$A$-hypergeometric systems and relative cohomology

TL;DR

This paper links the solutions of $A$-hypergeometric $\,D$-modules to relative cohomology of toric varieties, extending previous results and establishing the existence of rank one points in Calabi--Yau intersections.

Contribution

It demonstrates that the solution space of certain $A$-hypergeometric systems can be described via relative cohomology, generalizing earlier work and proving new existence results.

Findings

Solution space identified with relative cohomology groups

Generalization of previous results by Huang, Lian, Yau, and Zhu

Existence of rank one points for Calabi--Yau complete intersections

Abstract

We investigate the space of solutions to certain $A$-hypergeometric $\mathscr{D}$-modules, which were defined and studied by Gelfand, Kapranov, and Zelevinsky. We show that the solution space can be identified with certain relative cohomology group of the toric variety determined by $A$, which generalizes the results of Huang, Lian, Yau, and Zhu. As a corollary, we also prove the existence of rank one points for Calabi--Yau complete intersections in toric varieties.

Equations

$$\int (f_1\cdots f_r{)}^{-1}\frac{\mathrm{d}{t}_1}{t_1}\wedge \cdots \wedge \frac{\mathrm{d}{t}_n}{t_n}$$

$${\mathrm{Sol}}^0(M_{A,{\beta }_0})={\mathrm{H}}_n(X\setminus Y_x,(X\setminus Y_x)\cap D).$$

$$\{({\delta }_{1,i},\dots ,{\delta }_{n,i}):i=1,\dots ,n\}\cup \{(-1,\dots ,-1)\}\subset {\mathbb{Z}}^n$$

$$A_i=\left[\begin{array}{ccc}{a}_{i,1}& \cdots & a_{i,N_i}\\ w_{i,1}& \cdots & w_{i,N_i}\end{array}\right]=\left[\begin{array}{ccc}{\mu }_{i,1}& \cdots & {\mu }_{i,N_i}\end{array}\right]$$

$$(s,t)\mapsto (s^{a_{i,1}}{t}^{w_{i,1}},\dots ,s^{a_{i,N_i}}{t}^{w_{i,N_i}})$$

$${\beta }_0=(-1,\dots ,-1,0,\dots ,0)\in {\mathbb{C}}^r\times {\mathbb{C}}^n.$$

$${\sigma }_i:=x_i{|}_{({\mathbb{C}}^{\ast }{)}^n}=\sum _{w_{i,j}\in {\Delta }_i\cap {\mathbb{Z}}^n}{x}_{i,j}{t}^{w_{i,j}}$$

$${\int }_{\tilde{C}}\frac{1}{{\sigma }_1\cdots {\sigma }_r}\frac{d{t}_1}{t_1}\wedge \cdots \wedge \frac{d{t}_n}{t_n}$$

$${\beta }_0=(-1,\dots ,-1,0,\dots ,0)\in {\mathbb{C}}^r\times {\mathbb{C}}^n.$$

$$B_{ϵ}(-{\beta }_0/r)\cap H\subset {\mathbb{R}}_{\ge 0}A\cap H$$

$$z=(z_1,\dots ,z_{r+n})\in B_{ϵ}(-{\beta }_0/r)\cap H\Rightarrow (z_{r+1},\dots ,z_{r+n})\in B_{\delta }(0).$$

$$\sum t_i{a}_i=(q_1,\dots ,q_r,z_{r+1},\dots ,z_{r+n}),$$

$$z=\sum t_i{a}_i+\sum _{i=1}^r(1/r-q_i)b_i\in {\mathbb{R}}_{\ge 0}A.$$

$${\beta }_0=(-1,\dots ,-1,0,\dots ,0)\in {\mathbb{C}}^r\times {\mathbb{C}}^n,$$

$${\mathrm{Sol}}^0(M_{A,{\beta }_0},{\hat{\mathcal{O}}}_{V^{\vee },x})={\mathrm{H}}_n(U_x,U_x\cap D)={\mathrm{H}}_n(U_x^{\prime },U_x^{\prime }\cap D^{\prime }).$$

$$A=\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 2& -1& -1\\ 0& -1& 2& -1\end{array}\right],$$

$$\ast$$

$${\mathrm{f}}_i=\sum _{j=1}^n{b}_{ji}{\mathrm{e}}_j,i=1,\dots ,n.$$

$$c_k=\sum _{l=1}^n{d}_{lk}{\mathrm{f}}_l.$$

$$\left[\begin{array}{cc}{I}_r& 0\\ 0& B^{-1}\end{array}\right]\cdot A$$

$$R=\left[\begin{array}{cc}{I}_r& 0\\ 0& B^{-1}\end{array}\right],$$

$$\sum _{m\in (\Delta \cap M{)}_0}{a}_m{t}^m,a_m\in \mathbb{C}.$$

$$a_1{z}_1^3+a_2{z}_2^3+a_3{z}_3^3+a_4{z}_1{z}_2{z}_3=0,[z_1:z_2:z_3]\in {\mathbb{P}}^2.$$

$$A_0=\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 2& -1& -1\\ 0& -1& 2& -1\end{array}\right].$$

$$A_0^{\prime }=\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 0& 1& -1\\ 0& -1& 1& 0\end{array}\right].$$

$$B=\left[\begin{array}{cc}1& -2\\ 1& 1\end{array}\right],B^{-1}=\left[\begin{array}{cc}1/3& 2/3\\ -1/3& 1/3\end{array}\right].$$

$$\left[\begin{array}{cc}1& 0\\ 0& B^{-1}\end{array}\right]\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 2& -1& -1\\ 0& -1& 2& -1\end{array}\right]=\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 0& 1& -1\\ 0& -1& 1& 0\end{array}\right]$$

$${\mathrm{Sol}}^0(M_{A_0,{\beta }_0},{\hat{\mathcal{O}}}_{V^{\vee },x})={\mathrm{Sol}}^0(M_{A_0^{\prime },{\beta }_0},{\hat{\mathcal{O}}}_{V^{\vee },x})={\mathrm{H}}_n(U_x,U_x\cap D)$$

$$W$$

$${\mathrm{H}}_m(Z_i,Z_i-W)=R^{-m}{\Gamma }_c(Z_i,R{j}_{i\ast }{\omega }_W),i=1,2,$$

Full text

0. Introduction

An $A$-hypergeometric system, or a GKZ-system, is a certain holonomic system of linear partial differential equations determined by some combinatoric data. This notion was introduced by Gelfand–Kapranov–Zelevinsky [1989-Gelfand-Kapranov-Zelebinski-hypergeometric-functions-and-toral-manifolds]. It has a variety of applications in algebra, algebraic geometry, and mirror symmetry.

The purpose of this note is to give a cohomological interpretation of the space of local solutions to certain $A$-hypergeometric systems related to “period integrals”. Our theorem generalizes a result of Huang–Lian–Yau–Zhu [2016-Huang-Lian-Yau-Zhu-chain-integral-solutions-to-tautological-systems]. The theorem of Huang et. al. states that, for the special $A$-hypergeometric system associated with period integrals of Calabi–Yau hypersurfaces (see Remark 1.2) in a smooth projective toric variety, its solution space near a point $x$ can be identified with a relative homology group $\mathrm{H}_{n}(X\setminus{Y}_{x},(X\setminus Y_{x})\cap D)$, where $D$ is the union of all torus-invariant divisors of $X$, and $Y_{x}$ is a (possibly singular) Calabi–Yau hypersurface associated with the point $x$.

In this note, we relax both the Calabi–Yau condition and the smoothness assumption. We also give a uniform treatment to $A$-hypergeometric systems associated with complete intersections in a projective toric variety. In terms of the geometry of toric varieties, a special case of our result can be stated as follows.

Theorem**.**

Let $(X,L)$ be a polarized toric variety of dimension $n$ defined by a convex polytope $\Delta\subset\mathbb{Z}^{n}$ containing $\mathbf{0}$ as its interior point. Let $L=L_{1}\otimes\cdots\otimes L_{r}$ be a base-point-free decomposition of $L$. We consider the $A$-hypergeometric system $M_{A,\beta_{0}}$ on $V=\prod_{m=1}^{r}\mathrm{H}^{0}(X,L_{m})$ determined by “period integrals”

[TABLE]

where $f_{i}$ is the function on $(\mathbb{C}^{\ast})^{n}$ coming from a section of $L_{i}$. Then for any $x\in V$, we have

[TABLE]

Here, $D$ is the union of all the torus-invariant divisors; for $x=(f_{1},\ldots,f_{r})$, $Y_{x}=\cup_{m=1}^{r}\overline{(f_{m}=0)}$; $\mathrm{Sol}^{0}$ is the classical solution functor.

As an application, we show the existence of a “rank 1 point” (which was called a maximal degeneracy point in [1997-Hosono-Lian-Yau-maximal-degeneracy-points-of-gkz-systems]) for any family of Calabi–Yau complete intersections in any Gorenstein Fano toric variety, generalizing the theorem of Hosono–Lian–Yau [1997-Hosono-Lian-Yau-maximal-degeneracy-points-of-gkz-systems]. See Corollary 1.7.

The assumption that $\mathbf{0}$ is an interior point of $\Delta$ can not be removed from the hypotheses. This condition is also crucial for the existence of rank one point, Corollary 1.7.

Our main theorem, Theorem 1.6, can be applied to a more general situation than the one mentioned above. See Example 1.10 for such an application, where we deal with the $A$-hypergeometric system on the “simplified moduli space”. See Situation 1.1 and Hypothesis 1.4 for the conditions on the $A$-hypergeometric system in order to ensure the validity of Theorem 1.6.

Even if one is only interested in Calabi–Yau spaces, our theorem is desirable, because most toric varieties one meets in applications and computations are singular. For example, the hypotheses of the theorem of Huang et. al. are not satisfied by the the projective toric variety defined by the convex hull of

[TABLE]

(sometimes referred to as the “mirror $\mathbb{P}^{n}$”). Thus, one already needs the full strength of our theorem (1.6) to relate the solutions to the $A$-hypergeometric system and the cohomological objects attached to the so-called “mirror Calabi–Yau spaces”, the simplest type of Calabi–Yau spaces.

The note is organized as follows. Section 1 contains a more in-depth introduction to our theorem and the proof of the application on the existence of rank 1 point in some cases. Section 2 proves our situation fits into a picture studied by Reichelt [2014-Reichelt-laurent-polynomials-GKZ-hypergeometric-systems-and-mixed-hodge-modules], so that the $A$-hypergeometric system we are interested in is a Fourier transform of an extraordinary Gauss–Manin system. This is the key input for the rest of the paper. After reviewing some basics about $\mathscr{D}$-modules in Section 3, we show how to use exponential twists to compute cohomology of various spaces in Section 4. Section 5 proves a lemma relating Fourier transform of an extraordinary Gauss–Manin system and an exponentially twisted $\mathscr{D}$-module. The last section finishes the proof of the main theorem.

Acknowledgement

We thank An Huang, Bong Lian, Shing-Tung Yau and Chenglong Yu for introducing this question to us, and for the conversations they had with us. We are grateful to Jie Zhou, who explained to us his related work on the Hesse pencil of elliptic curves and the historic background of chain integral formulae.

T.-J. Lee is grateful to Professors Lian and Yau for the invitation to Harvard CMSA in April 2018, where this work was partially done.

D. Zhang is grateful to Baohua Fu and Xuanyu Pan for their hospitality during his visit to Academy of Mathematics and System Science in July 2018.

1. $A$-hypergeometric systems

We first introduce, in Situation 1.1, the necessary notation for the definition of the $A$-hypergeometric system we are interested in. Remark 1.2 will explain how to deduce a collection of data satisfying Situation 1.1 from toric geometry. After explaining a technical hypothesis which has a natural geometric meaning in 1.4, we state the main theorem of this note.

1.1**.**

We recall the definition of $A$-hypergeometric systems for complete intersections. Fix an integer $r>0$.

1. (1)

Let $V_{i}=\mathbb{C}^{N_{i}}$ be complex vector spaces of dimension $N_{i}$, $i=1,\ldots,r$. Set $N=N_{1}+\cdots+N_{r}$ and $V=V_{1}\times\cdots\times V_{r}$. 2. (2)

Let $x_{i,1},\cdots,x_{i,N_{i}}$ be a fixed coordinate system on the dual vector space ${V_{i}}^{\vee}$. Set $\partial_{i,j}=\partial/\partial x_{i,j}$, $1\leq j\leq N_{i}$. 3. (3)

Let $\{(s_{1},\ldots,s_{r},t_{1},\ldots,t_{n}):s_{j},~{}t_{i}\in\mathbb{C}^{\ast}\}$ be an algebraic torus of dimension $r+n$. We will write this torus as a product $(\mathbb{C}^{\ast})^{r}\times T$ with $\dim T=n$ 4. (4)

Let $A$ be an $(r+n)\times N$ matrix with integral entries. We shall write $A=[A_{1},\ldots,A_{r}]$, where $A_{i}$ is an $(r+n)\times N_{i}$ matrix. We write $A_{i}$ into columns

[TABLE]

where $a_{i,j}\in\mathbb{Z}^{r}$ and $\mu_{i,j}\in\mathbb{Z}^{r+n}$. We also assume that $\mathrm{rank}(A)=r+n$. 5. (5)

Let $\tau_{i}:(\mathbb{C}^{\ast})^{r}\times T\to V_{i}$ be the map defined by the submatrix $A_{i}$

[TABLE]

and $\bar{\tau_{i}}$ be the composition $(\mathbb{C}^{\ast})^{r}\times T\to V_{i}\setminus\{0\}\to\mathbb{P}V_{i}$. Let $\tau=(\tau_{1},\ldots,\tau_{r})$ and $\bar{\tau}=(\bar{\tau}_{1},\ldots,\bar{\tau}_{r})$. We assume that $\tau$ is injective. (This is not a serious restriction, but is required by Reichelt’s theorem 2.2. We show how to get rid of it in Lemma 1.9). 6. (6)

We assume $a_{i,j}=(\delta_{i,1},\ldots,\delta_{i,r})^{t}$ for $j=1,\ldots,N_{i}$. Here $\delta_{i,k}$ is the Kronecker delta. 7. (7)

Let $X^{\prime}$ be the closure of the image of $\bar{\tau}$. $X^{\prime}$ is a toric variety with maximal torus $T^{\prime}=\bar{\tau}(T)$. Let $\mathcal{L}_{i}^{\vee}$ be the pullback of the very ample line bundle on $\mathbb{P}V_{i}$ to $X^{\prime}$. Then for each $x=(x_{1},\ldots,x_{r})\in V^{\vee}=\prod_{i=1}^{r}V_{i}^{\vee}$, we denoted by $Y^{\prime}_{x_{i}}$ the subvariety in $X^{\prime}$ defined by $x_{i}=0$ (cf. Remark 1.2). We also set $Y^{\prime}_{x}:=\cup_{i=1}^{r}Y^{\prime}_{x_{i}}$ and $U^{\prime}_{x}:=X^{\prime}-Y^{\prime}_{x}$. 8. (8)

Let $X\to X^{\prime}$ be a toric resolution of singularities. Denote by $U_{x}$ and $Y_{x}$ the preimage of $U_{x}^{\prime}$ and $Y_{x}^{\prime}$ inside $X$ respectively.

Given $\beta\in\mathbb{C}^{r+n}$, the $A$-hypergeometric ideal $I_{A,\beta}$ is the left ideal of the Weyl algebra $\mathscr{D}=\mathbb{C}[x,\partial]$ on the dual vector space $V^{\vee}$ generated by the following two types of operators

• •

The “box operators”: $\partial^{\nu_{+}}-\partial^{\nu_{-}}$, where $\nu_{\pm}\in\mathbb{Z}_{\geq 0}^{N}$ satisfy $A\nu_{+}=A\nu_{-}$. Here for $m\in\mathbb{Z}_{\geq 0}^{N}$ we write $\partial^{m}=\partial_{1,1}^{m_{1,1}}\cdots\partial_{r,N_{r}}^{m_{r,N_{r}}}$.

• •

The “Euler operators”: $E_{l}-\beta_{l}$, where $E_{l}=\sum_{i,j}\langle\mu_{i,j},\mathrm{e}_{l}\rangle x_{i,j}\partial_{i,j}$. Here $\mathrm{e}_{l}=(\delta_{l,1},\ldots,\delta_{l,n+r})\in\mathbb{Z}^{r+n}$.

We shall only be interested in a special $\beta$, namely

[TABLE]

The $A$-hypergeometric system $M_{A,\beta}$ is the cyclic $\mathscr{D}$-module $\mathscr{D}/I_{A,\beta}$. As shown by Gelfand et. al. [1989-Gelfand-Kapranov-Zelebinski-hypergeometric-functions-and-toral-manifolds], (under our hypothesis on $a_{i,j}$) and Adolphson [1994-Adolphson-hypergemetric-functions-and-rings-generated-by-monomials] (in general), $M_{A,\beta}$ is a holonomic $\mathscr{D}$-module.

Remark 1.2**.**

The $\mathscr{D}$-module $M_{A,\beta_{0}}$ naturally arises from the study of period integrals on toric varieties. Let $(X,L)$ be an $n$-dimensional polarized toric variety defined by a full-dimensional polytope $\Delta\subset\mathbb{Z}^{n}$. We assume that $\mathbf{0}$ is an interior point of $\Delta$. In particular, the section of $L$ corresponding to $\mathbf{0}$ is supported on the union of all the torus-invariant divisors.

Suppose further that there exists a collection of base-point-free invertible sheaves $L_{i}$ on $X$ such that $L=L_{1}\otimes\cdots\otimes L_{r}$. Then the section polytope $\Delta_{i}$ of $L_{i}$ satisfies the condition that $\mathbf{0}\in\Delta_{i}$ (since $L_{i}$ contains some torus-invariant divisor as its section), and $\Delta=\Delta_{1}+\cdots+\Delta_{r}$.

By assumption, each $L_{i}$ determines a morphism $X\to\mathbb{P}(V_{i})$ with $V_{i}=\mathrm{H}^{0}(X,L_{i})^{\vee}$. Put $V=V_{1}\times\cdots\times V_{r}$. A generic element $x=(x_{1},\ldots,x_{r})\in V^{\vee}$ determines a smooth complete intersection $\cap_{i=1}^{r}Y_{x_{i}}$ in $X$. We can write

[TABLE]

(recall that $(t_{1},\ldots,t_{n})$ is the coordinate of the embedded torus). These data, together with $a_{i,j}=(\delta_{i,1},\ldots,\delta_{i,r})^{t}$, $1\leq i\leq r$ and $1\leq j\leq N_{i}:=\dim V_{i}$, gives a matrix $A$ which meets the requirements in Situation 1.1.

The $A$-hypergeometric system $M_{A,\beta_{0}}$ associated with the matrix $A$ just constructed then contains “period integrals” as its solutions. On the open embedded torus $(\mathbb{C}^{\ast})^{n}$, we can consider

[TABLE]

for some homology class $\tilde{C}\in\mathrm{H}_{n}((\mathbb{C}^{\ast})^{n}\cap(X\setminus\cup_{i=1}^{r}Y_{x_{i}}),\mathbb{Z})$. One can check directly that (1.3) is a solution to $M_{A,\beta_{0}}$ where

[TABLE]

This integral can be related to the homology of the affine part of the complete intersection $Y_{x_{1}}\cap\cdots\cap Y_{x_{r}}$ via the well-known residue method.

1.4**.**

Hypothesis. Throughout this note, we assume that the vector $-\beta_{0}$ lies in the interior of the cone generated by column vectors of $A$. This hypothesis is needed to ensure that we can apply a theorem of Reichelt [2014-Reichelt-laurent-polynomials-GKZ-hypergeometric-systems-and-mixed-hodge-modules]. See Lemma 2.3.

This hypothesis is not as technical as it may look. For instance, when $A$ is from “toric geometry” as in Remark 1.2, this hypothesis is always verified as long as $\mathbf{0}$ is an interior point of $\Delta$. This is a consequence of the following lemma.

Lemma 1.5**.**

Given a lattice polytope $\Delta$, not necessarily reflexive, with $\mathbf{0}\in\mathrm{int}(\Delta)$ and a decomposition $\Delta=\Delta_{1}+\cdots+\Delta_{r}$ with $\Delta_{i}$ being integral lattice polytopes, let $A$ be the integral matrix defined in Remark 1.2 from these data. Assume further that $\mathbf{0}\in\Delta_{i}$ for all $i$, then $-\beta_{0}$ lies in the interior of the cone generated by column vectors of $A$.

Proof.

We regard each column of $A$ as a vector in $\mathbb{R}^{r+n}$ and let $(x_{1},\ldots,x_{n+r})$ be the coordinates. Let $\mathbb{R}_{\geq 0}A$ be the cone generated by the columns of $A$. Looking at the intersection of $\mathbb{R}_{\geq 0}A$ with the affine hyperplane $H$ defined by $x_{1}+\cdots+x_{r}=1$. Note that all the column vectors of $A$ are sitting in this hyperplane. Then $-\beta_{0}$ lies in the interior of $\mathbb{R}_{\geq 0}A$ if and only if $-\beta_{0}/r$ lies in the interior of the convex hull determined by the column vectors of $A$.

We want to prove that there exists an $\epsilon>0$ such that

[TABLE]

($B_{\epsilon}(\rho)$ is the open ball of radius $\epsilon$ centered at $\rho$). First note that any $y\in\Delta$, there exist $0\leq t_{i}\leq 1$ such that $y=\sum_{i}t_{i}m_{i}$, where $m_{i}$ are vertexes of $\Delta$, since a polytope is the convex hull of its vertexes. Since $\mathbf{0}\in\mathrm{int}(\Delta)$, by rescaling $\Delta$, for any $R>0$ we may find a $\delta>0$ such that for each $y\in B_{\delta}(\mathbf{0})$ we have $y=\sum_{i}t_{i}m_{i}$ with $0\leq t_{i}<1/R$.

Now we can choose $\epsilon$ small enough such that

[TABLE]

We can find $0\leq t_{i}\leq 1/R$ such that $(z_{r+1},\ldots,z_{r+n})=\sum t_{i}m_{i}$. Let $a_{i}$ be the colume vector of $A$ corresponding to $m_{i}$. Then

[TABLE]

where $q_{i}=\sum_{j:m_{j}\in\Delta_{i}}t_{j}$. We can take $R$ large such that $q_{i}<1/r$ for all $i=1,\ldots,r$. Let $b_{i}$ be the column vector corresponding to $\mathbf{0}\in\Delta_{i}$. We have

[TABLE]

This completes the proof. ∎

Now we can state our main theorem.

Theorem 1.6**.**

Adapt the notation introduced in 1.1. Under Hypothesis 1.4 and the assumptions on $A$ made in 1.1, if

[TABLE]

then for any toric resolution $X\to X^{\prime}$ (in fact, we only need to assume that $X$ to be a smooth algebraic variety containing $T$ as an open dense subset, and the morphism $X\to X^{\prime}$ restricts to the identity on $T$) and any $x\in V^{\vee}$, we have

[TABLE]

Here $\mathrm{Sol}^{0}(-)=R^{0}\mathcal{H}om_{\mathscr{D}_{V^{\vee}}}(-,\widehat{\mathcal{O}}_{V^{\vee},x})$ is the underived solution functor of $\mathscr{D}$-modules which outputs the set of formal power series solutions of a $\mathscr{D}$-module around a point $x$.

When $X^{\prime}$ itself is smooth, $r=1$ and $Y_{x}$ are Calabi–Yau, (1.6) was proved by Huang et. al. [2016-Huang-Lian-Yau-Zhu-chain-integral-solutions-to-tautological-systems] using the general theory of [2016-Huang-Lian-Zhu-period-integrals-and-the-riemann-hilbert-correspondence]. The said theory, as we understood, requires the smoothness hypothesis in a crucial way.

An Huang has informed us, in a private communication, that he can prove (1.6) for mirror quintics. Related to our result is Jie Zhou’s work [2017-Zhou-gkz-hypergeometric-series-for-the-hesse-pencil-chain-integrals-and-orbifold-singularities]. He gives an explicit description of the solutions to the $A$-hypergeometric system associated with the matrix

[TABLE]

using relative homology classes on the Hesse pencil of elliptic curves.

An application

Let $\Delta_{i}$, $1\leq i\leq r$, be polytopes such that $\mathbf{0}\in\Delta_{i}$ for all $i$ and $\Delta=\Delta_{1}+\cdots+\Delta_{r}$. Assume that $\mathbf{0}$ is an interior point of $\Delta$. The integral points in $\Delta_{i}$ define an integral matrix $A$ as in Remark 1.2. We retain the notation in Situation 1.1 and let $\beta_{0}=(-1,\ldots,-1,0,\ldots,0)\in\mathbb{C}^{r}\times\mathbb{C}^{n}$.

Corollary 1.7** (Existence of rank 1 points).**

Let notation be as above. There exists a point $x\in V^{\vee}$ such that $\mathrm{Sol}^{0}(M_{A,\beta_{0}},\widehat{\mathcal{O}}_{V^{\vee},x})$ is of rank one.

Proof.

We choose $x$ to be the section corresponding to the lattice points $(0,\ldots,0)$, which exists since $\mathbf{0}\in\Delta_{i}$. The hypersurface $Y_{x}$ is just the union of all toric divisors and $U_{x}=T$. The assertion follows since $\mathrm{H}_{n}(T)$ is of rank one. ∎

The $r=1$ case was essentially proved by Hosono–Lian–Yau [1997-Hosono-Lian-Yau-maximal-degeneracy-points-of-gkz-systems] using another approach. The idea that one can use a certain cohomological interpretation to prove the existence of rank 1 point is due to Huang–Lian–Zhu [2016-Huang-Lian-Zhu-period-integrals-and-the-riemann-hilbert-correspondence].

Remark on the injectivity of $\tau$

In Situation 1.1 we assumed the map $\tau$ is injective. When $\beta=\beta_{0}$, we demonstrate below how to achieve this by modifying $A$ without changing the ambient $\mathscr{D}$-module $M_{A,\beta_{0}}$

Let $A$ be an $(r+n)\times N$ integral matrix. Assume that $A$ satisfies all the hypotheses in Situation 1.1, except the item (5), i.e., we do not assume that the toric mapping $\tau$ is injective. We impose the following condition on $A$:

[TABLE]

Let $C$ be the $n\times N$ integral matrix obtained by removing the first $r$ rows from $A$. Regarding $C$ as a map $\mathbb{C}^{N}\to\mathbb{C}^{n}$, $\mathrm{im}(C)$ is subgroup in $\mathbb{Z}^{n}$ of finite index. Let $\{\mathrm{f}_{1},\ldots,\mathrm{f}_{n}\}$ be a $\mathbb{Z}$-basis of $\mathrm{im}(C)$ and $\{\mathrm{e}_{1},\ldots,\mathrm{e}_{n}\}$ be the standard $\mathbb{Z}$-basis of $\mathbb{Z}^{n}$. They are related by

[TABLE]

$B=(b_{ji})$ is clearly an integral matrix and $B\in\mathrm{GL}_{n}(\mathbb{Q})$. Note that $B^{-1}$ may not be integral. However, $B^{-1}C$ is an integral matrix. To see this, if $c_{k}$ is the $k$th column of $C$, we write

[TABLE]

$d_{lk}\in\mathbb{Z}$ since $\{\mathrm{f}_{1},\ldots,\mathrm{f}_{n}\}$ is a $\mathbb{Z}$-basis of $\mathrm{im}(C)$. Put $D=(d_{lk})$. From the relation $C=BD$, we deduce that $B^{-1}C=D$ is integral and the columns of $D$ generate $\mathbb{Z}^{n}$. Now the condition ($\ast$ ‣ 1) ensures that the columns of the matrix

[TABLE]

generate $\mathbb{Z}^{r+n}$. We thus deduce that

Lemma 1.8**.**

Let $A$ be as above. Assume further that $A$ has property ($\ast$ ‣ 1). Let $\tau$ be the morphism defined by $A$. We denote by $\left.\tau\right|_{T}$ the pullback of $\tau$ via $T\to(\mathbb{C}^{\ast})^{r}\times T$, $t\mapsto(\mathbf{1},t)$. Let $B\in\mathrm{GL}_{n}(\mathbb{Q})$ such that the columns of $B$ form an integral basis for the image torus $T^{\prime}:=\mathrm{im}(\left.\tau\right|_{T})$. Define the $(r+n)\times(r+n)$ matrix $R$

[TABLE]

Then the columns of $RA$ generate $\mathbb{Z}^{r+n}$ as a $\mathbb{Z}$-module.

Lemma 1.9**.**

Let $A$ and $R$ be as in Lemma 1.8. We have $M_{A,\beta_{0}}=M_{RA,\beta_{0}}$.

Proof.

For a pair of non-negative integral vectors $\nu_{\pm}$, $A\nu_{+}=A\nu_{-}$ if and only if $RA\nu_{+}=RA\nu_{-}$. Thus we get the same collection of box operators. Since the first $r$ rows of $A$ and $RA$ are the same, the Euler operators induced by the $(\mathbb{C}^{\ast})^{r}$-action remain unchanged. The Euler operators associated with $T$ action and $T^{\prime}$ action differed by $B$. Since the character set $\beta_{0}$ has last $n$ entries equal to zero, The Euler equations associated with $T$ and $T^{\prime}$ are also the same. ∎

In [1993-Aspinwall-Greene-Morrison-the-monomial-divisor-mirror-map], Aspinwall, Greene, and Morrison introduced the “simplified moduli space” to approximate the (polynomial) moduli space of Calabi–Yau hypersurfaces in toric varieties. Given a lattice reflexive polytope $\Delta\subset M_{\mathbb{R}}\simeq\mathbb{R}^{n}$, the normal fan of $\Delta$ determines an $n$-dimensional toric variety with maximal torus $T=(\mathbb{C}^{\ast})^{n}$. Let $(\Delta\cap M)_{0}$ be the set of integral points which do not lie in the interior of any codimension one face of $\Delta$. We denote by $\mathbb{C}^{\#(\Delta\cap M)_{0}}$ the vector space of Laurent polynomials of the form

[TABLE]

The $T$-action on $(\Delta\cap M)_{0}$ induces a $T$-action on $\mathbb{C}^{\#(\Delta\cap M)_{0}}$. Let $\mathbb{C}^{\ast}$ act on $\mathbb{C}^{\#(\Delta\cap M)_{0}}$ by an overall scaling. The “simplified moduli space” is defined by the GIT quotient $\mathbb{C}^{\#(\Delta\cap M)_{0}}\sslash(\mathbb{C}^{\ast}\times T)$.

Let $A_{0}$ be the integral matrix whose columns consist of the vectors of the form $(1,m)^{t}$ with $m\in(\Delta\cap M)_{0}$. The “period integrals” on the space $\mathbb{C}^{\#(\Delta\cap M)_{0}}$ as well as on its GIT quotient $\mathbb{C}^{\#(\Delta\cap M)_{0}}\sslash(\mathbb{C}^{\ast}\times T)$ can be studied by the GKZ system $M_{A_{0},\beta_{0}}$ with $\beta_{0}=(-1,0,\ldots,0)$ as usual. The matrix $A_{0}$ satisfies the condition ($\ast$ ‣ 1) (with $r=1$). We can apply our method in Lemma 1.8 to solve the GKZ system $M_{A_{0},\beta_{0}}$. Here is an example.

Example 1.10**.**

Let $\Delta$ be the convex hull of $(-1,-1)$, $(2,-1)$ and $(-1,2)$ in $\mathbb{R}^{2}$. $\Delta$ can be identified with the section polytope of $-K_{\mathbb{P}^{2}}$. $(\Delta\cap M)_{0}$ consists of four integral points: $(-1,-1)$, $(2,-1)$, $(-1,2)$ and $(0,0)$. The vector space of Laurent polynomials $\mathbb{C}^{\#(\Delta\cap M)_{0}}$ is the Dwork family in $\mathbb{P}^{2}$:

[TABLE]

Let $\beta_{0}=(-1,0,0,0)$ as before. The corresponding $A_{0}$ matrix in the GKZ system is

[TABLE]

The columns of $A_{0}$ do not generate $\mathbb{Z}^{3}$. However, we can perform row operations on the last two rows of $A_{0}$ to get

[TABLE]

The matrices $B$ and $B^{-1}$ in this case are

[TABLE]

and we have

[TABLE]

From the discussion above,

[TABLE]

with $X$ being a resolution of “mirror $\mathbb{P}^{2}$” and $U_{x}=X\setminus Y_{x}$.

Independence of relative homology.

As we have been asked several times, we should explain the (trivial) fact that the relative homology groups displayed above are independent of the choice of the resolution, i.e., the second equality in (1.6). We choose to use the language of sheaves to show this.

Let $W$ be a (locally quasi-compact, Hausdorff) topological space (in our case $W=U_{x}\cap T$). Assume that there is a commutative diagram

[TABLE]

of (locally quasi-compact, Hausdorff) topological spaces, in which $j_{1}$ and $j_{2}$ are open embeddings and $\varphi$ is proper. Then we know

[TABLE]

where $\omega_{W}$ is the dualizing complex of $W$ (this can be also served as the definition of the relative homology groups). But then

[TABLE]

This proves that the relative homology groups are independent of the choice of the $Z_{i}$’s. Taking $Z_{i}$ to be $U_{x}$ and $U^{\prime}_{x}$ respectively proves the independence of relative homology groups in (1.6).

2. $\beta_{0}$ is semi-nonresonant

The proof of (1.6) contains an input: a comparison theorem between an extraordinary Gauss–Manin system and the $A$-hypergeometric system, proved by Reichelt [2014-Reichelt-laurent-polynomials-GKZ-hypergeometric-systems-and-mixed-hodge-modules] (see also the results of Walther and Schulze [2009-Schulze-Walther-hypergeometric-d-modules-and-twisted-gauss-manin-systems]). In this section we verify that our parameter $\beta_{0}$ is semi-nonresonant, thus we can apply Reichelt’s theorem.

Definition 2.1**.**

Let notation be as in Situation 1.1. We say $\beta$ is semi-nonresonant if

[TABLE]

where the union is taken over all the faces $F$ of $A$. Recall that a face $F$ of $A$ is a subset of columns of $A$ that minimizing some nonzero linear functional on the cone generated by $A$. $\mathbb{Q}_{\geq 0}A$ denotes the $\mathbb{Q}_{\geq 0}$-span of the columns of $A$ and $\mathbb{C}F$ the $\mathbb{C}$-span of $F$.

Theorem 2.2** (Reichelt).**

In Situation 1.1, if $\beta=(\beta_{i})\in\mathbb{C}^{r+n}$ is semi-nonresonant, then we have

[TABLE]

Here $\mathrm{FT}$ stands for the Fourier–Laplace transform of $\mathscr{D}$-modules and $\mathcal{O}_{(\mathbb{C}^{\ast})^{r}\times T}^{\beta}$ is the cyclic $\mathscr{D}$-module

[TABLE]

For the proof, see [2014-Reichelt-laurent-polynomials-GKZ-hypergeometric-systems-and-mixed-hodge-modules]*Proposition 1.14.

The following lemma enables us to apply Reichelt’s theorem in our setting.

Lemma 2.3**.**

In situation 1.1, under Hypothesis 1.4, the parameter $\beta_{0}$ is semi-nonresonant.

Proof.

Suppose on the contrary that $\beta_{0}\in\bigcup_{F}(\mathbb{Z}^{r+n}\cap\mathbb{Q}_{\geq 0}A)+\mathbb{C}F$. We can find $m\in(\mathbb{Z}^{r+n}\cap\mathbb{Q}_{\geq 0}A)$, $c\in\mathbb{C}$ and $f\in F$ such that $\beta_{0}=m+c\cdot f$. Then we have

[TABLE]

By our Hypothesis 1.4, $m+(-\beta_{0})$ is contained in the interior of the cone generated by column vectors of $A$, but $-c\cdot f$ is in the boundary. This is a contradiction: we pick a nonzero linear functional $h:\mathbb{R}^{r+n}\to\mathbb{R}$ defining $F$, we then deduce from the minimizing property that $h(-c\cdot f)<h(m+(-\beta_{0}))$. ∎

3. Functors on $\mathscr{D}$-modules

To fix notation used throughout this note, we recall some notions in algebraic $\mathscr{D}$-modules.

Let $X$ be a smooth variety and $\mathscr{D}_{X}$ be the sheaf of differential operators on $X$. By a $\mathscr{D}_{X}$-module on $X$ we always mean a left $\mathscr{D}_{X}$-module. Let $D^{b}_{h}(\mathscr{D}_{X})$ be the bounded derived categories of $\mathscr{D}$-modules over $X$ with holonomic cohomology sheaves. Let $D^{b}_{rh}(\mathscr{D}_{X})$ be the derived category of complex of $\mathscr{D}_{X}$-modules with regular holonomic cohomology sheaves. One can define the duality functor, denoted by $\mathcal{M}\mapsto\mathbb{D}\mathcal{M}$, on $D_{h}^{b}(\mathscr{D}_{X})$. Let $f:X\to Y$ be a morphism between smooth varieties. One can define the following functors

• •

For a complex $\mathcal{M}\in D_{h}^{b}(\mathscr{D}_{X})$, let $f_{+}(\mathcal{M}):=Rf_{\ast}(\mathrm{dR}_{X/Y}(\mathcal{M}))$, where $\mathrm{dR}_{X/Y}$ is the relative de Rham functor.

• •

For a complex $\mathcal{N}\in D_{h}^{b}(\mathscr{D}_{Y})$, let $f^{!}\mathcal{N}:=f^{\ast}\mathcal{N}[\dim X-\dim Y]$, where $f^{\ast}$ is the derived pullback on the category of quasi-coherent $\mathcal{O}_{Y}$-modules.

Note that these functors can be defined on the category of $\mathscr{D}$-modules without the holonomic condition. Nonetheless, all the functors $\mathbb{D}$, $f_{+}$ and $f^{!}$ preserve the holonomicity. We put

• •

$f^{+}:=\mathbb{D}_{X}f^{!}\mathbb{D}_{Y}$, and

• •

$f_{!}:=\mathbb{D}_{Y}f_{+}\mathbb{D}_{X}$.

$f^{+}$ is the left adjoint of $f_{+}$ and $f_{!}$ is the left adjoint of $f^{!}$.

When $f$ is a smooth morphism, or more generally non-characteristic with respect to a holonomic $\mathscr{D}$-module $\mathcal{M}$, we have $f^{\ast}\mathcal{M}=f^{!}\mathcal{M}[\dim Y-\dim X]=f^{+}\mathcal{M}[\dim X-\dim Y]$. Finally, given a cartisian diagram

[TABLE]

with all varieties are smooth, then we have

[TABLE]

Let $S\subset X$ be a (possibly singular) subscheme of $X$ and $\mathscr{I}_{S}$ be the corresponding ideal sheaf. For a $\mathcal{O}_{X}$-module $\mathcal{F}$ on $X$, we define

[TABLE]

The quasi-coherent $\mathcal{O}_{X}$-module $\Gamma_{[S]}(\mathcal{F})$ inherits a $\mathscr{D}_{X}$-module structure and we can consider its right derived functor $R\Gamma_{[S]}$. When $\mathcal{M}$ is a complex with holonomic cohomology sheaves, so is $R\Gamma_{[S]}(\mathcal{M})$. Let $j:X\setminus S\to X$ be the open embedding. For $\mathcal{M}\in D^{b}_{h}(\mathscr{D}_{X})$ we have the distinguished triangle

[TABLE]

Let $i:S\to X$ be the closed embedding. In case $S$ is smooth, we have $R\Gamma_{[S]}(\mathcal{M})\simeq i_{+}i^{!}\mathcal{M}$ and the distinguished triangle (3.1) becomes

[TABLE]

Therefore we shall sometimes abuse notation and use $i_{+}i^{!}$ instead of $R\Gamma_{[S]}$ even when $S$ is singular.

The proofs of the above said results can be found in [2003-Baldassarri-DAgnolo-on-dwork-cohomology-and-algebraic-d-modules].

4. Cohomology computation

In this section, we explain how to use exponential twists to compute cohomology groups. There are three topics:

• •

how to use an exponential twisted de Rham cohomology (Dwork cohomology) to compute the cohomology of a complete intersection,

• •

how to use an exponential twisted de Rham cohomology to compute cohomology of the complement of anormal-crossing divisor, and finally

• •

what is the sheaf-theoretic mechanism for computing relative cohomology.

The main result is Lemma 4.8.

Definition 4.1**.**

Let $\gamma:Z\to\mathbb{A}^{1}$ be a morphism between smooth algebraic varieties. We define the exponential $\mathscr{D}$-module on $Z$ to be

[TABLE]

This is a holonomic, but generally irregular, $\mathscr{D}$-module on $Z$.

4.3**.**

Let $X$ be a smooth algebraic variety and $\pi:E\to X$ be a rank $r$ vector bundle. Let $\sigma:X\to E^{\vee}$ be a section of the dual bundle and $\Sigma$ be the reduced zero scheme of $\sigma$. We consider the following commutative diagram, which will be used frequently throughout this note.

[TABLE]

In this diagram, $\iota$ the embedding of the zero section, $\gamma$ is the natural pairing, $i$ is the closed embedding, $\varepsilon$ is the pullback of $\sigma$, and $F=\gamma\circ\varepsilon$.

Definition 4.4**.**

Let notation be as in Situation 4.3. For a holomonic complex $\mathcal{M}\in D^{b}_{h}(\mathscr{D}_{X})$, we define the Dwork complex of $\mathcal{M}$, denoted by $\mathrm{Dw}_{E/X}(\mathcal{M})$ or simply $\mathrm{Dw}(\mathcal{M})$, to be

[TABLE]

A priori, $\mathrm{Dw}_{E/X}(\mathcal{M})$ seems to be irregular, but the following theorem implies it is in fact regular.

Theorem 4.6**.**

In Situation 4.3, we have an isomorphism

[TABLE]

for any object $\mathcal{M}\in D^{b}_{h}(\mathscr{D}_{X})$.

The theorem was obtained by many people in various situations: by Katz [1968-Katz-on-the-differential-equations-satisfied-by-period-matrices] when $X$ is affine and $E$ is the trivial line bundle, by Adolphson–Sperber [2000-Adolphson-Sperber-dwork-cohomology-de-rham-cohomology-and-hypergeometric-functions] when $X$ is the affine space but $E$ can have higher rank, and by Dimca–Maaref–Sabbah–Saito [2000-Dimca-Maaref-Sabbah-Saito-dwork-cohomology-and-algebraic-d-modules], independently Baldassarri–D’Agnolo [2003-Baldassarri-DAgnolo-on-dwork-cohomology-and-algebraic-d-modules], when $X$ and $E$ are both general. The reader is referred to [2003-Baldassarri-DAgnolo-on-dwork-cohomology-and-algebraic-d-modules] for a proof of the theorem as its proof is more related to our note.

For us, what is important is that the isomorphy for a general $\mathcal{M}$ is obtained from the isomorphy for $\mathcal{O}_{X}$ by tensoring with $\mathrm{id}_{\mathcal{M}}$. This point is clear from Baldassarri–D’Agnolo’s proof we just cited, i.e., the following diagram commutes

[TABLE]

In the diagram the horizontal arrows are the canonical isomorphisms.

4.7**.**

Next let us explain how to compute the cohomology of the complement of a normal crossing divisor using an exponential twist. Consider the following situation.

• •

Let $X$ be a smooth algebraic variety.

• •

Let $\mathcal{L}_{1},\ldots,\mathcal{L}_{r}$ be $r$ invertible sheaves on $X$.

• •

Let $\mathbb{L}_{m}=\mathit{Spec}(\mathrm{Sym}^{\bullet}\mathcal{L}_{m}^{\vee})$, for $m=1,\ldots,r$. Let $\mathbb{L}_{m}^{\vee}$ be the dual of $\mathbb{L}_{m}$.

• •

Let $\mathbb{L}=\mathbb{L}_{1}\times_{X}\cdots\times_{X}\mathbb{L}_{r}$. This is a vector bundle of rank $r$. Let $\pi:\mathbb{L}\to X$ be the bundle projection.

• •

Let $\sigma_{m}\in\mathrm{H}^{0}(X,\mathcal{L}_{m}^{\vee})$. Let $H_{m}=V(\sigma_{m})\subset X$, and $H=\bigcup_{m=1}^{r}H_{m}$. Let $\rho:U=X\setminus H\to X$ be the open immersion. For a subset $I\subset\{1,2,\ldots,r\}$, let $H_{I}=\bigcap_{m\in I}H_{m}$. For an integer $k$, define $H_{(k)}=\coprod_{\#{I}=k}H_{I}$. Let $i_{k}:H_{(k)}\to X$ be the natural morphism induced by the inclusions.

• •

The sections $\sigma_{m}$ define a function $F:\mathbb{L}\to\mathbb{A}^{1}$.

• •

Let $Z_{m}\subset\mathbb{L}$ be the fiber product

[TABLE]

where $[0_{m}]$ is the zero section of $\mathbb{L}_{m}$. Then $Z_{m}$ are Cartier divisors on $\mathbb{L}$.

• •

For $I\subset\{1,2,\ldots,r\}$, let $Z_{I}=\bigcap_{m\in I}Z_{m}$. Then $Z_{I}$ are vector bundles over $X$ of rank $r-\#I$. Hence $Z=Z_{1}\cup\ldots\cup{}Z_{m}$ is a simple normal divisor of $\mathbb{L}$. Let $Z_{(k)}=\coprod_{\#I=k}Z_{I}$. Let $\iota_{k}:Z_{(k)}\to\mathbb{L}$ be the natural morphism.

• •

Let $\theta:\mathbb{L}^{\circ}\to\mathbb{L}$ be the open immersion whose complement is $Z$.

Lemma 4.8**.**

Let notation be as in Situation 4.7. Let $\mathcal{M}$ be a bounded complex of $\mathscr{D}_{X}$-modules with regular singular, holonomic cohomology sheaves. Assume that either $r=1$, or $r>1$ but assume further that the $H_{i}$’s meet transversely. Then there is an isomorphism

[TABLE]

Proof.

Let $\mathcal{K}$ be a regular holonomic complex of $\mathscr{D}_{\mathbb{L}}$-modules. Using the covariant Riemann–Hilbert correspondence, that $Z$ is a normal crossing divisor, and results in topology, we infer that the complex $\theta_{!}\theta^{!}\mathcal{K}$ can be computed by the following double complex

[TABLE]

Since $\pi$ is affine, and since the complexes above underlie complexes of quasi-coherent $\mathcal{O}_{\mathbb{L}}$-modules, $\pi_{+}$ has no higher direct images. Thus, if we take $\mathcal{K}=\pi^{\ast}\mathcal{M}$, we see

[TABLE]

is computed by the direct image

[TABLE]

In what follows, we shall compute the complexes $\pi_{+}(\iota_{k+}\iota_{k}^{+}\pi^{\ast}\mathcal{M}\otimes\exp(F))$. We claim that there is an isomorphism

[TABLE]

Using projection formula, we have

[TABLE]

Since $Z_{(k)}$ is a disjoint union, we can treat each $Z_{I}$ separately. We shall only consider the case $I=\{r-k+1,\ldots,r\}$. The proofs for the rest cases are the same, but one needs to use different index sets. The variety $Z_{I}$ is the same as the bundle $\mathbb{L}_{1}\times_{X}\cdots\times_{X}\mathbb{L}_{r-k}$. Let $\pi_{k}$ be the bundle projection $\mathbb{L}_{1}\times_{X}\cdots\times_{X}\mathbb{L}_{r-k}\to X$.

To simplify the notation, set $\nu_{k}=\iota_{k}|_{Z_{I}}$. Note that $\pi\circ\nu_{k}=\pi_{k}$. Moreover, the function $F|_{Z_{I}}$ is identified with the function $F_{k}:\mathbb{L}_{1}\times_{X}\cdots\times_{X}\mathbb{L}_{r-k}\to\mathbb{A}^{1}$ defined by the sections $\sigma_{1},\ldots,\sigma_{r-k}$. Then on $Z_{I}$ we have

[TABLE]

In the last step we used the isomorphism provided by Theorem 4.6. This completes the proof of (4.10). Thus, the direct image $\pi_{+}((\theta_{!}\theta^{!}(\pi^{\ast}\mathcal{M}))\otimes\exp(F))$ is computed by the a double complex of the form

[TABLE]

The arrows $a_{m}(\mathcal{M})$ are obtained by pushing forward the arrows in (4.9). On the other hand, there are canonical arrows

[TABLE]

[These canonical arrows are from the distinguished triangle

[TABLE]

associated with a closed embedding $u$ of spaces (whose complement open immersion denoted by $v$).]

We do not know whether $a_{m}(\mathcal{M})=b_{m}(\mathcal{M})$. They probably do equal. But we only need to know that they differ by a constant on each direct summand. To see this, we first assume $\mathcal{M}=\mathcal{O}_{X}$. In this case, on each $H_{I}$, we know that $\mathrm{Hom}_{\mathscr{D}_{H_{I}}}(\mathcal{O}_{H_{I}},\mathcal{O}_{H_{I}})=\mathbb{C}^{c}$ where $c$ is the number of connected components of $H_{I}$. Let $u:H_{I}\to H_{J}$ be the inclusion for some $J$ such that $\#J-\#I=1$. Then $a_{m}(\mathcal{O}_{X})$ has a direct summand which is induced by the shift of an arrow

[TABLE]

But by the previous equality, such an arrow, if nonzero, is unique up to a nonzero scalar on each component. Since $a_{m}(\mathcal{M})$ is obtained by $a_{m}(\mathcal{O}_{X})\otimes\mathrm{id}_{\mathcal{M}}$ by the proof of Theorem 4.6, we conclude that $a_{m}(\mathcal{M})$ agrees with the canonical one up to a scalar on each of its direct summand.

To conclude, we can deduce from the above that the double complex (4.11) is isomorphic to the double complex

[TABLE]

When $r=1$, we have won. Assume below that $r>1$. Applying the duality, we see (4.12) is the dual of the complex

[TABLE]

which is the complex that computes $\rho_{!}\rho^{!}\mathbb{D}\mathcal{M}$. This can be seen, again, by applying the covariant Riemann–Hilbert correspondence. Under Riemann–Hilbert correspondence the complex becomes the standard Mayer–Vietories type complex computing the cohomology of a normal crossing variety — this is the reason that we assumed, when $r>1$, that $\cup H_{i}$ is a normal crossing divisor. Taking Verdier dual again, we conclude that (4.12) computes $\rho_{+}\rho^{+}\mathcal{M}$. This completes the proof. ∎

4.13**.**

Relative cohomology. We review the mechanism of computing relative cohomology in terms of the language of $\mathscr{D}$-modules (via the Riemann–Hilbert correspondence).

Let $X$ be a smooth, proper, algebraic variety. Let $b:T\to X$ be an affine open immersion with a complement divisor $D$, possibly singular. Let $Y$ be a Cartier divisor on $X$ with complement $U$. Assume that $Y$ is smooth. Let $Y_{T}=Y\cap T$, $Y_{D}=Y\cap D$, etc. Let $\mathcal{M}\in D^{b}(\mathscr{D}_{X})$. We form the following diagram

[TABLE]

There is a complex of $\mathscr{D}$-modules that “computes” the relative cohomology of $\rho^{!}\mathcal{M}$ on $U$ with respect to $U_{D}$ (in the sense that it corresponds to the complex whose cohomology is relative cohomology, via the Riemann–Hilbert correspondence): this is $c_{!}c^{!}(\rho^{!}\mathcal{M})$.

Let us explain this if the reader is not familiar to this formalism. In fact if $\rho^{!}\mathcal{M}$ corresponds to a constructible complex $F$ under the covariant Riemann–Hilbert functor, then there is a distinguished triangle

[TABLE]

(taking $R\Gamma(U,-)$ recovers the usual long exact sequence of sheaf cohomology groups). The corresponding version of the above distinguished triangle in the language of $\mathscr{D}$-modules is

[TABLE]

Note that $\rho$ is an open embedding, which implies $\rho^{!}=\rho^{\ast}$ and $\rho_{T}^{!}=\rho_{T}^{\ast}$. Manipulating with base change functors we have

[TABLE]

We can put the discussion above in the relative setting. Assume that all schemes in the diagram (4.14) are schemes over a smooth $\mathbb{C}$-variety $B$ and let $p:X\to B$ be the structure morphism. Then the “variation” of the relative cohomology groups (up to a shift) of the pair $(U,U\cap D)$ is computed by the complex

[TABLE]

where $\mathrm{RH}$ stands for the covariant Riemann–Hilbert correspondence. Recall that its Verdier dual is the derived solution complex: $(\text{Verdier duality})\circ\mathrm{RH}=\mathrm{Sol}$.

5. A lemma

In this section, we prove a lemma which will be used to explain how the Dwork complexes (and its siblings used in the previous section) can be related to the Fourier–Laplace transform of a certain complexes.

5.1**.**

Setting. Retain the notation set up in Situation 4.7. In addition, we make the following assumptions.

• •

Assume that $X$ is proper of pure dimensional $n$.

• •

Assume that each $\mathcal{L}_{i}^{\vee}$ is generated by its global sections. Put $V_{i}=\mathrm{H}^{0}(X,\mathcal{L}_{i}^{\vee})^{\vee}$. We have morphisms $X\to\mathbb{P}(V_{i})$ and their product $X\to\prod_{i=1}^{r}\mathbb{P}(V_{i})$. Let $V=V_{1}\times\cdots\times V_{r}$.

• •

We will regard each $V_{i}$ as an algebraic variety, which is $\mathrm{Spec}(\mathrm{Sym}^{\bullet}(V_{i}^{\vee}))$.

• •

For each $i$, we denote by $\pi_{i}$ the projection morphism $\pi_{i}:\mathbb{L}_{i}\to X$,

• •

For each $i$, we have a morphism $b_{i}:\mathbb{L}_{i}\to V_{i}$, which contracts the zero section to the origin. This can be obtained as follows. $V_{i}^{\vee}\otimes\mathcal{O}_{X}\to\mathcal{L}_{i}^{\vee}$ gives a morphism of algebraic varieties $\mathbb{L}_{i}\to V_{i}\times X$. Then $b_{i}$ is the composition $\mathbb{L}_{i}\to V_{i}\times X\to V_{i}$. Let $b:\mathbb{L}\to V$ be their product.

• •

Let $\mathbb{L}_{i}^{\circ}$ be the subset of $\mathbb{L}_{i}$ with its zero section removed. Thus $\mathbb{L}_{1}^{\circ}\times_{X}\cdots\times_{X}\mathbb{L}_{r}^{\circ}$ is just $\mathbb{L}^{\circ}$. Let $\theta_{i}:\mathbb{L}_{i}^{\circ}\to\mathbb{L}_{i}$ be the open inclusion. Thus $\theta:\mathbb{L}^{\circ}\to\mathbb{L}$ is the relative product of $\theta_{i}$.

• •

Let $\widetilde{\mathbb{L}}$ be the pullback of $\mathbb{L}$ via the natural projection $X\times V^{\vee}\to X$. $\widetilde{\mathbb{L}}^{\vee}$ is defined in the same manner. Let $\tilde{\sigma}:X\times V^{\vee}\to\widetilde{\mathbb{L}}^{\vee}$ be the universal section. We will write $\tilde{\sigma}=(\tilde{\sigma}_{1},\ldots,\tilde{\sigma}_{r})$. $Y_{i}=\{\tilde{\sigma}_{i}=0\}$, $1\leq i\leq r$, are the universal hypersurface determined by $\tilde{\sigma}$. Note that $\cup_{i=1}^{r}Y_{i}$ is a simple normal crossing divisor.

• •

All the morphisms obtained via the pullback along $X\times V^{\vee}\to X$ are denoted by the same symbol with a ‘tilde’. For instance, the map $\widetilde{\mathbb{L}}\to X\times V^{\vee}$ will be denoted by $\tilde{\pi}$ according to our convention.

• •

Put $\iota=b\circ\theta:\mathbb{L}^{\circ}\to V$.

These data form the following commutative diagram (cf. §4.3).

[TABLE]

In this diagram,

• •

$\widetilde{\varepsilon}$ is the pullback of $\widetilde{\sigma}$,

• •

$\gamma$ is the canonical dual pairing, and

• •

$F=\gamma\circ\widetilde{\varepsilon}$.

Lemma 5.2**.**

Let notation be as in §5.1. Let $\mathcal{M}$ be a holonomic complex of $\mathscr{D}$-modules on $\mathbb{L}^{\circ}$. Then

[TABLE]

Here $\mathrm{pr}_{V^{\vee}}:\mathbb{L}\times V^{\vee}\to V^{\vee}$ and $\mathrm{pr}_{\mathbb{L}}:\mathbb{L}\times V^{\vee}\to\mathbb{L}$ are projections.

Proof.

Look at the following commutative diagram

[TABLE]

Since $b$ is proper, $b_{+}=b_{!}$. Thus by the base change theorem, and the commutativity of the entire lower rectangle, we have

[TABLE]

Hence

[TABLE]

as claimed. ∎

6. Proof of the main theorem

We prove our main theorem in this section.

Given an $(r+n)\times N$ integral matrix $A$, the columns of $A$ give rise to a morphism $\tau:(\mathbb{C}^{\ast})^{r}\times T\to\prod_{i=1}^{r}\mathbb{P}V_{i}$ as in Situation 1.1. Let $X^{\prime}\subset\mathbb{P}V_{1}\times\cdots\times\mathbb{P}V_{r}$ be the closure of the image. It is a possibly singular toric variety. Choose a toric desingularization $X\to X^{\prime}$. Each $X\to\mathbb{P}V_{i}$ determines a base point free line bundle $\mathcal{L}_{i}^{\vee}$. We are now in the situation of 5.1. Let us retain the notation there.

We have a sequence of maps.

[TABLE]

On the one hand, owing to our hypothesis on $A$, Reichelt’s result in [2014-Reichelt-laurent-polynomials-GKZ-hypergeometric-systems-and-mixed-hodge-modules]*Proposition 1.14 implies

[TABLE]

On the other hand, applying Lemma 5.2 to $\mathcal{M}=j_{!}\mathcal{O}_{(\mathbb{C}^{\ast})^{r}\times T}$, we obtain

[TABLE]

To proceed, we consider the following commutative diagram.

[TABLE]

Note the all vertical maps are smooth. We have $\mathrm{pr}^{\ast}_{\bullet}=\mathrm{pr}^{!}_{\bullet}[-\dim V^{\vee}]=\mathrm{pr}^{+}_{\bullet}[\dim V^{\vee}]$. An iterated application of the projection formula to this diagram yields

[TABLE]

To compare this with the objects on $X$, we identify $((\mathbb{C}^{\ast})^{r}\times T)\times V^{\vee}$ with $\widetilde{\mathbb{L}}^{\circ}|_{T}$ and look at the following commutative diagram.

[TABLE]

Since $\mathcal{O}_{\widetilde{\mathbb{L}}^{\circ}|_{T}}=(\tilde{\theta}_{T})^{+}\mathcal{O}_{\widetilde{\mathbb{L}}|_{T}}=(\tilde{\theta}_{T})^{+}(\tilde{\pi}_{T})^{+}\mathcal{O}_{T\times V^{\vee}}[r]$, the last quantity in (6.3) can be transformed into

[TABLE]

Plugging the displayed equation above and (6.3) into (6.2) yields

[TABLE]

We recapitulate that the $\mathrm{pr}_{V^{\vee}+}$ appeared above is the projection from $\mathbb{L}\times V^{\vee}$ to $V^{\vee}$.

Now we can apply Lemma 4.8, by

• •

taking $X$ in the lemma to be $X\times V^{\vee}$,

• •

taking $\mathbb{L}$ in the lemma to be $\widetilde{\mathbb{L}}$, and

• •

taking $\mathcal{M}$ in the lemma to be $\beta_{!}\mathcal{O}_{T\times V^{\vee}}$.

We also use $\mathrm{pr}_{V^{\vee}}$ to denote the projection from $X\times V^{\vee}$ to $V^{\vee}$. Thus (by abuse of notation) $\mathrm{pr}_{V^{\vee}}=\mathrm{pr}_{V^{\vee}}\widetilde{\pi}$. Thus Lemma 4.8 implies that

[TABLE]

where $\rho:X\times V^{\vee}-(\bigcup_{i=1}^{r}Y_{i})\to X\times V^{\vee}$ is the open immersion. By (6.5), we have

[TABLE]

By (4.15), the Riemann–Hilbert partner of the $\mathscr{D}$-module $\mathrm{pr}_{V^{\vee}+}\rho_{+}\rho^{!}\beta_{!}\mathcal{O}_{T\times V^{\vee}}$ is $R\mathrm{pr}_{V^{\vee}*}(\left.\beta_{!}\mathbb{C}[n]\right|_{U})$. For each $x\in V^{\vee}$, let $i_{x}$ be the closed immersion. Then we have

[TABLE]

Taking the zeroth cohomology of the displayed complex and applying the duality between homology and cohomology, we have

[TABLE]