪ {p = k − l, q = n − k + l − 1, 1 ≤ l ≤ k − 1} ∪ {p = k − l, q = n − k + l, 1 ≤ l ≤ k − 1}, there is an isomorphism

k n−k n n−k+1 n (M/JM) → F H (X0 − Y0)/F H (X0 − Y0).

Proof. The proof follows directly from section 6 of [4]. Since Y + D is simple normal

n−k n crossing, the Hodge to de-Rham spectral sequence degenerate at E1-page. So F H (X0 −

n−k+1 n ∼ k n−k Y0)/F H (X0 −Y0) = H (X, ΩX (log(Y +D))). There is an exact sequence (Theorem 6.2 in [4])

n−k n−k d n−k+1 n−k+1 d 0 → ΩX (log(Y + D)) → ΩX,D(Y ) −→ ΩX,D (2Y )/ΩX,D (Y ) −→

d n n ··· −→ ΩX,D((k + 1)Y )/ΩX,D(kY ) → 0

k n−k k−1 n−k The assumption (3.35) implies H (X, ΩX,D(Y )) = H (X, ΩX,D(Y )) = 0,

k−l n−k+l n−k+l H (X, ΩX,D ((l + 1)Y )/ΩX,D (lY )) = 0, 1 ≤ l ≤ k − 1

k−l−1 n−k+l n−k+l H (X, ΩX,D ((l + 1)Y )/ΩX,D (lY )) = 0, 1 ≤ l ≤ k − 2 and also subjectivity of the maps

0 n−1 0 n−1 n−1 H (X, ΩX,D(kY )) → H (X, ΩX,D(kY )/ΩX,D((k − 1)Y ))

0 n 0 n n−1 H (X, ΩX,D((k + 1)Y )) → H (X, ΩX,D((k + 1)Y )/ΩX,D((kY )).

47 A standard spectral sequence argument shows

H0(X, Ωn−1 ((k + 1)Y )) Hk(X, Ωn−k(log(Y + D))) ∼= X,D (3.37) X 0 n 0 n−1 H (X, ΩX,D(kY )) + dH (X, ΩX,D(kY ))

n−1 ∼ Using isomorphism ΩX,D = TX (− log D) ⊗ KX (D) and the same calculation in the proof of Proposition 3.2, we prove the theorem.

Remark 3.21. The vanishing condition (3.35) with p > q is proved in [10]. See also [10] or [56] for the log version of Lefschetz theorem for the Hodge structures other than middle dimension case.

Remark 3.22. When X is log parallelizable, the condition (3.35) always holds and M =

0 k+1 ⊕k≥0H (X,L ) is the maximal homogeneous ideal in R. See [9] for classiﬁcation of log parallelizable varieties. In toric case, this is Theorem 6.9 in [4]. The proof given here is the same as the proof in [4].

3.6 Hodge conjecture for very generic hypersurfaces

This section is another application of the Jacobian ring for generalized ﬂag varieties. We prove that Hodge conjecture holds for very generic hypersurfaces in ﬂag varieties with the vanishing conditions in section 3.1.

Theorem 3.23 ([15], Corollary 7.5.2). Let X = G/P be a generalized ﬂag variety with odd dimension n = 2k + 1. Let L be an ample line bundle on X satisfying vanishing conditions

0 k (3.8) for p, q, l in the range 1 ≤ p ≤ k, q = n − p, l = k − p + 1 and H (X,KX ⊗ L ) 6= 0. Then for f ∈ H0(X,L) outside a countable union of proper subvarieties, we have for the hypersurface Yf

k,k k,k 2k H (Yf , Q) = Im{H (X, Q) → H (Yf , Q)} (3.38)

48 Proof. The vanishing condition (3.8) for 1 ≤ p ≤ k, q = n − p, l = k − p + 1 implies that

0 k+1 k 2k αk : H (X,KX ⊗ L ) → F H (Yf ) is surjective. According to Theorem 7.5.1, Corollary 7.5.2 in [15], we only need to check

0 0 k 0 k+1 H (X,L) ⊗ H (X,KX ⊗ L ) → H (X,KX ⊗ L ) (3.39)

0 k 0 k+1 0 is surjective. Since H (X,KX ⊗ L ), H (X,KX ⊗ L ) and H (X,L) are irreducible

0 k representations of g, the multiplication map is surjective if H (X,KX ⊗L ) is not zero.

Since the Hodge conjecture holds for generalized ﬂag variety, we have

Theorem 3.24. Under the same assumption in Theorem 3.23, then Hodge conjecture holds for hypersurfaces Yf outside a countable union of proper subvarieties in the linear system |L|.

Remark 3.25. The Hodge conjecture for very generic hypersurfaces in toric varieties with certain combinatorial property is proved in [13]. The proof reduces to the surjectivity of similar map (3.39) in toric Jacobian ring. We hope similar results hold for certain log homogenous varieties from the Jacobian ring in section 3.5.

4 Arithmetic properties of period integrals

4.1 Introduction

We ﬁrst recall some notations and deﬁnitions for period integral

49 4.1.1 Notations

1. Let X be a smooth semi-Fano variety of dimension n over C. In this section X is toric variety or partial ﬂag variety G/P with P parabolic subgroup in a semisimple algebraic group G.

∨ 0 −1 2. We denote V = H (X, ωX ) to be the space of anticanonical sections.

∨ 3. For any nonzero section s ∈ V , the zero locus Ys is a Calabi-Yau hypersurface in X.

∨ 4. Let B be the set of s ∈ V such that Ys is smooth. Then B is Zariski open subset of

∨ V and there is a family of smooth Calabi-Yau varieties π : Y → B with Ys as ﬁbers.

∼ −1 5. The section s induces the adjunction formula ωYs = ωX ⊗ ωX |Ys . The constant

0 function 1 on the right hand side corresponds to a canonical section ωs ∈ H (Ys, ωYs ).

1 In other words, the section ωs is the residue of rational form s . Putting ωs together, 0 we get a canonical section of R π∗(ωY/B), denoted by ω.

Deﬁnition 4.1 (Period integral). Consider the local system L on B formed by Hn−1(Ys, Q),

n−1 which is the dual of R π∗QY . For any ﬂat section γ of L on U ⊂ B an open subset, the R period integral Iγ is deﬁned by γ ω.

Deﬁnition 4.2 (Picard-Fuchs system). Let DV ∨ be the sheaf of linear diﬀerential operators

∨ generated by Der(V ). Then any section D of DV ∨ acts on the de-Rham coholomogy sheaf

n−1 • R π∗(Ω (Y/B)) via restriction to B and Gauss-Manin connection ∇. Deﬁne the sheaf of Picard-Fuchs system for ω to be PF (U) = {D ∈ DV ∨ (U)|∇(D|U∩B)ω = 0}. Period integrals are solutions to Picard-Fuchs system. In other words, we have DIγ = 0 for any D ∈ PF (U).

50 ∨ ∨ Consider the classical solution sheaf Sol = HomDV ∨ (DV /P F, OV ). The solution rank at ∨ a point s ∈ V is deﬁned to be the dimension of the stalk Sols. We will consider the points in V ∨ having solution rank 1.

Deﬁnition 4.3 (Special point s0). There exist special solution-rank-1 points when X is toric or G/P . The theorem we will state is for those special solution-rank-1 points. We

characterize s0 up to scaling in terms of its zero locus Ys0 as follows. If X is toric variety, we consider the stratiﬁcation of X by the torus action. Then Ys0 is the union of toric invariant divisors. If X = G/P , we consider the stratiﬁcation of X by projected Richardson varieties,

see [40]. Then Ys0 is the union of projected Richardson divisors. Especially, if P is a Borel subgroup, the divisor Ys0 is the union of Schubert divisors and opposite Schubert divisors. They are proven to have solution rank 1 from Gel’fand-Kapranov-Zelevinski systems and tautological systems by Huang-Lian-Zhu [34]. The unique solution at s0 is realized as a

period integral Iγ0 over invariant cycle γ0. The special point s0 is known as the large complex structure limit in the moduli of toric hypersurfaces. We expect the same result holds for ﬂag varieties.

n Example 4.4. Let X = P with homogenous coordinate [x0, ··· , xn]. Then V is identiﬁed with space of homogenous polynomials of degree n + 1. In this case, the special solution- rank-1 point s0 = x0 ··· xn.

Definition 4.5 (Hasse-Witt matrix). Next we define the Hasse-Witt matrix. Let k be a perfect field of characteristic p. Assume π : Y → S is a smooth family of Calabi-Yau

0 variety over k with relative dimension n − 1. Let ω be a trivializing section of R π∗(ωY/S).

∗ n−1 Let ω be the dual section of R π∗(OY ). The p-th power endomorphism of OY induces

n−1 n−1 ∗ ∗ a p-semilinear map R π∗(OY ) → R π∗(OY ) sending ω to aω . Then H-Wp = a as a

∗ section of OS is the Hasse-Witt matrix under the basis ω . The choice of ω for Calabi-Yau

51 hypersurfaces is made by adjunction formula similar to period integral.

4.1.2 Statement of the theorem

Now we state our main theorem. When X is toric or G/P , it has an integral model over Z.

0 −1 Let s0 ∈ H (X,KX ) be the special solution-rank-1 point chosen in deﬁnition 4.3. Then

0 −1 s0 can be extended as a basis s0 ··· sN of H (X,KX ). Let a0 ··· aN be the dual basis for s0, ··· sN . Suitable choices of s0 ··· sN are still basis considering the p-reduction of X. See section 4.2 and section 4.3 for the details of choice of si and the integration cycle γ0. There exists the following truncation relation between Hasse-Witt invariants of hypersurfaces over

Fp and period integrals. It can also be viewed as a relation between mod p solutions to Picard-Fuchs systems and solution over C.

Theorem 4.6. 1. The Hasse-Witt matrix H-Wp deﬁned above are polynomials of aI of degree p − 1.

2. The period integral Iγ0 deﬁned above can be extended as holomorphic functions at s0

and has the form 1 P ( aI ), where P ( aI ) is a Taylor series of a1 , ··· , aN with integer a0 a0 a0 a0 a0 coeﬃcients.

3. They satisfy the following truncation relation

1 (p−1) aI p−1 H-Wp = P ( ) mod p (4.1) a0 a0

where (p−1)P ( aI ) is the truncation of P at degree p − 1. a0

Remark 4.7. In characteristic p, the conjugate spectral sequence provides a horizontal ﬁl- tration for relative de-Rham cohomology. In particular, the Hasse-Witt matrix gives part

n−1 of the coordinate for the projection of R π∗(OY ) to the horizontal subbundle in de-Rham

52 cohomology. This is how Katz [36] proved elements in Hasse-Witt matrix satisfy Picard- Fuchs equations. On the other hand, period integrals give the coordinate of horizontal sections in relative de-Rham cohomology over C. The above relations suggest that the horizontal subbundle provided by conjugate spectral sequence can approximate the horizontal section in characteristic zero near some degeneration point when p → ∞.

Remark 4.8. The Hasse-Witt matrices count rational point on Calabi-Yau hypersurfaces mod p by Fulton’s ﬁxed point formula [24]. In the case of Calabi-Yau hypersurfaces in toric varieties, the relation between point counting and period integrals has been studied by Candelas, de la Ossa and Rodriguez-Villegas [14].

Next we state the relation between period integrals and unit roots of zeta-function of toric hypersurfaces. Let a0 = 1. The formal power series

P (aI ) g(aI ) = p (4.2) P (aI ) lies in lim Z [a ··· a , ((p−1)P (a ))−1]/psZ [a ··· a , ((p−1)P (a ))−1] and satisﬁes Dwork ←−s→∞ p I1 IN I p I1 IN I congruences (ps−1) (P (aI )) s g(aI ) ≡ s−1 mod p . (4.3) (p −1) p (P )((aI ) ) P Theorem 4.9. Let aI ∈ Fq. Assume the hypersurface Y deﬁned by aI sI is smooth and

H-Wp(aI ) 6= 0. Then there exists exactly one p-adic unit root in the factor of zeta function

n−1 of Y corresponding to Frobenius action on Hcris (Y ). It is given by the formula

p pr−1 g(âI )g(âI ) ··· g(âI ) (4.4)

p with aˆI being the Teichm¨uller lifting under aI → aI .

Similar unit root formulas for general-type toric hypersurfaces and Calabi-Yau hyper-

53 surfaces in G/P is given in 4.5 and 4.7.

4.2 Local expansions and Hasse-Witt matrices

Now we prove a algorithm of calculating Hasse-Witt matrices of hypersurfaces of X in terms of local expansions of the sections. The key ingredient is to related the Hasse-Witt operator of Calabi-Yau or general type families Y to the Cartier operator on X. Then we apply the algorithm to toric and generalized ﬂag varieties. Especially this recovers the algorithm for Pn. We make the following assumptions for this section.

1. Let Xn be a smooth projective variety deﬁned over k and satisﬁes Hn(X, O) = Hn−1(X, O) = 0

2. Let L be an base point free line bundle on X and V ∨ = H0(X,L) 6= 0 and W ∨ =

0 ∨ ∨ ∨ ∨ ∨ ∨ ∨ H (X,L ⊗ KX ) 6= 0. Let a1 , a2 ··· aN and e1 ··· er be basis of V and W .

3. Consider the smooth hypersurfaces Y over S,→ V ∨ − {0}. Let X be X × S and i: Y → X be the embedding. The projections to S are denoted by π

(p) 4. Let FS be the absolute Frobenius on S and X = X ×FS S the ﬁber product.

(p) Then we have absolute Frobenius FX the relative Frobenius FX /S : X → X . Denote W : X (p) → X and π(p) : X (p) → S to be the projections. The corresponding diagram for family Y is deﬁned in a similar way.

Consider the following diagram

∗ −1 f 0 W L OX (p) i∗OY(p) 0

p−1 f F F (4.5)

−1 f 0 FX /S∗L FX /S∗OX i∗FY/S∗OY 0

54 p−1 ∗ −1 −1 ∗ ∗ −1 ∗ −1 ∼ −p The map f : W L → FX /S∗L is induced by FX /S W L = FX L = L multiplied by f p−1. This induces the diagram

n−1 (p) n (p) ∗ −1 R π∗ (OY(p) ) R π∗ (W L )

F f p−1 (4.6)

n−1 (p) n −1 R π∗ (FY/S∗OY ) R π∗(L )

The two horizontal maps are isomorphism. The left vertical map is the Hasse-Witt operator

∗ n−1 ∼ n−1 (p) n−1 (p) ∼ n−1 H-W: FS (R π∗(OY )) = R π∗ (OY(p) ) → R π∗ (FY/S∗OY ) = R π∗OY .

∨ ∨ 0 0 ∗ Deﬁnition 4.10. The basis e1 ··· er of H (X,KX ⊗ L) induces a basis of R π (ωY/S) by

n−1 residue map and dual basis e1 ··· er of R π∗(OY ) under Serre duality. The Hasse-Witt

∗ P matrix aij is deﬁned by H-W(FS (ei)) = j aijej.

Let CX/S : ωX /S → ωX (p)/S be the top Cartier operator. For any coherent sheaf M on X , the Grothendieck duality

∼ FX /S∗Hom(M, ωX /S) = Hom(FX /S∗M, ωX (p)/S) (4.7)

is related to CX/S by the natural pairing

FX /S∗Hom(M, ωX /S) ⊗ FX /S∗M → ωX (p)/S (4.8)

−p −p ∼ ∗ −1 sending g ⊗ m to CX/S(g(m)). Consider M = L . Since FX /S∗L = W L , we have

−p ∼ ∗ −1 FX /S∗Hom(L , ωX /S) = Hom(W L , ωX (p)/S) (4.9)

55 Then we have a morphism induced by multiplication by f p−1

−1 −p ∼ ∗ −1 FX /S∗Hom(L , ωX /S) → FX /S∗Hom(L , ωX /S) = Hom(W L , ωX (p)/S). (4.10)

After taking R0π(p) on both sides, we have an morphism

0 −1 0 ∗ −1 R π∗Hom(L , ωX /S) → R π∗Hom(W L , ωX (p)/S). (4.11)

This the dual of

n (p) ∗ −1 n −1 R π∗ (W L ) → R π∗(L ) (4.12)

Now we can conclude the dual of Hasse-Witt matrix is given by the following algorithm.

P I Let (t1, ··· , tn) be local coordinate of X at a point x. Denote g(t) = aI t to be a formal

P J power series. Then deﬁne τ(g) = J aI t with I = (p − 1, ··· , p − 1) + pJ. Fix a local ∨ ei trivialization section ξ of L on an Zariski open section U containing x. Then any section ξ is a section of ωX |U and has the form hi(t)dt1 ∧ dt2 · · · ∧ dtn. Under the same trivilization,

∨ p−1 ei f the section ξp has the form as gi(t)dt1 ∧ dt2 · · · ∧ dtn. Then we claim τ(gi) has the form P τ(gi) = j ajihj.

Theorem 4.11. The matrix aij deﬁned above is the Hasse-Witt matrix under the basis

∨ ∨ e1 ··· er .

Notice that τ is p−1-semilinear τ(hpg) = hτ(g). So the same algorithm works if we use a rational section ξ. Now we specialize this algorithm to toric hypersurfaces or Calabi-Yau hypersurfaces. Let X be a smooth complete toric variety deﬁned by a fan σ. The 1-dimensional primitive P vectors v1, ··· vN correspond to toric divisors Di. Assume L = O( aiDi) with ai ≥ 1. n ˚ 0 Let ∆ = {v ∈ R |hv, vii ≥ −ai} and ∆ the interior of ∆. Then H (X,L) has a basis

56 Zn 0 ∨ ˚ Zn corresponding to uI ∈ ∆ ∩ and H (X,L ⊗ KX ) has basis ei identiﬁed with ui ∈ ∆ ∩ .

P uI 0 Let f = aI t be the Laurent series representing the universal section of H (X,L) and

p−1 P u f = Aut . Then we have

Corollary 4.12. The Hasse-Witt matrix of hypersurface family over |L| under the basis

∨ ˚ Zn ei ∈ ∆ ∩ is given by aij = Apuj −ui .

n Proof. After an action of SL(n, Z), we can assume v1 ··· vn is the standard basis of Z .

Then t1, ··· , tn is an aﬃne chart on X. We choose a section of L⊗KX to be s0 corresponding

˚ n dt1∧dt2···∧dtn s0 to origin in ∆ ∩ Z and a meromorphic section of KX to be θ = . Let s = be t1···tn θ a meromorphic section of L. Then

∨ ui ei t = dt1 ∧ dt2 · · · ∧ dtn. (4.13) s t1 ··· tn

If we view f as a section in H0(X,L), then

f X = t ··· t a tuI (4.14) s 1 n I I

∨ p−1 ei f Hence sp = gidt1 ∧ dt2 · · · ∧ dtn with

ui P uI p−1 t ( aI t ) X g = I = A tu+ui−1. (4.15) i t ··· t u 1 n u

Here 1 = (1, ··· , 1). So

X v τ(gi) = Aut (4.16) v

57 with u + ui − 1 = pv + (p − 1)1. On the other hand, we have

X uj −1 τ(gi) = ajit . (4.17) j

So aij = Apuj −ui .

Remark 4.13. When X is Pn, Corollary 4.12 gives the same algorithm as Katz [36]. In [67], Vlasenko deﬁnes the higher Hasse-Witt matrices for Laurent polynomial f. When f is a homogenous polynomial of degree d, the ﬁrst matrix α1 in [67] mod p is the Hasse- Witt matrix for hypersurface Y in Pn. The p-adic limit of the matrices is conjectured to

n−1 give the Frobenius matrix of the unit root part of Hcris (Y ), which is a dual analogue of matrices by Katz [37]. Corollary 4.12 proves that α1 mod p is also the Hasse-Witt matrix for toric hypersurfaces. Hence it is natural to generalize Vlasenko’s conjecture to toric hypersurfaces.

−1 If X is any smooth variety satisfying the assumptions in this section and L = KX , then we have a Calabi-Yau family. In this case, the algorithm coincides with the criterion for

0 Frobenius splitting of X respect to Y . The basis of H (X,L⊗KX ) is chosen to be constant function 1. The Hasse-Witt matrix is a function a on S. We can choose the trivializing

−1 section of L to be (dt1 ∧ dt2 · · · ∧ dtn) . The local algorithm in this case it the following.

−1 Corollary 4.14. Let f = g(t)(dt1 ∧ dt2 · · · ∧ dtn) . Then the Hasse-Witt matrix a is

p−1 p−1 given by τ(g ). More explicitly a is the coeﬃcient of (t1 ··· tn) in local expansion of (g(t))p−1.

p−1 0 1−p Remark 4.15. For any closed point s ∈ S(k), the corresponding section fs ∈ H (X, ωX ) determines a Frobenius splitting of X compatibly with Ys if and only if a(s) 6= 0. It is also equivalent to Ys being Frobenius split. Especially, Corollary 4.12 implies the well-known

58 fact that toric variety X is Frobenius split compatibly with torus invariant divisors. See Chapter 1 of [11].

Proof of Theorem 4.6 for toric X. Following the previous notations, let X be smooth com-

−1 P 0 plete toric variety and L = KX = OX ( i Di). Then the basis of H (X,L) is identiﬁed

n with the integral points uI in the polytope ∆ = {v ∈ R |hv, vii ≥ −1}. The universal

P uI section f(t) = aI t with u0 = (0, ··· , 0). Then H-Wp is the coeﬃcient of constant term in f p−1 according to Corollary 4.12 or 4.14. On the other hand, the period integral

1 Z dt ∧ · · · ∧ dt I = √ 1 n (4.18) γ n (2π −1) γ t1 ··· tnf(t)

along the cycle γ : |t1| = |t2| = · · · |tn| = 1 is the coeﬃcient of constant term in the Laurent expansion of f −1. So

1 ∞ k a a X k X I1 k1 Il kl Iγ = (1 + (−1) ( ) ··· ( ) ) (4.19) a0 k1, k2, ··· , kl a0 a0 k=1 k u +···+k u =0,P k =k,I 6=0 1 I1 l Il j j and

p−1 p − 1 a a p X X I1 k1 Il kl H-Wp = a0(1 + ( ) ··· ( ) ) k1, k2, ··· , kl, p − 1 − k a0 a0 k=1 k u +···+k u =0,P k =k,I 6=0 1 I1 l Il j j (4.20) Then apply the congruence relation

 p − 1 k ≡ (−1)k mod p (4.21) k1, k2, ··· , kl, p − 1 − k k1, k2, ··· , kl in the two expansions to get the conclusion.

Remark 4.16 (General toric hypersurfaces). The same argument also applies to general-type

59 toric hypersurfaces. The entries in Hasse-Witt matrix are truncations of period integrals. The results for hypersurfaces in Pn are proved by Adolphson and Sperber in [2]. We follow

0 the notations in Corollary 4.12. The sections s ∈ H (X,L ⊗ KX ) determines a section of

0 R π∗(Y, ωY/B) via residue map and we can deﬁne period integral of s in a similar way as

0 −1 Calabi-Yau hypersurfaces. Let s0 ∈ H (X,KX ) be the large complex structure limit point

∨ 0 with zero locus equal to the union of Di. Let ei be the basis of H (X,L⊗KX ) corresponding

˚ Zn ∨ 0 P uI to ui ∈ ∆ ∩ and denote si = s0 ⊗ ei ∈ H (X,L). Let f = aI t be the universal section of L. In the Laurent series expression of f, the section si deﬁned above is identiﬁed

ui ∨ with multi-index t . The period integral of ei along the cycle γ : |t1| = |t2| = · · · |tn| = 1 near sj is given by 1 Z tui dt ∧ · · · ∧ dt I = √ 1 n (4.22) γ,i n (2π −1) γ t1 ··· tnf(t) and it is equal to the coeﬃcient of t−ui in the Laurent expansion of f −1. On the other

∨ ∨ hand, the ijth entry aij of the Hasse-Witt matrix under the basis e1 ··· er is given by the coeﬃcient of tpuj −ui in the Laurent expansion of f p−1. So we have the following

1. The function aij on S are polynomials of aI of degree p − 1.

1 aI 2. The period integral Iγ,i is a holomorphic functions at sj and has the form Pi( ), aj aj

aI aI where Pi( ) is a Taylor series of with integer coeﬃcients. aj aj

3. They satisﬁes the following truncation relation

1 (p−1) aI p−1 aij = (Pi( )) mod p (4.23) aj aj

(p−1) aI where (Pi( )) is the truncation of P at degree p − 1. aj

60 Since the period integral of ui t dt1 ∧ · · · ∧ dtn ωi = t1 ··· tnf(t) satisfies the corresponding Gel’fand-Kapranov-Zelevinski hypergeometric differential system, the entries aij of Hasse-Witt matrices are mod p solutions to the same differential system. See [1] for mod p solutions to general hypergeometric systems. In [2], Adolphson and Sperber also proved the generic invertibility of Hasse-Witt matrices for hypersurfaces in Pn. Similar idea gives the same result for toric hypersurfaces.

Corollary 4.17. The Hasse-Witt matrices for generic smooth toric hypersurface are not degenerate. In other words, the determinant det(aij) 6= 0.

(p−1) aI 1 Proof. Consider the determinant of matrix (Bij) = ( (Pi( a ))) = ( p−1 aij). The entry j aj

(p−1) aI (Pi( )) has the form aj

p−1 X X aI aI (−1)k ( 1 ) ··· ( k ). (4.24) aj aj k=0 u +···+u =(k+1)u −u I1 Ik j i

The indices Il are not required to be distinct. The constant term in Bij = δij. Now we prove the constant term in det B is 1. Let be a permutation of r-elements. Assume

1 2 aI1 aI aI2 aI aIr aIr 1 ··· k1 · 1 ··· k2 ··· 1 ··· kr (4.25) a1 a1 a2 a2 ar ar

be a constant term appearing in the product B(1)1 ··· B(r)r. Then all indices Im appearing ˚ n in the numerator correspond to interior integer points ui ∈ ∆ ∩ Z and satiesfy

uIi + ··· + uIi + u(i) = (ki + 1)ui. (4.26) 1 ki

˚ n Consider the vertex ul of the convex polytope generated by all ui ∈ ∆ ∩ Z . Since the

61 l l convex expression for such ul is unique, the indices I1 = ··· = kl = (l) = l. Hence other terms in the product does not involve ul. We can delete the vertices and consider the convex polytope generated by the remaining ui and get (i) = i inductively. Then the only constant term is 1.

4.3 Generalized ﬂag vareities

Now we prove similar proposition for generalized ﬂag variety X = G/P using Corollary 4.14. There is a natural candidate for large complex structure limit in Calabi-Yau hypersurfaces family of G/P , which is the union of codimension one stratum of projections of

Richardson varieties, denoted by Y0. See [40] for the deﬁnition of Y0 and [34] for indentiﬁca- tion of Y0 as solution rank 1 point of Picard-Fuchs system. In the proof of toric Calabi-Yau

n families, we only used the following fact. There is an aﬃne chart (t1 ··· tn) ∈ AZ on XZ

−1 with s0 = t1 ··· tn(dt1 ··· dtn) . So we expect that Y0 = Y1 +···+Yn +W ,where Y1, ··· ,Yn has complete intersections at some point x and Z is an eﬀective divisor outside x. But this only happens in some special cases, for example, projective spaces and Grassmannian G(2, 4). In general, the projections of Richardson varieties are not intersecting transversely to one point. We need the Bott-Samelson-Demazure-Hansen type resolution of projections of Richardson varieties to lift the anticanonical sections to rational anticanonical sections. This construction is also used in the proof of Frobenius splitting for projections of Richard- son varieties, see [40]. Now let ψ : Z → X be a proper birational morphism between smooth varieties Z and

∼ ∗ X over k. Let ωZ = ψ ωX + E, where E is a Weil divisor supported on exceptional divisor.

∗ −1 ∼ −1 Then we have ψ ωX = ωZ + E inducing the isomorphism

−1 ∼ ∗ −1 ∼ −1 ψ∗(ωZ + E) = ψ∗(ψ ωX ) = ωX . (4.27)

62 This isomorphism is induced by pulling back anticanonical sections on X to anticanonical sections on Z with poles along E and hence ﬁts in the commutative diagram of sheaves

1−p 1−p F∗(ωX ) ψ∗F∗(ωZ ((p − 1)E))

τˆ τˆ (4.28)

OX ψ∗(OZ (E))

P I 1 Hereτ ˆ is the same trace map induced by Cartier operator as follows. If σ = I fI t (dt ∧ n 1−p 1−p P J · · · ∧ dt ) is a local section of ωX , thenτ ˆ(F∗(σ)) = J fI t with I = (p − 1, ··· , p − 1−p 1) + pJ. The mapτ ˆ: F∗(ωZ ((p − 1)E)) → OZ (E) is deﬁned as followsτ ˆ(F∗(σ)) =

1 p 1 p 1 p τˆ(F∗( ηp η σ)) =τ ˆ( η F∗(η σ)) = η τˆ(F∗(η σ)). Here η is local deﬁning section of E. Then p 1−p p η σ is a holomorphic section of ωZ andτ ˆ(F∗(η σ)) is deﬁned the same as X. After taking global sections, we reduce the calculation of Hasse-Witt matrix to Z. If we have a section

0 −1 s0 ∈ H (X, ωZ (E)) with the desired property as toric case, then similar conclusion follows. Note that we need to take into account meromorphic sections. When E is union of some coordinate hypersurfaces at a point x, the same formula forτ ˆ in terms of Laurent expansion of σ in local coordinates still applies. Now we apply the discussion above to Bott-Samelson-Demazure-Hansen varieties. They arise as resolutions of singularities of Schubert varieties and projections of Richardson varieties. See [11], section 2, [7], or [40]. First we ﬁx some notations. Let G be simple complex Lie group with Lie algebra g. Fix upper Borel subgroup B = B+ and lower Borel

− subgroup B of G. Denote the simple roots by α1 ··· αl. Let W be the Weyl group and

si ∈ W the simple reﬂection generated by αi. Let w = si1 ··· sin be a reduced expression for

w ∈ W and we denote it by w = (si1 , ··· , sin ). Let Pij be the minimal parabolic subgroup corresponding to simple root αij . Then the Bott-Samelson variety Zw is deﬁned to be

n n Pi1 × · · · × Pil /B . Here the right action by B is deﬁned by (p1, ··· , pn) · (b1, ··· , bn) =

63 −1 −1 (p1b1, b1 p2b2, ··· , bn−1pnbn). The image of (p1, ··· , pn) under the quotient map is denoted by [p1, ··· , pn].

We now recall some basic properties of Bott-Samelson variety. The map ψw : Zw →

G/B deﬁned by [p1, ··· , pn] 7→ p1 ··· pn is a birational map to the Schubert variety Xw = ˆ n−1 BwB/B. Let Zw(j) = Pi1 × · · · Pij · · · × Pil /B be a divisor of Zw via the embedding

[p1, ··· , pˆj, ··· , pn] 7→ [p1, ··· , 1, ··· , pn]. The boundary of Zw is deﬁned to be ∂Zw =

Zw(1) + ··· + Zw(n). These components have normal crossing intersection at [1, ··· , 1]. Let

L(λ) = G ×B k−λ be the equivariant line bundle on G/B associated to character λ and

∗ ∼ Lw(λ) = ψwL(λ). Then ωZw = OZw (−∂Zw)⊗Lw(−ρ) with ρ being the sum of fundamental weights.

From the previous discussion for toric case, we are looking for a special section s0 ∈

0 −1 H (X, ωX ) and suitable aﬃne chart on Zw. Since the Picard group of G/B is generated by opposite Schubert divisors, we have a section σ of L(ρ) vanishing exactly along all opposite

∗ Schubert divisors. Lets ˜0 be the tensor product of ψwσ and canonical section of OZw (∂Zw). Thens ˜ vanishes along ∂Z and preimage of opposite Schubert divisors. Let U − be the 0 w ij negative unipotent root subgroup P ∩ U −. The natural map U − × · · · × U − → Z gives ij i1 in w n an aﬃne neighborhood of [1, ··· , 1] which is isomorphic to A with coordinate (t1, ··· , tn).

Then Zw(j) on this aﬃne chart is deﬁned by tj = 0. The image of this chart under ψ is inside

id − the opposite Schubert cell C = B B/B. Sos ˜0 vanishes with simple zero along coordinate

−1 hyperplanes on this chart. After rescaling, we can takes ˜0 = t1 ··· tn(dt1 ∧ · · · ∧ dtn) . Let

P W be the set of minimal representatives in cosets W/WP and wP be the longest element

P in W with reduced expression wP . Then ψ : ZwP → G/P is birational and it is an isomorphism restricted to ZwP − ∂ZwP → BwP B/B → BwP P/P .The exceptional divisor is supported on ∂ZwP . Next we identifys ˜0 with a special anticanonical form deﬁned on v v X = G/P . Let Xw = X ∩ Xw be the intersection of Schubert variety Xw = BwB/B

64 with opposite Schubert variety Xv = B−vB/B. The image in X = G/P is denoted by

v Πw. This forms a stratification of X. The codimension one strata form an anticanonical divisor Π1 + ··· + Πs. See [40], section 3. (Note that our notations for Schubert variety and opposite Schubert variety are different from [40].) So there is an anticanonical section s0 vanishing to the first order along Y0 = Π1 ∪ · · · ∪ Πs.

∗ Lemma 4.18. The two anticanonical sections are related by s˜0 = ψ (s0) up to a rescaling.

∗ Proof. We compare the two divisors (˜s0) and (ψ (s0)). Since σ vanishes along opposite

Schubert divisors on G/B, thens ˜0 vanishes along the preimage of opposite Schubert divisors

id under ψw and ∂Zw. Let CwP be the Schubert cell and C the opposite Schubert cell. The restriction of ψwP : ZwP −∂ZwP → CwP is an isomorphism. Let Di be divisors supported on

id P −1 P Cw −C . Then (˜s0) = ψ (Di)+ Zw (j). The restriction of projection Cw → G/P P i wP j P P is also isomorphism on its image. The divisors Πj are exactly the complement of the image

id ∗ P −1 of Cw ∩ C . So we have (ψ (s0))| −1 = ψw (Di). The exceptional locus of P ψwP (CwP ) i P ∗ P −1 P ψ is supported on ∂Z . So (ψ (s0)) = ψ (Di) + njZw (j) as a meromorphic wP i wP j P anticanonical section. Since (˜s ) and (ψ∗(s )) are linear equivalent, then P Z and 0 0 j wP (j) P n Z are linear equivalent. On the other hand, the divisors Z ··· Z form a j j wP (j) wP (1) wP (n) basis for Pic(ZwP ), see [11] Excercise 3.1.E (3). So nj = 1.

So the Hasse-Witt invariants have similar expansion algorithm as toric case according to the discussion above. On the other hand, the period integral near s0 can also be calculated by pulling-back to Zω. The cycle γ : |t1| = |t2| = · · · |tn| = 1 has nontrivial image in

R 1 Hn(X − Ys ) since the integral of 6= 0. This is the unique invariant cycle near 0 γ s˜0 n s0 since dim Hc (X − Y0) = 1. According to Theorem 1.4 in [34], the period integral R 1 = R ψ∗( 1 ) is the unique holomorphic solution to the Picard-Fuchs system near ψ∗γ f γ f s0. So we proved Theorem 4.6 for generalized ﬂag variety X = G/P . Note that the

65 0 −1 basis of H (X, ωX ) including s0 can also be written down explicitly in terms of standard monomials, see [12]. This method also gives a way to calculate the power series expansions of period integrals of hypersurfaces in G/P .

Remark 4.19. The anticanonical form s0 appears in [57] and [40]. In [57], the form s0 is

id id constructed on torus chart of the open Richardson cell Rw = C ∩ Cw and glued together

id by coordinate transformations. We use the construction in [40] that the complement of Rw

∗ −1 is an anticanonical divisor. Lemma 4.18 proves that ψ (s0) = t1 ··· tn(dt1 ∧ · · · ∧ dtn) on the aﬃne coordinate of Zω, which is the local formula on torus chart appeared in [57]. This gives an explanation of the footnote in section 7 of [57]. The cycle γ appears in [58] 7.1 for complete ﬂag variety G/B, in [54] Theorem 4.2 for Grassmannians and in [47] 12.4 for general G/P .

Now we give some explicit examples of the resolution and the anticanonical form s0 under the resolution.

Example 4.20. Let X be Grassmannian G(2, 4). Then X = G/P with G = SL(4)   ????      ????    and P = { }. The Weyl group is S4 and WP = S2 × S2. The element  0 0 ??      0 0 ?? 4 wp = (13)(24) = (23)(34)(12)(23) = s2s3s1s2. So Zw = P1 × P2 × P3 × P4/B with       ???? ???? ????              0 ???   0 ???   t3 ???        P1 = { }, P2 = { }, P3 = { } and P4 =  0 t ??   0 0 ??   0 0 ??   1            0 0 0 ? 0 0 t2 ? 0 0 0 ?

66     ???? a b 1 0          0 ???   c d 0 1      { }. The largest Schubert cell is { }P/P with coordinates  0 t ??   1 0 0 0   4        0 0 0 ? 0 1 0 0 A4 (a, b, c, d). The aﬃne coordinate (t1, ··· , t4) ∈ on ZwP is

        1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0                  0 1 0 0   0 1 0 0   t3 1 0 0   0 1 0 0          {[  ,   ,   ,  ]}.  0 t 1 0   0 0 1 0   0 0 1 0   0 t 1 0   1       4          0 0 0 1 0 0 t2 1 0 0 0 1 0 0 0 1

So the map ψ : ZwP → X under these local charts is given by

1 t + t 1 1 a = , b = − 1 4 , c = , d = − (4.29) t1t3 t1t2t3t4 t1 t1t2

Recall the anticanonical section s0 in [34] is given in terms of standard monomials as follows.   a a a a  11 12 13 14  Let   be the basis of any two plane. The Pl¨ucker coordinates xij are a21 a22 a23 a24 the determinant of i, j columns. The section s0 = x12x23x34x14. In coordinate of Schubert

−1 cell, we have s0 = −ad(ad−bc)(da∧db∧dc∧dd) . A direct calculation using (4.29) shows

∗ −1 0 that ψ s0 = t1t2t3t4(dt1dt2dt3dt4) . The other sections of H (X,L) can also be written as homogenous polynomials of xij of degree 4.

Remark 4.21. The proof for both toric and ﬂag varieties only depends on the the following

fact. There is a torus chart (t1, ··· , tn) on the complement of Ys0 with s0 = t1 ··· tn(dt1 ∧

−1 · · · ∧ dtn) on the chart. So Theorem main can hold for more general ambient spaces X.

This also implies the Frobenius splitting of X compatibly with Y0.

67 4.4 Complete intersections

We further discuss the algorithm for Hasse-Witt matrix for complete intersections.

n 1. Let X be a smooth projective variety deﬁned over k. Let L1, ··· Ls be line bundles on

i s−i X and E = ⊕iLi. Assume the following vanishing conditions H (X,KX ⊗ ∧ E) =

i−1 s−i H (X,KX ⊗ ∧ E) = 0 for i = 1, ··· , s.

∨ 0 ∨ 0 2. Let Vi = H (X,Li) 6= 0 and W = H (X, det E ⊗ KX ) 6= 0. We further assume the zero locus of a generic element of V ∨ = H0(X,E) is smooth with codimension s. Let

∨ ∨ ∨ e1 ··· er be a basis of W . Let fi be the universal section of Li and f = (f1, ··· , fs) be universal section of E.

3. Consider the family of complete intersections deﬁned by f over the smooth locus S,→ V ∨ − {0}. Let X be X × S and i: Y → X is the embedding of universal family. The projections to S are denoted by π.

(p) 4. Let FS be the absolute Frobenius on S and X = X ×FS S the ﬁber product. Then

(p) we have absolute Frobenius FX the relative Frobenius FX /S : X → X . Denote W : X (p) → X and π(p) : X (p) → S to be the projections. The corresponding diagram for family Y is deﬁned in a similar way.

We repeat the argument in the hypersurfaces using the Koszul resolution

s ∨ s−1 ∨ ∨ 0 → ∧ E → ∧ E → · · · → E → OX → i∗OY → 0. (4.30)

Standard spectral sequence argument together with the vanishing assumptions gives an

n−s n −1 isomorphism R π∗(OY ) → R π∗((det E) ). The maps in the Koszul resolution is given as follows. Identify the section of ∧kE∨ with the sections (f ∨ ) of ⊕ L∨ ⊗· · ·⊗L∨ j1,··· ,jk j1,··· ,jk j1 jk

68 ∨ ∨ with ordered set j1, ··· , jk, such that j1, ··· , jk are distinct and fj ,··· ,j = f 0 0 if 1 k j1,··· ,jk j , ··· , j is a permutation of j0 , ··· , j0 with signature 1. Then (f ∨ ) is mapped to 1 k 1 k j1,··· ,jk+1 (f ∨ ) = (P f ∨ f ). We have similar commutative diagram as (4.5). j1,··· ,jk j j1,··· ,jk,j j

∗ ∨ f 0 W det E ··· OX (p) i∗OY(p) 0

p−1 f F F (4.31)

∨ f 0 FX /S∗ det E ··· FX /S∗OX i∗FY/S∗OY 0

The map f p−1 : W ∗ ∧k E∨ → F ∧k E∨ is induced by multiplication (f ∨ ) 7→ X /S∗ j1,··· ,jk ((f ∨ )pf p−1 ··· f p−1). So we have commutative diagram j1,··· ,jk j1 jk

n−s (p) n (p) ∗ ∨ R π∗ (OY(p) ) R π∗ (W det E )

F f p−1 (4.32)

n−s n ∨ R π∗(FY/S∗OY ) R π∗(det E ) with horizontal maps being isomorphisms. The left vertical map is the Hasse-Witt operator

∗ n−s ∼ n−s (p) n−s (p) ∼ n−s H-W: FS (R π∗(OY )) = R π∗ (OY(p) ) → R π∗ (FY/S∗OY ) = R π∗OY . So we ∨ ∨ have similar deﬁnition of Hasse-Witt matrix under basis e1 ··· er .

∨ ∨ 0 0 ∗ Deﬁnition 4.22. The basis e1 ··· er of H (X,KX ⊗ det E) induces a basis of R π (ωY/S)

n−s by residue map and dual basis e1 ··· er of R π∗(OY ) under Serre duality. The Hasse-Witt

∗ P matrix aij is deﬁned by H-W(FS (ei)) = j aijej.

The same argument in hypersurfaces case gives us the algorithm of computing Hasse-

p−1 p−1 Witt matrix in terms of local expansion of f. Now f is replaced by (f1 ··· fs) . Under

∨ ei the trivialization ξ of det E under local coordinates (t1, ··· , tn), the section ξ has the form ∨ p−1 ei (f1···fs) hi(t)dt1 ∧ dt2 · · · ∧ dtn and ξp has the form as gi(t)dt1 ∧ dt2 · · · ∧ dtn. Then τ(gi) P has the form τ(gi) = j ajihj.

69 R Ω R Ω 0 On the other hand, the period integral has the form Res = 0 where γ is γ f1···fs γ f1···fs a cycle in the complement of {f1 ··· fs = 0}. So it is the same form as hypersurfaces with f replaced by f1 ··· fs. The section f1 ··· fs also deﬁnes a subfamily of hypersurfces in the linear system | det E|. Both Hasse-Witt matrices and period integrals can be calculated with the same algorithm applied to this subfamily. So the truncation relation still holds for complete intersections in both toric variety and ﬂag variety. For example, the statement for toric Calabi-Yau hypersurfaces is as follows. Let X be a smooth toric variety and

−1 KX = D1 + ··· + Ds be a partition of toric invariant divisors. Let Li = O(Di). Let fij

0 be a basis of H (X,Li) consisting of monomials with fi0 the deﬁning section of Di. The P universal section is f = ( j bijfij)i. Then the period integral of the unique invariant cycle

1 b1j1 ···bsjs b1j1 ···bsjs near f0 = (fi0) has the form P ( ), in which P ( ) is a Taylor series of b10···bs0 b10···bs0 b10···bs0 b ···b 1j1 sjs with integer coeﬃcients. The degree-(p − 1) truncation (p−1)P ( bij ) multiplied by b10···bs0 bi0 p−1 (b10 ··· bs0) is a degree-(p − 1) polynomial of b1j1 ··· bsjs and gives the Hasse-Witt matrix for the Calabi-Yau complete intersection family.

4.5 Frobenius matrices of toric hypersurfaces

Now we give a proof of the conjecture in [67] for toric hypersurfaces. First we state the conjecture. The notations follow [39]. Let k be a perfect field of characteristic p. Let W (k) be the ring of Witt vectors of k. Denote σ : W → W be the absolute Frobenius automorphism of W . For any W -scheme Z, let Z0 = Z ⊗W k be the reduction mod p. Let S = Spec(R) be an affine W -scheme. Let R = lim R/psR and S = Spf(R ). We fix a ∞ ←− ∞ ∞ Frobenius lifting on R and also denote it by σ, which is a ring endomorphism σ : R → R such that σ(a) = ap mod pR. Let X be a smooth complete toric variety defined by a fan. The 1-dimensional primitive vectors v1, ··· vN correspond to toric divisors Di. Assume

70 P n ˚ L = O( kiDi) with ki ≥ 1. Let ∆ = {v ∈ R |hv, vii ≥ −ki} and ∆ the interior of ∆.

0 n 0 Then H (X,L) has a basis corresponding to uI ∈ ∆ ∩ Z and H (X,L ⊗ KX ) has basis

∨ ˚ Zn P uI ei identiﬁed with ui ∈ ∆ ∩ . Let f = aI t , aI ∈ R be a Laurent series representing

0 a section of H (X,L). Let (αs)i,j be a matrix with ij-th entry equal to the coeﬃcient of

s s tp uj −ui in (f(t))p −1. The endmorphism σ is also extended entry-wisely to matrices. It is proved in [67] that αs satisﬁes the following congruence relations

Theorem 4.23 (Theorem 1 in [67]). 1. For s ≥ 1,

s−1 αs ≡ α1 · σ(α1) ··· σ (α1) mod p.

2. Assume α1 is invertible in R∞. Then

−1 −1 s αs+1 · σ(αs) ≡ αs · σ(αs−1) mod p .

3. Under the condition of (2), for any derivation D : R → R, we have

m m −1 m m −1 s+m D(σ (αs+1)) · σ (αs+1) ≡ D(σ (αs)) · σ (αs) mod p .

Suppose that f deﬁnes a smooth hypersurface π : Y → S. We assume Y satisﬁes the

j i condition (HLF) in [39], the Hodge cohomology groups H (Y, ΩY/S) are locally free R- modules for i + j = n − 1. We also assume the pair (X,Y ) satisﬁes (HLF), the Hodge

j i cohomology groups H (X, ΩX/S(log Y )) are locally free R-modules for i + j = n. Consider n−1 ∼ n−1 the F -crystal Hcris (Y0/S∞) = HDR (Y/S) ⊗R R∞. We further assume the family Y/S satisfy condition HW (n − 1) in [39], which says for any s0 : R0 → K with K perfect ﬁeld,

n−1 (p) n−1 the Hasse-Witt operator H (Y , O (p) ) → H (Y , O ) is an automorphism. Notice s0 s0 Ys0 Ys0

71 u t i dt1∧···∧dtn that α1 mod p is the Hasse-Witt matrix under the dual basis of ωi = Res ∈ t1···tnf(t) 0 n−1 H (Y, ΩY/S ) according to Corollary 4.12. The condition HW (n − 1) can be checked using

α1. In particular, this condition implies that α1 mod p is invertible. The unit-root F -

n−1 crystal U0 ⊂ Hcris (Y0/S∞) and slope ≤ n − 2 sub-crytal U≤n−2 are deﬁned under the

n−1 n−1 assumption. The quotient Qn−1 = Hcris /U≤n−2 is isomorphic to the p -twist of the dual

∨ U0 to U0. The projection of ωi to Qn−1 gives a dual basis of U0. In [67], the Frobenius matrix and connection matrix of U0 are conjectured to be the limits of matrices in Theorem 4.23.

Conjecture 4.24 ([67]). The Frobenius matrix is the p-adic limit

−1 F = lim αs+1σ(αs) . (4.33) s→∞

The connection matrix is given by

−1 ∇D = lim D(αs)(αs) . (4.34) s→∞

Now we give the proof of this conjecture under an additional assumption on (X,L).

P Theorem 4.25. Let (X,L) be a smooth toric variety with line bundle L = O( kiDi). Let pi be toric invariant points corresponding to top dimensional cone in the fan decomposition.

If a generic section of L does not vanish at some pi, then Conjecture 4.24 is true.

The assumption in the theorem can be checked from toric data, or replaced by the equivalent assumption on the polytope of |L|. Let pi be the intersection of D1 ··· Dn. Under a transformation of SL(n, Z), we can assume the corresponding cone is generated

Rn P uI by standard basis of . Let f = I aI t as before. Then under a trivialization of L,

k1 kn P uI the universal section f is t1 ··· tn ( I aI t ). So a generic section f does not vanish at

72 (t1, ··· , tn) = (0, ··· , 0) means (−k1, ··· , −kn) is a vertex of ∆. The assumption the theorem is equivalent to that at least one of the vertices of ∆ is the intersection of hyperplanes

n hv, vii = −ki, 1 ≤ i ≤ n with vi ··· vn generating a cone of X. This is satisﬁed by X = P with L = O(d), d ≥ n + 1.

Proof. The proof follows the ideas in Katz’s proof of Theorem 6.2 [39]. Consider the F -

n ∼ n crystal constructed by logarithmic crystalline cohomology Hcris(X0,Y0) = HDR(X,Y ) ⊗

R∞. From the long exact sequence

n n n−1 · · · → HDR(X) → HDR(X,Y ) → HDR (Y )(−1) → · · · (4.35)

k k , k and HDR(X) is concentrated in H 2 2 , the corresponding subcrystal U≤n−1 and quotient

n Qn are also deﬁned on Hcris(X0,Y0) by taking the inverse image of U≤n−2 subcrystal in

n−1 Hcris (Y0) and n ∼ n−1 Qn(Hcris(X0,Y0)) = Qn−1(Hcris (Y0))(−1).

Here H(−1) means the Frobenius action is multiplied by p. We also have isomorphism

n 0 n Hcris(X0,Y0) = (H (X, ΩX/S(Y ))⊗R∞)⊕Un−1. So we only need to consider the Frobenius

u t i dt1∧···∧dtn matrix acting on projections of log n-forms ωi = onto Qn. We can assume the t1···tnf(t) n primitive vectors v1, ··· vn are the standard basis of R . The cone generated by v1 ··· vn

n deﬁnes an aﬃne coordinate (t1, ··· , tn) on X which is isomorphic to A . First we assume Y is away from (t1, ··· , tn) = 0 and consider the formal expansion map along (t1, ··· , tn) = 0

n n P : HDR(X,Y ) ⊗ R∞ → HDR(R∞[[t1, ··· , tn]]/R∞). (4.36)

Similar as Katz’s proof of Theorem 6.2 [39], we have the following conjecture

73 Conjecture 4.26. U≤n−1 is the kernel of formal expansion map.

Actually a weaker statement that U≤n−1 is contained in the the kernel can imply that

U≤n−1 is the kernel, see remark 4.27. The conjecture might be proved by log version of the theory of de Rham-Witt following Katz’s proof. We will ﬁrst give the proof of Theorem 4.25 assuming the conjecture and state the method to get around the conjecture at the

1 1 1 end. Assume the local expansion of exist in R[[t1, ··· , tn]][ , ··· , ] and has the form f t1 tn

1 X = A tu. (4.37) f u u

1 1 Notice that f may not have an inverse in R[[t1, ··· , tn]][ , ··· , ]. We can consider the t1 tn localization of R by inverting the coeﬃcient au0 of the vertex u0 = (−k1, ··· , −kn). Let au0 = 1, then

1 t−u0 −u0 X k X uI −u0 k X u = = t (1 + (−1) ( aI t ) ) = Aut . (4.38) P uI −u0 f 1 + u 6=u aI t I 0 k uI 6=u0 u

So the local expansion of ωi has the form

ui t dt1 ∧ · · · ∧ dtn dt1 ∧ · · · ∧ dtn X ω = = · A tu+ui (4.39) i t ··· t f(t) t ··· t u 1 n 1 n u with all u + ui > 0. Assume the Frobenius action on ωi is in the form

(σ) X F (ωi ) ≡ fijωj mod U≤n−1 (4.40) j

74 and the connection of ∇D has the form

X ∇(D)(ωi) ≡ ∇(D)ijωj mod U≤n−1 (4.41) j

Assume Conjecture 4.26 is true, then

(σ) X n F (ωi ) = fijωj in HDR(R∞[[t1, ··· , tn]]/R∞) (4.42) j and

X n ∇(D)(ωi) = ∇(D)ijωj in HDR(R∞[[t1 ··· tn]]/R∞). (4.43) j

n According to the Frobenius action on HDR(R∞[[t1, ··· , tn]]/R∞), we compare the coeﬃcient of tpkv for multi-index v ∈ Zn and v > 0

n X k p σ(A 0 ) ≡ f A 00 mod p (4.44) ui ij uj j

0 k 00 with p(ui + ui) = p v = uj + uj. On the other hand, we compare the expansions of ps−1 P ˜s u 1 f = u Aut and f

X s 1 X X A˜s tu = f p = ( Bstu)( A tu). (4.45) u f u u u u

˜s P s So Au = u0+u00=u Bu0 Au00 . We can extend σ to any Laurent series with coeﬃcients in R by σ(tu) = tpu. Since σ(f) = f p + pg, then

σ(f ps−1 ) = σ(f)ps−1 = (f p + pg)ps−1 ≡ f ps mod ps. (4.46)

75 ps−1−1 P ˜s−1 u Let f = u Au t , then

X s 1 X X σ(A˜s−1)tpu ≡ f p σ( ) = ( Bstu)( σ(A )tpu) mod ps. (4.47) u f u u u u u

˜s−1 P s s ˜s−1 So σ(A ) ≡ 0 00 B 0 σ(A 00 ) mod p . Now we compute σ(α ) = σ(A s−1 ) u u +pu =pu u u s−1 im p um−ui s 0 00 s−1 in terms of Au and Bu. It is the sum of Bu0 σ(Au00 ) with u + pu = p(p um − ui). The

s factor Bu0 is the sum of the terms

 ps ak1 ··· akl (4.48) I1 Il k1, k2, ··· , kl

0 with k1uI1 +···+kluIl = u . Denote νp to be the p-adic valuation. Let k = min{νp(k1) ··· νp(kl)}

k s 0 00 and p v = p um − u = p(u + ui) in (4.44), then

X k σ(A 00 ) ≡ f A 00 mod p (4.49) u ij uj j

0 00 s with u + uj = p um − uj. Since the p-adic valuation of multinomial has estimate

 ps νp ≥ s − min{νp(k1) ··· νp(kl)}, (4.50) k1, k2, ··· , kl then

 s s p k1 kl X p k1 kl s a ··· a σ(Au00 ) ≡ fij a ··· a Au00 mod p . (4.51) k , k , ··· , k I1 Il k , k , ··· , k I1 Il j 1 2 l j 1 2 l

So we have

X s B 0 σ(A 00 ) ≡ f B 0 A 00 mod p (4.52) u u ij u uj j

76 0 00 s−1 0 00 s with u + u = p um − ui and u + uj = p um − uj. Summing all such terms implies

n s p σ(αs−1) ≡ (fij)αs mod p (4.53)

n −1 −1 s−n and p (fij) ≡ αsσ(αs−1) mod p .

s Similar calculation as [39] and congruence relation D(Bu0 ) ≡ 0 mod p imply

s D(αs) ≡ (∇(D)ij)αs mod p . (4.54)

If the coeﬃcient of the vertex u0 is zero, we regard aI as formal variables and the universal hypersurface family. Then we can prove the result on an open subset of S and the p-adic limit formulas holds on the open subset. Since Vlasenko proved the congruence relations in Theorem 4.23 without any constraints on the coeﬃcients, the p-adic limits always exist. So the limits coincide with Frobenius matrices and connection matrices because they are equal restricted to an open subset of S.

l(n−1) Now we state the proof without assuming Conjecture 4.26. We claim p P (U≤n−1) ⊂

ln n p HDR(R∞[[t1, ··· , tn]]/R∞). Applying Katz’s argument of extension of scalars, we only need to prove this when R is the Witt vectors of a perfect ﬁeld. The Frobenius action on

n−1 −1 ˜ ˜ ˜ n−1 U≤n−1 divides p . So there exists σ -linear map F on U≤n−1 such that FF = F F = p .

n On the other hand, the Frobenius action on each element in HDR(R∞[[t1 ··· tn]]/R∞) has a n n−1 n−1 ˜ n n factor p . So p P (U≤n−1) = P (p U≤n−1) = P (F FU≤n−1) ⊂ p HDR(R∞[[t1, ··· , tn]]/R∞)

l(n−1) ln n and l iterations give p P (U≤n−1) ⊂ p HDR(R∞[[t1, ··· , tn]]/R∞). Multiplying both (4.42) and (4.43) by pl(n−1), we get

l(n−1) (σ) l(n−1) X ln n p F (ωi ) = p fijωj mod p in HDR(R∞[[t1, ··· , tn]]/R∞) (4.55) j

77 and

l(n−1) l(n−1) X ln n p ∇(D)(ωi) = p ∇(D)ijωj mod p in HDR(R∞[[t1 ··· tn]]/R∞). (4.56) j

Let s = nl, similar congruence relation as (4.44) still holds for k ≤ s

n+l(n−1) l(n−1) X k p σ(A 0 ) ≡ p f A 00 mod p (4.57) ui ij uj j

0 k 00 with p(ui + ui) = p v = uj + uj and v > 0. The same argument shows

n+l(n−1) l(n−1) s p σ(αs−1) ≡ p (fij)αs mod p (4.58) and

l(n−1) l(n−1) s p D(αs) ≡ p (∇(D)ij)αs mod p . (4.59)

l(n−1) −1 Dividing both sides by p and letting l → ∞, we see that the subsquence αsσ(αs−1)

−1 and D(αs)(αs) converges to the Frobenius matrix and connection matrix.

Remark 4.27. If U≤n−1 is contained in the the kernel of formal expansion map, then it is exactly the kernel. We only need to show the restriction of expansion map on

0 n n H (X, ΩX/S(Y )) ⊗ R∞ → HDR(R∞[[t1 ··· tn]]/R∞) is injective. This can be proved by similar argument in the proof and invertibility of αs.

Remark 4.28. The proof also gives a weaker version of the second and third congruence relations in Vlasenko’s Theorem 4.23. The ﬁrst congruence relation αs ≡ α1 · σ(αs−1) mod p can also be proved geometrically using the argument in Theorem 4.11 and Corollary 4.12.

(ps) n−1 s n−1 We can consider the s-iterated Hasse-Witt operation H (Y0 , O (p ) ) → H (Y0, OY0 ). Y0 Using similar commutative diagram 4.5 with the ﬁrst vertical map L−1 → L−1 being re-

78 placed by the composition L−1 → L−ps → L−1 with ξ 7→ ξps · f ps−1, we can see the matrix for s-iterated Hasse-Witt operation is given by αs mod p. Hence αs ≡ α1 ·σ(αs−1) mod p.

4.6 Unit root of toric Calabi-Yau hypersurfaces and periods

Now we discuss the relation between unit roots of zeta functions and period integrals for toric Calabi-Yau hypersurfaces. Let f be a Laurent series deﬁning toric Calabi-Yau hypersurfaces. For the sake of simplicity, let a0 = 1 be the constant term (or the coeﬃcient of interior point of ∆). The unique holomorphic period integral at the special solution-1 point or ”large complex structure limit” is Iγ = the constant term of the expansion

1 1 X k X uI k = = 1 + (−1) ( aI t ) . (4.60) P uI f 1 + u 6=0 aI t I k uI 6=0

It can be written as formal power series of aI with constant term being 1 and denoted by

P (aI ). Then

(ps−1) s αs ≡ (P (aI )) mod p (4.61) because of the congruence

 s p − 1 k k s s ≡ (−1) mod p . (4.62) k1, k2, ··· , kl, p − 1 − k k1, k2, ··· , kl

Then s (p −1)(P (a )) α σ(α )−1 ≡ I mod ps (4.63) s s−1 (ps−1−1) (P )(σ(aI )) according to Vlasenko’s congruences without any geometric constraints. So the p-adic limit

P (aI ) p exists in R∞ and equal to the Frobenius matrix. We can ﬁx σ(aI ) = a . Then the P (σ(aI )) I

79 formal power series

P (aI ) g(aI ) = p (4.64) P (aI ) has p-adic limit in lim Z [a ··· a , α−1]/psZ [a ··· a , α−1] and it satisﬁes Dwork ←−s→∞ p I1 IN 1 p I1 IN 1 congruences s (p −1)(P (a )) g(a ) ≡ I mod ps. (4.65) I (ps−1−1) (P )(σ(aI )) Especially it is related to Hasse-Witt matrix by

P (aI ) (p−1) p ≡ (P (aI )) mod p. (4.66) P (aI )

r Let q = p and aI ∈ Fq deﬁning a smooth Calabi-Yau variety Y0 over Fq. Assume the Hasse-

(p−1) Witt matrix (P (aI )) mod p is not zero. Then there exist exactly one p-adic unit root

n−1 in the factor of zeta function of Y0 corresponding to Frobenius action on Hcris (Y0). It is given by

p pr−1 g(âI )g(âI ) ··· g(âI ) (4.67)

witha ˆI being the Teichm¨ullerlifting under σ. For example, if aI has lifting as an integer,

ps thena ˆI = lims→∞ aI .

Remark 4.29. In [67], the following result about unit roots is proved. When S0 = Spec(Fq)

n r−1 and Y0 is a smooth hypersurface in P , let Φ = F · σ(F ) ··· σ (F ). Then the eigenvalues of Φ are unit roots of zeta-function of Y0. The conjecture proved above implies that the multiplicities of unit roots are also equal. The proof in [67] uses Stienstra’s result on formal groups [63, 64]. See also [69] for the unit root formula for Dwork family using formal groups following Stienstra. It might be possible to give a proof of the conjecture by this approach.

80 4.7 Frobenius matrices for Calabi-Yau hypersurfaces

Now we discuss the algorithm of Frobenius matrix of Calabi-Yau hypersurfaces in terms of local expansion. Let X be a smooth Fano variety over S with ample line bundle L. Let

0 0 −1 f ∈ H (X,L) deﬁne a smooth hypersurface Y in X. Let ωi be a basis of H (X,L ⊗ KX ).

0 This induces a basis of H (Y,KY ) via adjunction formula. The Hasse-Witt matrix under this basis in terms of local coordinate is given by the algorithm in Section 4.2. Fix a section p: S → X on ambient space X such that Y is away from p. Let (t1, ··· , tn) be the formal coordinate of X at p. The proof of Theorem 4.25 depends on ωi having the following

wi form in local expansion. There is a trivialization ξ of L along p such that ξ has the form s u s p −1 t i dt1∧···∧dtn p uj −ui f . The matrix (αs)ij is deﬁned to be the coeﬃcients of t in ps−1 . This t1···tn ξ −1 −1 applies to L = KX with trivialization ξ = (dt1 ∧ · · · ∧ dtn) . So we have the following

−1 ps−1 Proposition 4.30. Let f = g(t)(dt1 ∧ · · · ∧ dtn) and αs = the coeﬃcient of (t1 ··· tn) in the local expansion of gps−1. Then similar congruence relations in Vlasenko’s theorem (Theorem 4.23) still stand

1. For s ≥ 1,

p αs ≡ α1 · (αs−1) mod p.

2. Under the condition HW (n − 1) and g(0) 6= 0, we have

−1 −1 s−n αn(s+1) · σ(αn(s+1)−1) ≡ αns · σ(αns−1) mod p .

3. Under the condition of (2), for any derivation D : R → R, we have

−1 −1 s D(σ(αn(s+1))) · σ(αn(s+1)) ≡ D(σ(αns)) · σ(αns) mod p .

81 −1 The p-adic limit αns · σ(αns−1) gives the Frobenius action on the unit root part U0 of

n−1 Hcris (Y0) under the basis induced by residue map.

4.7.1 Unit root of Calabi-Yau hypersurfaces in G/P

Now we discuss the algorithm for Frobenius matrix of the unit root part of Calabi-Yau hypersurfaces in X = G/P . Consider the aﬃne chart An on the Bott-Samelson desingu-

Gn larization of G/P in Section 4.3. This induces a torus chart m on G/P . We consider the

1 1 formal polydisc R∞[[t1, ··· , tn]][ , ··· , ] instead of R∞[[t1, ··· , tn]] in the formal expan- t1 tn sion map in the proof of Theorem 4.25. The same method gives the algorithm of the unit root part of the Frobenius action.

0 −1 Theorem 4.31. Let f ∈ H (X,KX ) be an anticanonical form deﬁning an smooth Calabi-

−1 Yau hypersurface Y . Assume f has the form f = g(t)(dt1 ∧ · · · ∧ dtn) with g(t) ∈

1 1 R[t1, ··· , tn][ , ··· , ] in the torus chart as above. Assume the hypersurface Y is away t1 tn ps−1 from the image of (t1, ··· , tn) = 0. Let αs be the coeﬃcient of (t1 ··· tn) in the local

ps−1 expansion of g . The Hasse-Witt matrix is given by α1 mod p. The same congruence relations and Frobenius matrix in proposition 4.30 holds.

−k1 −tk 0 0 Proof. The function g(t) = t1 ··· tn g (t) for some ki ≥ 0 and g (t) ∈ R[t1, ··· , tn] does

n not vanish at (t1, ··· , tn) = 0. This is because the torus chart extends to an map on A

k and ki are the multiplicity of exceptional divisor. So the index p v appearing in the proof of Theorem 4.25 still has positive components. The rest of the proof is the same as toric case.

In Section 4.3, the period integral for the Calabi-Yau hypersurfaces in G/P is reduced to similar algorithm on the torus chart. So the same argument in Section 4.6 implies similar

82 relations between periods and the unit root of zeta-function of Calabi-Yau hypersurface defined on finite fields.

Remark 4.32. We require the non-vanishing condition in Theorem 4.25 and 4.31 to discuss the local expansion map. But the deﬁnition of matrix αs and congruence relations do not require this condition. So there might be a proof for general cases not depending on local expansions.

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