ON VANISHING THEOREMS FOR LOCAL SYSTEMS ASSOCIATED TO LAURENT POLYNOMIALS

Abstract

We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results of Gelfand et al. [Generalized Euler integrals and$A$-hypergeometric functions, Adv. Math. 84 (1990), 255–271] to various directions. In the course of the proof, some properties of vanishing cycles of perverse sheaves and twisted Morse theory are used.

1 Introduction

The study of the cohomology groups of local systems is an important subject in algebraic geometry, hyperplane arrangements, topology and hypergeometric functions of several variables. Many mathematicians are interested in the conditions for which we have their concentrations in the middle degrees. (For a review of this subject, see, for example, [4, Section 6.4].) Here, let us consider this problem in the following situation. Let $B=\{b(1),b(2),\ldots ,b(N)\}\subset \mathbb{Z}^{n-1}$ be a finite subset of the lattice $\mathbb{Z}^{n-1}$ . Assume that the affine lattice generated by $B$ in $\mathbb{Z}^{n-1}$ coincides with $\mathbb{Z}^{n-1}$ . For $z=(z_{1},\ldots ,z_{N})\in \mathbb{C}^{N}$ , we consider Laurent polynomials $P(x)$ on the algebraic torus $T_{0}=(\mathbb{C}^{\ast })^{n-1}$ defined by $P(x)=\sum _{j=1}^{N}z_{j}x^{b(j)}$ ( $x=(x_{1},\ldots ,x_{n-1})\in T_{0}=(\mathbb{C}^{\ast })^{n-1}$ ). Then, for $c=(c_{1},\ldots ,c_{n})\in \mathbb{C}^{n}$ , we obtain a possibly multivalued function $P(x)^{-c_{n}}x_{1}^{c_{1}-1}\cdots x_{n-1}^{c_{n-1}-1}$ on $W=T_{0}\setminus P^{-1}(0)$ . It generates the rank-one local system

(1.1) $$\begin{eqnarray}{\mathcal{L}}=\mathbb{C}_{W}P(x)^{-c_{n}}x_{1}^{c_{1}-1}\cdots x_{n-1}^{c_{n-1}-1}\end{eqnarray}$$

on $W$ . Under the nonresonance condition (see Definition 3.1) on $c\in \mathbb{C}^{n}$ , Gelfand et al. [10] proved that we have the concentration

(1.2) $$\begin{eqnarray}H^{j}(W;{\mathcal{L}})\simeq 0\quad (j\not =n-1)\end{eqnarray}$$

for nondegenerate Laurent polynomials $P(x)$ . This result was obtained as a byproduct of their study on the integral representations of $A$ -hypergeometric functions in [10]. Since their proof of this concentration heavily relies on the framework of the ${\mathcal{D}}$ -module theory, it is desirable to prove it more directly. In this paper, by applying the twisted Morse theory to perverse sheaves, we extend the result of Gelfand–Kapranov–Zelevinsky to various directions.

First, in Theorem 3.3, we relax the nondegeneracy condition on $P(x)$ by replacing it with a weaker one (see Definition 3.2). We thus extend the result of [10] to the case where the hypersurface $P^{-1}(0)\subset T_{0}$ may have isolated singular points in $T_{0}$ . In fact, in Theorem 3.3, we relax also the condition that $B$ generates $\mathbb{Z}^{n-1}$ to a weaker one that the dimension of the convex hull $\unicode[STIX]{x1D6E5}\subset \mathbb{R}^{n-1}$ of $B$ in $\mathbb{R}^{n-1}$ is $n-1$ . In Theorem 3.11, we extend these results to more general local systems associated to several Laurent polynomials. Namely, we obtain a vanishing theorem for arrangements of toric hypersurfaces with isolated singular points. Our proofs of Theorems 3.3 and 3.11 are very natural and are obtained by taking (possibly singular) “minimal” toric compactifications of $T_{0}$ . In order to work on such singular varieties, we use our previous idea in the proof of [7, Lemma 4.2]. See Section 3 for the details. Moreover, in Theorem 5.1 (assuming the nondegeneracy of Gelfand et al. [10] for Laurent polynomials), we relax the nonresonance condition of $c\in \mathbb{C}^{n}$ in Theorem 3.11 by replacing it with the much weaker one $c\notin \mathbb{Z}^{n}$ . To prove Theorem 5.1, we first perturb Laurent polynomials by multiplying monomials. Then, we apply the twisted Morse theory to the real-valued functions associated to them by using some standard properties of vanishing cycles of perverse sheaves. See Sections 4 and 5 for the details. In the course of the proof of Theorem 5.1, we obtain also the following result which might be of independent interest. Let $Q_{1},\ldots ,Q_{l}$ be Laurent polynomials on $T=(\mathbb{C}^{\ast })^{n}$ , and for $1\leqslant i\leqslant l$ denote by $\unicode[STIX]{x1D6E5}_{i}\subset \mathbb{R}^{n}$ the Newton polytope $NP(Q_{i})$ of $Q_{i}$ . Set $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6E5}_{1}+\cdots +\unicode[STIX]{x1D6E5}_{l}$ .

Theorem 1.1. Let ${\mathcal{L}}$ be a nontrivial local system of rank one on $T=(\mathbb{C}^{\ast })^{n}$ . Assume that for any $1\leqslant i\leqslant l$ we have $\operatorname{dim}\unicode[STIX]{x1D6E5}_{i}=n$ , and the subvariety

(1.3) $$\begin{eqnarray}Z_{i}=\{x\in T\mid Q_{1}(x)=\cdots =Q_{i}(x)=0\}\subset T\end{eqnarray}$$

of $T$ is a nondegenerate complete intersection. Then, for any $1\leqslant i\leqslant l$ , we have the concentration

(1.4) $$\begin{eqnarray}H^{j}(Z_{i};{\mathcal{L}})\simeq 0\quad (j\not =n-i).\end{eqnarray}$$

Moreover, we have

(1.5) $$\begin{eqnarray}\operatorname{dim}H^{n-i}(Z_{i};{\mathcal{L}})=\mathop{\sum }_{\substack{ m_{1},\ldots ,m_{i}\geqslant 1 \\ m_{1}+\cdots +m_{i}=n}}\operatorname{Vol}_{\mathbb{Z}}(\underbrace{\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{1}}_{m_{1}\text{-times}},\ldots ,\underbrace{\unicode[STIX]{x1D6E5}_{i},\ldots ,\unicode[STIX]{x1D6E5}_{i}}_{m_{i}\text{-times}}),\end{eqnarray}$$

where $\operatorname{Vol}_{\mathbb{Z}}(\underbrace{\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{1}}_{m_{1}\text{-times}},\ldots ,\underbrace{\unicode[STIX]{x1D6E5}_{i},\ldots ,\unicode[STIX]{x1D6E5}_{i}}_{m_{i}\text{-times}})\in \mathbb{Z}$ is the normalized $n$ -dimensional mixed volume with respect to the lattice $\mathbb{Z}^{n}\subset \mathbb{R}^{n}$ .

Note that this result can be considered as a refinement of the classical Bernstein–Khovanskii–Kushnirenko theorem (see [13]). On the other hand, Matusevich et al. [21] and Saito et al. [26] studied the condition on the parameter vector $c\in \mathbb{C}^{n}$ , for which the corresponding local system of $A$ -hypergeometric functions is non-rank-jumping. They also relaxed the nonresonance condition of $c\in \mathbb{C}^{n}$ . It would be an interesting problem to study the relationship between Theorem 5.1 and their results.

2 Preliminary results

In this section, we recall basic notions and results which are used in this paper. In this paper, we essentially follow the terminology of [4], [12], etc. For example, for a topological space $X$ , we denote by $\mathbf{D}^{b}(X)$ the derived category whose objects are bounded complexes of sheaves of $\mathbb{C}_{X}$ -modules on $X$ . We denote by $\mathbf{D}_{c}^{b}(X)$ the full subcategory of $\mathbf{D}^{b}(X)$ consisting of constructible objects. Let $\unicode[STIX]{x1D6E5}\subset \mathbb{R}^{n}$ be a lattice polytope in $\mathbb{R}^{n}$ . For an element $u\in \mathbb{R}^{n}$ of (the dual vector space of) $\mathbb{R}^{n}$ , we define the supporting face $\unicode[STIX]{x1D6FE}_{u}\prec \unicode[STIX]{x1D6E5}$ of $u$ in $\unicode[STIX]{x1D6E5}$ by

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{u}=\left\{v\in \unicode[STIX]{x1D6E5}\mid \langle u,v\rangle =\min _{w\in \unicode[STIX]{x1D6E5}}\langle u,w\rangle \right\},\end{eqnarray}$$

where for $u=(u_{1},\ldots ,u_{n})$ and $v=(v_{1},\ldots ,v_{n})$ we set $\langle u,v\rangle =\sum _{i=1}^{n}u_{i}v_{i}$ . For a face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ , we set

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE})=\overline{\{u\in \mathbb{R}^{n}\mid \unicode[STIX]{x1D6FE}_{u}=\unicode[STIX]{x1D6FE}\}}\subset \mathbb{R}^{n}.\end{eqnarray}$$

Then, $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE})$ is an $(n-\operatorname{dim}\unicode[STIX]{x1D6FE})$ -dimensional rational convex polyhedral cone in $\mathbb{R}^{n}$ . Moreover, the family $\{\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE})\mid \unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E5}\}$ of cones in $\mathbb{R}^{n}$ thus obtained is a subdivision of $\mathbb{R}^{n}$ . We call it the dual subdivision of $\mathbb{R}^{n}$ by $\unicode[STIX]{x1D6E5}$ . If $\operatorname{dim}\unicode[STIX]{x1D6E5}=n$ , it satisfies the axiom of fans (see [8] and [22], etc.). We call it the dual fan of $\unicode[STIX]{x1D6E5}$ .

Let $\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{p}\subset \mathbb{R}^{n}$ be lattice polytopes in $\mathbb{R}^{n}$ , and let $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6E5}_{1}+\cdots +\unicode[STIX]{x1D6E5}_{p}\subset \mathbb{R}^{n}$ be their Minkowski sum. For a face $\unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E5}$ of $\unicode[STIX]{x1D6E5}$ , by taking a point $u\in \mathbb{R}^{n}$ in the relative interior of its dual cone $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE})$ , we define the supporting face $\unicode[STIX]{x1D6FE}_{i}\prec \unicode[STIX]{x1D6E5}_{i}$ of $u$ in $\unicode[STIX]{x1D6E5}_{i}$ . Then, it is easy to see that $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{1}+\cdots +\unicode[STIX]{x1D6FE}_{p}$ . Now, we recall Bernstein–Khovanskii–Kushnirenko’s theorem [13].

Definition 2.1. Let $g(x)=\sum _{v\in \mathbb{Z}^{n}}c_{v}x^{v}$ be a Laurent polynomial on the algebraic torus $T=(\mathbb{C}^{\ast })^{n}$ ( $c_{v}\in \mathbb{C}$ ).

1. (1) We call the convex hull of $\operatorname{supp}(g):=\{v\in \mathbb{Z}^{n}\mid c_{v}\neq 0\}\subset \mathbb{Z}^{n}\subset \mathbb{R}^{n}$ in $\mathbb{R}^{n}$ the Newton polytope of $g$ and denote it by $NP(g)$ .

2. (2) For a face $\unicode[STIX]{x1D6FE}\prec NP(g)$ of $NP(g)$ , we define the $\unicode[STIX]{x1D6FE}$ -part $g^{\unicode[STIX]{x1D6FE}}$ of $g$ by $g^{\unicode[STIX]{x1D6FE}}(x):=\sum _{v\in \unicode[STIX]{x1D6FE}}c_{v}x^{v}$ .

Definition 2.2. (see [14], [23], etc.)

Let $g_{1},g_{2},\ldots ,g_{p}$ be Laurent polynomials on $T=(\mathbb{C}^{\ast })^{n}$ . Set $\unicode[STIX]{x1D6E5}_{i}=NP(g_{i})$ $(i=1,\ldots ,p)$ and $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6E5}_{1}+\cdots +\unicode[STIX]{x1D6E5}_{p}$ . Then, we say that the subvariety $Z=\{x\in T=(\mathbb{C}^{\ast })^{n}\mid g_{1}(x)=g_{2}(x)=\cdots =g_{p}(x)=0\}$ of $T=(\mathbb{C}^{\ast })^{n}$ is a nondegenerate complete intersection if for any face $\unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E5}$ of $\unicode[STIX]{x1D6E5}$ the $p$ -form $dg_{1}^{\unicode[STIX]{x1D6FE}_{1}}\wedge dg_{2}^{\unicode[STIX]{x1D6FE}_{2}}\wedge \cdots \wedge dg_{p}^{\unicode[STIX]{x1D6FE}_{p}}$ does not vanish on $\{x\in T=(\mathbb{C}^{\ast })^{n}\mid g_{1}^{\unicode[STIX]{x1D6FE}_{1}}(x)=\cdots =g_{p}^{\unicode[STIX]{x1D6FE}_{p}}(x)=0\}$ .

Definition 2.3. Let $\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{n}$ be lattice polytopes in $\mathbb{R}^{n}$ . Then, their normalized $n$ -dimensional mixed volume $\operatorname{Vol}_{\mathbb{Z}}(\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{n})\in \mathbb{Z}$ is defined by the formula

(2.3) $$\begin{eqnarray}\operatorname{Vol}_{\mathbb{Z}}(\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{n})=\frac{1}{n!}\mathop{\sum }_{k=1}^{n}(-1)^{n-k}\mathop{\sum }_{\substack{ I\subset \{1,\ldots ,n\} \\ \sharp I=k}}\operatorname{Vol}_{\mathbb{Z}}\left(\mathop{\sum }_{i\in I}\unicode[STIX]{x1D6E5}_{i}\right),\end{eqnarray}$$

where $\operatorname{Vol}_{\mathbb{Z}}(\cdot )=n!\operatorname{Vol}(\cdot )\in \mathbb{Z}$ is the normalized $n$ -dimensional volume with respect to the lattice $\mathbb{Z}^{n}\subset \mathbb{R}^{n}$ .

Theorem 2.4. [13]

Let $g_{1},g_{2},\ldots ,g_{p}$ be Laurent polynomials on $T=(\mathbb{C}^{\ast })^{n}$ . Assume that the subvariety $Z=\{x\in T=(\mathbb{C}^{\ast })^{n}\mid g_{1}(x)=g_{2}(x)=\cdots =g_{p}(x)=0\}$ of $T=(\mathbb{C}^{\ast })^{n}$ is a nondegenerate complete intersection. Set $\unicode[STIX]{x1D6E5}_{i}=NP(g_{i})$ $(i=1,\ldots ,p)$ . Then, we have

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D712}(Z)=(-1)^{n-p}\mathop{\sum }_{\substack{ m_{1},\ldots ,m_{p}\geqslant 1 \\ m_{1}+\cdots +m_{p}=n}}\operatorname{Vol}_{\mathbb{Z}}(\underbrace{\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{1}}_{m_{1}\text{-times}},\ldots ,\underbrace{\unicode[STIX]{x1D6E5}_{p},\ldots ,\unicode[STIX]{x1D6E5}_{p}}_{m_{p}\text{-times}}),\end{eqnarray}$$

where $\operatorname{Vol}_{\mathbb{Z}}(\underbrace{\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{1}}_{m_{1}\text{-times}},\ldots ,\underbrace{\unicode[STIX]{x1D6E5}_{p},\ldots ,\unicode[STIX]{x1D6E5}_{p}}_{m_{p}\text{-times}})\in \mathbb{Z}$ is the normalized $n$ -dimensional mixed volume with respect to the lattice $\mathbb{Z}^{n}\subset \mathbb{R}^{n}$ .

3 A vanishing theorem for local systems

Let $B=\{b(1),b(2),\ldots ,b(N)\}\subset \mathbb{Z}^{n-1}$ be a finite subset of the lattice $\mathbb{Z}^{n-1}$ . Let $\unicode[STIX]{x1D6E5}\subset \mathbb{R}^{n-1}$ be the convex hull of $B$ in $\mathbb{R}^{n-1}$ . Assume that $\operatorname{dim}\unicode[STIX]{x1D6E5}=n-1$ . For $z=(z_{1},\ldots ,z_{N})\in \mathbb{C}^{N}$ , we define a Laurent polynomial $P(x)$ on $T_{0}=(\mathbb{C}^{\ast })^{n-1}$ by $P(x)=\sum _{j=1}^{N}z_{j}x^{b(j)}$ ( $x=(x_{1},\ldots ,x_{n-1})\in T_{0}=(\mathbb{C}^{\ast })^{n-1}$ ). Then, for $c=(c_{1},\ldots ,c_{n})\in \mathbb{C}^{n}$ , the possibly multivalued function $P(x)^{-c_{n}}x_{1}^{c_{1}-1}\cdots x_{n-1}^{c_{n-1}-1}$ on $W=T_{0}\setminus P^{-1}(0)$ generates the local system

(3.1) $$\begin{eqnarray}{\mathcal{L}}=\mathbb{C}_{W}P(x)^{-c_{n}}x_{1}^{c_{1}-1}\cdots x_{n-1}^{c_{n-1}-1}.\end{eqnarray}$$

Set $a(j)=(b(j),1)\in \mathbb{Z}^{n}$ ( $1\leqslant j\leqslant N$ ) and $A=\{a(1),a(2),\ldots ,a(N)\}\subset \mathbb{Z}^{n}$ . Then, $K=\mathbb{R}_{+}A\subset \mathbb{R}^{n}$ is an $n$ -dimensional closed convex polyhedral cone in $\mathbb{R}^{n}$ . For a face $\unicode[STIX]{x1D6E4}\prec K$ of $K$ , let $\operatorname{Lin}(\unicode[STIX]{x1D6E4})\simeq \mathbb{C}^{\operatorname{dim}\unicode[STIX]{x1D6E4}}\subset \mathbb{C}^{n}$ be the $\mathbb{C}$ -linear subspace of $\mathbb{C}^{n}$ generated by $\unicode[STIX]{x1D6E4}$ .

Definition 3.1. (Gelfand et al. [10, page 262])

We say that the parameter vector $c\in \mathbb{C}^{n}$ is nonresonant (with respect to $A$ ) if for any face $\unicode[STIX]{x1D6E4}\prec K$ of $K$ such that $\operatorname{dim}\unicode[STIX]{x1D6E4}=n-1$ we have $c\notin \{\mathbb{Z}^{n}+\operatorname{Lin}(\unicode[STIX]{x1D6E4})\}$ .

The following definition is essentially weaker than the usual (Kouchnirenko) nondegeneracy (see [14], [23], etc.).

Definition 3.2. We say that the Laurent polynomial $P(x)=\sum _{j=1}^{N}z_{j}x^{b(j)}$ is “weakly” nondegenerate if for any face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ such that $\operatorname{dim}\unicode[STIX]{x1D6FE}<\operatorname{dim}\unicode[STIX]{x1D6E5}=n-1$ , the hypersurface

(3.2) $$\begin{eqnarray}\biggl\{x\in T_{0}=(\mathbb{C}^{\ast })^{n-1}\mid P^{\unicode[STIX]{x1D6FE}}(x)=\mathop{\sum }_{j:b(j)\in \unicode[STIX]{x1D6FE}}z_{j}x^{b(j)}=0\biggr\}\subset T_{0}\end{eqnarray}$$

is smooth and reduced.

Let $\unicode[STIX]{x1D704}:W=T_{0}\setminus P^{-1}(0){\hookrightarrow}T_{0}$ be the inclusion map, and set ${\mathcal{M}}=R\unicode[STIX]{x1D704}_{\ast }{\mathcal{L}}\in \mathbf{D}_{c}^{b}(T_{0})$ . Then, the following theorem generalizes one of the results in Gelfand et al. [10] to the case where the hypersurface $P^{-1}(0)\subset T_{0}$ may have isolated singular points.

Theorem 3.3. Assume that $\operatorname{dim}\unicode[STIX]{x1D6E5}=n-1$ , the parameter vector $c\in \mathbb{C}^{n}$ is nonresonant and the Laurent polynomial $P(x)$ is weakly nondegenerate. Then, there exists an isomorphism

(3.3) $$\begin{eqnarray}H_{c}^{j}(T_{0};{\mathcal{M}})\simeq H^{j}(T_{0};{\mathcal{M}})\simeq H^{j}(W;{\mathcal{L}})\end{eqnarray}$$

for any $j\in \mathbb{Z}$ . Moreover, we have the concentration

(3.4) $$\begin{eqnarray}H^{j}(W;{\mathcal{L}})\simeq 0\quad (j\not =n-1).\end{eqnarray}$$

Proof. Let $\unicode[STIX]{x1D6F4}_{0}$ be the dual fan of $\unicode[STIX]{x1D6E5}$ in $\mathbb{R}^{n-1}$ , and let $X$ be the (possibly singular) toric variety associated to it. Then, there exists a natural action of $T_{0}$ on $X$ whose orbits are parametrized by the faces of $\unicode[STIX]{x1D6E5}$ . For a face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ , denote by $X_{\unicode[STIX]{x1D6FE}}\simeq (\mathbb{C}^{\ast })^{\operatorname{dim}\unicode[STIX]{x1D6FE}}$ the $T_{0}$ -orbit associated to $\unicode[STIX]{x1D6FE}$ . Note that $X_{\unicode[STIX]{x1D6E5}}\simeq T_{0}$ is the unique open dense $T_{0}$ -orbit in $X$ and its complement $X\setminus X_{\unicode[STIX]{x1D6E5}}$ is the union of $X_{\unicode[STIX]{x1D6FE}}$ for $\unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E5}$ such that $\operatorname{dim}\unicode[STIX]{x1D6FE}<n-1$ . Let $i:X_{\unicode[STIX]{x1D6E5}}\simeq T_{0}{\hookrightarrow}X$ be the inclusion map. Then, by the weak nondegeneracy of $P(x)$ , the closure $S=\overline{i(P^{-1}(0))}\subset X$ of the hypersurface $i(P^{-1}(0))\subset i(T_{0})$ in $X$ intersects $T_{0}$ -orbits $X_{\unicode[STIX]{x1D6FE}}$ in $X\setminus X_{\unicode[STIX]{x1D6E5}}$ transversely. Moreover, by the nonresonance of $c\in \mathbb{C}^{n}$ , for any $\unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E5}$ such that $\operatorname{dim}\unicode[STIX]{x1D6FE}=n-2$ , the monodromy of the local system ${\mathcal{L}}$ around the codimension-one $T_{0}$ -orbit $X_{\unicode[STIX]{x1D6FE}}\subset X$ in $X$ is nontrivial. Indeed, let $\unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E5}$ be such a facet of $\unicode[STIX]{x1D6E5}$ . We denote by $\unicode[STIX]{x1D6E4}$ the facet of the cone $K=\mathbb{R}_{+}A$ generated by $\unicode[STIX]{x1D6FE}\times \{1\}\subset K$ . Let $\unicode[STIX]{x1D708}\in \mathbb{Z}^{n-1}\setminus \{0\}$ be the primitive inner conormal vector of the facet $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}\subset \mathbb{R}^{n-1}$ , and set

(3.5) $$\begin{eqnarray}m=\min _{v\in \unicode[STIX]{x1D6E5}}\langle \unicode[STIX]{x1D708},v\rangle =\min _{v\in \unicode[STIX]{x1D6FE}}\langle \unicode[STIX]{x1D708},v\rangle \in \mathbb{Z}.\end{eqnarray}$$

Then, the primitive inner conormal vector $\widetilde{\unicode[STIX]{x1D708}}\in \mathbb{Z}^{n}\setminus \{0\}$ of the facet $\unicode[STIX]{x1D6E4}$ of $K\subset \mathbb{R}^{n}$ is explicitly given by the formula

(3.6) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D708}}=\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D708}\\ -m\end{array}\right)\in \mathbb{Z}^{n}\setminus \{0\},\end{eqnarray}$$

and the condition $c=(c_{1},\ldots ,c_{n-1},c_{n})\notin \{\mathbb{Z}^{n}+\operatorname{Lin}(\unicode[STIX]{x1D6E4})\}$ is equivalent to the one

(3.7) $$\begin{eqnarray}m(\unicode[STIX]{x1D6FE}):=\left\langle \unicode[STIX]{x1D708},\left(\begin{array}{@{}c@{}}c_{1}-1\\ \vdots \\ c_{n-1}-1\end{array}\right)\right\rangle -m\cdot c_{n}\quad \notin \mathbb{Z}.\end{eqnarray}$$

We can easily see that the order of the (multivalued) function $P(x)^{-c_{n}}x_{1}^{c_{1}-1}\cdots x_{n-1}^{c_{n-1}-1}$ along the codimension-one $T_{0}$ -orbit $X_{\unicode[STIX]{x1D6FE}}\subset X$ in $X$ is equal to $m(\unicode[STIX]{x1D6FE})\notin \mathbb{Z}$ . Then, by constructing suitable distance functions as in the proof of [7, Lemma 4.2], we can show that for the open embedding $i:T_{0}{\hookrightarrow}X$ we have

(3.8) $$\begin{eqnarray}(Ri_{\ast }{\mathcal{M}})_{p}\simeq 0\quad \text{for any}~p\in X\setminus i(T_{0}),\end{eqnarray}$$

as follows. Let us first assume that the point $p\in X\setminus i(T_{0})$ lies in a $0$ -dimensional $T_{0}$ -orbit $X_{\unicode[STIX]{x1D6FE}}$ . Let $U_{\unicode[STIX]{x1D6FE}}\subset X$ be an $(n-1)$ -dimensional affine toric variety containing $\{p\}=X_{\unicode[STIX]{x1D6FE}}$ , and regard it as a subvariety of $\mathbb{C}_{\unicode[STIX]{x1D701}}^{l}$ for some $l$ . Let $a=(a_{1},\ldots ,a_{n-1})\in \mathbb{Z}^{n-1}$ be the coordinate of the vertex $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ , and define a (nontrivial) rank-one local system $\widetilde{{\mathcal{L}}}$ on $T_{0}$ by

(3.9) $$\begin{eqnarray}\widetilde{{\mathcal{L}}}=\mathbb{C}_{T_{0}}x_{1}^{c_{1}-c_{n}a_{1}-1}\cdots x_{n-1}^{c_{n-1}-c_{n}a_{n-1}-1}.\end{eqnarray}$$

Then, on a neighborhood of the point $p$ in $U_{\unicode[STIX]{x1D6FE}}\subset \mathbb{C}_{\unicode[STIX]{x1D701}}^{l}$ , $Ri_{\ast }{\mathcal{M}}$ is isomorphic to $Ri_{\ast }\widetilde{{\mathcal{L}}}$ . Next, as in the proof of [7, Lemma 4.2], we construct a real-valued function $\unicode[STIX]{x1D711}$ on $\mathbb{C}_{\unicode[STIX]{x1D701}}^{l}$ whose level sets $\unicode[STIX]{x1D6FA}_{t}=\{\unicode[STIX]{x1D701}\in \mathbb{C}^{l}\mid \unicode[STIX]{x1D711}(\unicode[STIX]{x1D701})<t\}$ ( $t\in \mathbb{R}_{{>}0}$ ) satisfy the conditions $\bigcap _{t>0}\unicode[STIX]{x1D6FA}_{t}=\{p\}=X_{\unicode[STIX]{x1D6FE}}$ and $(\bigcup _{t>0}\unicode[STIX]{x1D6FA}_{t})\cap T_{0}=T_{0}$ , and use it to show the isomorphism

(3.10) $$\begin{eqnarray}0\simeq R\unicode[STIX]{x1D6E4}(T_{0};\widetilde{{\mathcal{L}}})\overset{{\sim}}{\longrightarrow }(Ri_{\ast }\widetilde{{\mathcal{L}}})_{p}\end{eqnarray}$$

by the twisted Morse theory. We thus obtain the isomorphism $(Ri_{\ast }{\mathcal{M}})_{p}\simeq 0$ . When the point $p\in X\setminus i(T_{0})$ lies in a $T_{0}$ -orbit $X_{\unicode[STIX]{x1D6FE}}$ such that $\operatorname{dim}X_{\unicode[STIX]{x1D6FE}}=\operatorname{dim}\unicode[STIX]{x1D6FE}>0$ , by taking a normal slice of $X_{\unicode[STIX]{x1D6FE}}$ in $X$ , we can reduce the problem to the case where $\operatorname{dim}X_{\unicode[STIX]{x1D6FE}}=0$ . We thus obtain an isomorphism $i_{!}{\mathcal{M}}\simeq Ri_{\ast }{\mathcal{M}}$ in $\mathbf{D}_{c}^{b}(X)$ . Applying the functor $R\unicode[STIX]{x1D6E4}_{c}(X;\cdot )=R\unicode[STIX]{x1D6E4}(X;\cdot )$ to it, we obtain the desired isomorphisms

(3.11) $$\begin{eqnarray}H_{c}^{j}(T_{0};{\mathcal{M}})\simeq H^{j}(T_{0};{\mathcal{M}})\simeq H^{j}(W;{\mathcal{L}})\end{eqnarray}$$

for $j\in \mathbb{Z}$ . Now, recall that $T_{0}$ is an affine variety, and ${\mathcal{M}}\in \mathbf{D}_{c}^{b}(T_{0})$ is a perverse sheaf on it (up to some shift). Then, by Artin’s vanishing theorem for perverse sheaves over affine varieties (see [4, Corollaries 5.2.18 and 5.2.19], etc.), we have

(3.12) $$\begin{eqnarray}H_{c}^{j}(T_{0};{\mathcal{M}})\simeq 0\quad \text{for}~j<\operatorname{dim}T_{0}=n-1\end{eqnarray}$$

and

(3.13) $$\begin{eqnarray}H^{j}(T_{0};{\mathcal{M}})\simeq 0\quad \text{for}~j>\operatorname{dim}T_{0}=n-1,\end{eqnarray}$$

from which the last assertion immediately follows. This completes the proof. ◻

By Theorem 2.4, we obtain the following corollary of Theorem 3.3.

Corollary 3.4. In the situation of Theorem 3.3, let $p_{1},\ldots ,p_{r}\in P^{-1}(0)$ be the (isolated) singular points of $P^{-1}(0)\subset T_{0}$ , and for $1\leqslant i\leqslant r$ , let $\unicode[STIX]{x1D707}_{i}>0$ be the Milnor number of $P^{-1}(0)$ at $p_{i}$ . Then, we have

(3.14) $$\begin{eqnarray}\operatorname{dim}H^{n-1}(W;{\mathcal{L}})=\operatorname{Vol}_{\mathbb{ Z}}(\unicode[STIX]{x1D6E5})-\mathop{\sum }_{i=1}^{r}\unicode[STIX]{x1D707}_{i}.\end{eqnarray}$$

Proof. By multiplying a monomial $x^{a}$ ( $a\in \mathbb{Z}^{n-1}$ ) to $P(x)$ , we may assume that the Newton polytope $\unicode[STIX]{x1D6E5}$ of $P$ contains the origin $0\in \mathbb{R}^{n-1}$ . Then, by Sard’s theorem, the generic fiber $P^{-1}(t)\subset T_{0}$ ( $t\not =0$ ) of the map $P:T_{0}\longrightarrow \mathbb{C}$ is a nondegenerate hypersurface of $T_{0}$ in the sense of Definition 2.2. Hence, it follows from Theorem 2.4 that its Euler characteristic $\unicode[STIX]{x1D712}(P^{-1}(t))$ is equal to $(-1)^{n-1-1}\operatorname{Vol}_{\mathbb{Z}}(\unicode[STIX]{x1D6E5})=(-1)^{n-2}\operatorname{Vol}_{\mathbb{Z}}(\unicode[STIX]{x1D6E5})$ . It is also well known that we have

(3.15) $$\begin{eqnarray}\unicode[STIX]{x1D712}(P^{-1}(0))=\unicode[STIX]{x1D712}(P^{-1}(t))-(-1)^{n-2}\mathop{\sum }_{i=1}^{r}\unicode[STIX]{x1D707}_{i}.\end{eqnarray}$$

For the open set $W=T_{0}\setminus P^{-1}(0)$ of $T_{0}$ , by $\unicode[STIX]{x1D712}(T_{0})=0$ , we thus obtain the equality

(3.16) $$\begin{eqnarray}(-1)^{n-1}\unicode[STIX]{x1D712}(W)=\operatorname{Vol}_{\mathbb{ Z}}(\unicode[STIX]{x1D6E5})-\mathop{\sum }_{i=1}^{r}\unicode[STIX]{x1D707}_{i}.\end{eqnarray}$$

Moreover, by applying the Mayer–Vietoris argument to the rank-one local system ${\mathcal{L}}$ , we have $\unicode[STIX]{x1D712}(W)=\sum _{j\in \mathbb{Z}}(-1)^{j}\operatorname{dim}H^{j}(W;{\mathcal{L}})$ . Then, the assertion follows immediately from Theorem 3.3.◻

We can generalize Theorem 3.3 to the case where the hypersurface $S=\overline{i(P^{-1}(0))}\subset X$ has (stratified) isolated singular points $p$ also in $T_{0}$ -orbits $X_{\unicode[STIX]{x1D6FE}}\subset X\setminus i(T_{0})$ as follows. For such a point $p\in S\cap X_{\unicode[STIX]{x1D6FE}}$ of $S$ , let us show that we have the vanishing $(Ri_{\ast }{\mathcal{M}})_{p}\simeq 0$ in general. First, consider the case where the codimension of $X_{\unicode[STIX]{x1D6FE}}$ in $X$ is one. The question being local, it suffices to consider the case where $X=\mathbb{C}_{y}^{n-1}\supset X_{\unicode[STIX]{x1D6FE}}=\{y_{n-1}=0\}$ , $S=\{f(y)=0\}\ni p=0$ , $T_{0}=\mathbb{C}^{n-1}\setminus \{y_{n-1}=0\}$ , $i:\mathbb{C}^{n-1}\setminus \{y_{n-1}=0\}{\hookrightarrow}\mathbb{C}^{n-1}$ and

(3.17) $$\begin{eqnarray}{\mathcal{L}}=\mathbb{C}_{\mathbb{C}^{n-1}\setminus \{f(y)\cdot y_{n-1}=0\}}f(y)^{\unicode[STIX]{x1D6FC}}y_{n-1}^{\unicode[STIX]{x1D6FD}}\end{eqnarray}$$

for $\unicode[STIX]{x1D6FC}=-c_{n}$ and some $\unicode[STIX]{x1D6FD}\in \mathbb{C}$ . (By the notation in the proof of Theorem 3.3, we have $\unicode[STIX]{x1D6FD}=m(\unicode[STIX]{x1D6FE})$ .) Here, $f(y)$ is a polynomial on $\mathbb{C}^{n-1}$ such that $S=f^{-1}(0)$ has a (stratified) isolated singular point at $p=0\in S\cap X_{\unicode[STIX]{x1D6FE}}$ . Moreover, for the inclusion map $\unicode[STIX]{x1D704}:\mathbb{C}^{n-1}\setminus \{f(y)\cdot y_{n-1}=0\}{\hookrightarrow}\mathbb{C}^{n-1}\setminus \{y_{n-1}=0\}$ , we have ${\mathcal{M}}\simeq R\unicode[STIX]{x1D704}_{\ast }{\mathcal{L}}$ . By the nonresonance of $c\in \mathbb{C}^{n}$ , we have $\unicode[STIX]{x1D6FD}=m(\unicode[STIX]{x1D6FE})\notin \mathbb{Z}$ , and there exists an isomorphism

(3.18) $$\begin{eqnarray}i_{!}(\mathbb{C}_{\mathbb{C}^{n-1}\setminus \{y_{n-1}=0\}}y_{n-1}^{\unicode[STIX]{x1D6FD}})\overset{{\sim}}{\longrightarrow }Ri_{\ast }(\mathbb{C}_{\mathbb{C}^{n-1}\setminus \{y_{n-1}=0\}}y_{n-1}^{\unicode[STIX]{x1D6FD}}).\end{eqnarray}$$

Set ${\mathcal{N}}=i_{!}(\mathbb{C}_{\mathbb{C}^{n-1}\setminus \{y_{n-1}=0\}}y_{n-1}^{\unicode[STIX]{x1D6FD}})$ . Then, ${\mathcal{N}}$ is a perverse sheaf (up to some shift) on $X=\mathbb{C}^{n-1}$ and satisfies the condition $\unicode[STIX]{x1D713}_{f}({\mathcal{N}})_{p}\simeq \unicode[STIX]{x1D719}_{f}({\mathcal{N}})_{p}$ (using Equation (3.18)), where

(3.19) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{f},\unicode[STIX]{x1D719}_{f}:\mathbf{D}_{c}^{b}(X)\longrightarrow \mathbf{D}_{c}^{b}(\{f=0\})\end{eqnarray}$$

are the nearby and vanishing cycle functors associated to $f$ respectively (see [4], etc.). By the $t$ -exactness of the functor $\unicode[STIX]{x1D719}_{f}$ , the constructible sheaf $\unicode[STIX]{x1D719}_{f}({\mathcal{N}})$ on $S=f^{-1}(0)$ is perverse (up to some shift). Moreover, by our assumption, its support is contained in the point $\{p\}=\{0\}\subset X=\mathbb{C}^{n-1}$ . This implies that we have the concentration

(3.20) $$\begin{eqnarray}H^{j}\unicode[STIX]{x1D713}_{f}({\mathcal{N}})_{p}\simeq H^{j}\unicode[STIX]{x1D719}_{f}({\mathcal{N}})_{p}\simeq 0\quad (j\not =n-2).\end{eqnarray}$$

Namely, for the Milnor fiber $F_{p}$ of $f$ at $p=0\in \mathbb{C}^{n-1}$ , we have

(3.21) $$\begin{eqnarray}H^{j}(F_{p};{\mathcal{N}})\simeq H^{j}\unicode[STIX]{x1D713}_{f}({\mathcal{N}})_{p}\simeq 0\quad (j\not =n-2).\end{eqnarray}$$

Let $B(p;\unicode[STIX]{x1D700})\subset \mathbb{C}^{n-1}$ be a small open ball in $\mathbb{C}^{n-1}$ centered at $p=0$ , and for $0<\unicode[STIX]{x1D702}\ll \unicode[STIX]{x1D700}$ set

(3.22) $$\begin{eqnarray}G=\{y\in \overline{B(p;\unicode[STIX]{x1D700})}\mid 0<|f(y)|<\unicode[STIX]{x1D702}\}.\end{eqnarray}$$

Then, in order to show the vanishing $(Ri_{\ast }{\mathcal{M}})_{p}\simeq 0$ , it suffices to prove the one $R\unicode[STIX]{x1D6E4}(G;Ri_{\ast }{\mathcal{M}})\simeq 0$ for the constructible sheaf

(3.23) $$\begin{eqnarray}(Ri_{\ast }{\mathcal{M}})|_{G}\simeq ({\mathcal{N}}|_{G})\otimes _{\mathbb{C}_{G}}(f|_{G})^{-1}{\mathcal{L}}^{\prime }\end{eqnarray}$$

on $G$ , where ${\mathcal{L}}^{\prime }$ is the rank-one local system on the punctured disk $D_{\unicode[STIX]{x1D702}}^{\ast }=\{t\in \mathbb{C}\mid 0<|t|<\unicode[STIX]{x1D702}\}\subset \mathbb{C}$ generated by the function $t^{\unicode[STIX]{x1D6FC}}$ . By the projection formula, we have

(3.24) $$\begin{eqnarray}R\unicode[STIX]{x1D6E4}(G;Ri_{\ast }{\mathcal{M}})\simeq R\unicode[STIX]{x1D6E4}(D_{\unicode[STIX]{x1D702}}^{\ast };R(f|_{G})_{\ast }({\mathcal{N}}|_{G})\otimes _{\mathbb{C}_{D_{\unicode[STIX]{x1D702}}^{\ast }}}{\mathcal{L}}^{\prime }).\end{eqnarray}$$

Note that $H^{j}R(f|_{G})_{\ast }({\mathcal{N}}|_{G})\simeq 0$ ( $j\not =n-2$ ), and $H^{n-2}R(f|_{G})_{\ast }({\mathcal{N}}|_{G})$ is a local system on $D_{\unicode[STIX]{x1D702}}^{\ast }$ whose stalks are isomorphic to $H^{n-2}(F_{p};{\mathcal{N}})\simeq H^{n-2}\unicode[STIX]{x1D713}_{f}({\mathcal{N}})_{p}$ . Hence, in order to show the vanishing $(Ri_{\ast }{\mathcal{M}})_{p}\simeq 0$ , it suffices to prove that the monodromy operator $\unicode[STIX]{x1D6F7}:H^{n-2}\unicode[STIX]{x1D713}_{f}({\mathcal{N}})_{p}\overset{{\sim}}{\longrightarrow }H^{n-2}\unicode[STIX]{x1D713}_{f}({\mathcal{N}})_{p}$ does not have the eigenvalue $\exp (-2\unicode[STIX]{x1D70B}i\unicode[STIX]{x1D6FC})$ . For this purpose, we use the results in [18, Section 5]. Let $\unicode[STIX]{x1D6E4}_{+}(f)\subset \mathbb{R}_{+}^{n-1}$ be the convex hull of $\bigcup _{v\in \operatorname{supp}(f)}(v+\mathbb{R}_{+}^{n-1})$ in $\mathbb{R}_{+}^{n-1}$ . We call it the Newton polyhedron of $f$ at the origin $p=0\in \mathbb{C}^{n-1}$ .

Definition 3.5. (see [14], [23], etc.)

We say that $f$ is Newton nondegenerate at the origin $p=0\in \mathbb{C}^{n-1}$ if for any compact face $\unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E4}_{+}(f)$ of $\unicode[STIX]{x1D6E4}_{+}(f)$ the hypersurface $\{y\in (\mathbb{C}^{\ast })^{n-1}\mid f^{\unicode[STIX]{x1D6FE}}(y)=0\}$ of $(\mathbb{C}^{\ast })^{n-1}$ is smooth and reduced.

For each subset $I\subset \{1,2,\ldots ,n-1\}$ , we set

(3.25) $$\begin{eqnarray}\mathbb{R}_{+}^{I}=\{v=(v_{1},\ldots ,v_{n-1})\in \mathbb{R}_{+}^{n-1}\mid v_{i}=0~\text{for any}~i\notin I\}\simeq \mathbb{R}_{+}^{\sharp I}.\end{eqnarray}$$

Let $\unicode[STIX]{x1D6FE}_{1}^{I},\ldots ,\unicode[STIX]{x1D6FE}_{n(I)}^{I}\prec \unicode[STIX]{x1D6E4}_{+}(f)\cap \mathbb{R}_{+}^{I}$ be the compact facets of $\unicode[STIX]{x1D6E4}_{+}(f)\cap \mathbb{R}_{+}^{I}$ . For $1\leqslant i\leqslant n(I)$ , denote by $d_{i}^{I}\in \mathbb{Z}_{{>}0}$ the lattice distance of $\unicode[STIX]{x1D6FE}_{i}^{I}$ from the origin $0\in \mathbb{R}_{+}^{I}$ , and let $u_{i}^{I}=(u_{i,1}^{I},\ldots ,u_{i,n-1}^{I})\in \mathbb{R}_{+}^{I}\cap \mathbb{Z}^{n-1}$ be the unique (nonzero) primitive vector which takes its minimum exactly on $\unicode[STIX]{x1D6FE}_{i}^{I}$ . For simplicity, we set $\unicode[STIX]{x1D6FF}_{i}^{I}:=u_{i,n-1}^{I}$ . Finally, we define a finite subset $E_{p}\subset \mathbb{C}$ of $\mathbb{C}$ by

(3.26) $$\begin{eqnarray}E_{p}=\mathop{\bigcup }_{I:I\ni n-1}\mathop{\bigcup }_{i=1}^{n(I)}\{\unicode[STIX]{x1D706}\in \mathbb{C}\mid \unicode[STIX]{x1D706}^{d_{i}^{I}}=\exp (2\unicode[STIX]{x1D70B}\sqrt{-1}\unicode[STIX]{x1D6FD}\cdot \unicode[STIX]{x1D6FF}_{i}^{I})\}.\end{eqnarray}$$

Then, the following result is a special case of [18, Theorem 5.5].

Proposition 3.6. In the above situation, assume moreover that $f$ is Newton nondegenerate at the origin $p=0\in \mathbb{C}^{n-1}$ . Then, the set of the eigenvalues of the monodromy operator $\unicode[STIX]{x1D6F7}:H^{n-2}\unicode[STIX]{x1D713}_{f}({\mathcal{N}})_{p}\overset{{\sim}}{\longrightarrow }H^{n-2}\unicode[STIX]{x1D713}_{f}({\mathcal{N}})_{p}$ is contained in $E_{p}$ .

Corollary 3.7. Assume that $\operatorname{dim}\unicode[STIX]{x1D6E5}=n-1$ , $c\in \mathbb{C}^{n}$ is nonresonant, $\exp (-2\unicode[STIX]{x1D70B}\sqrt{-1}\unicode[STIX]{x1D6FC})=\exp (2\unicode[STIX]{x1D70B}\sqrt{-1}c_{n})\notin E_{p}$ and $f$ is Newton nondegenerate at the origin $p=0\in \mathbb{C}^{n-1}$ . Then, we have $(Ri_{\ast }{\mathcal{M}})_{p}\simeq 0$ .

In fact, by [18, Theorem 5.5], we can generalize this corollary to the case where the codimension of the $T_{0}$ -orbit $X_{\unicode[STIX]{x1D6FE}}$ in $X_{\unicode[STIX]{x1D6FE}}\subset X\setminus i(T_{0})$ containing the (stratified) isolated singular point $p$ of $S$ is larger than one. We leave the precise formulation to the reader and omit the details here. In this way, our Theorem 3.3 can be generalized to the case where $S$ has (stratified) isolated singular points $p$ also in $T_{0}$ -orbits $X_{\unicode[STIX]{x1D6FE}}\subset X\setminus i(T_{0})$ . In particular, we have the following result. For a face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ , let $L_{\unicode[STIX]{x1D6FE}}\simeq \mathbb{R}^{\operatorname{dim}\unicode[STIX]{x1D6FE}}$ be the linear subspace of $\mathbb{R}^{n-1}$ parallel to the affine span of $\unicode[STIX]{x1D6FE}$ in $\mathbb{R}^{n-1}$ , and consider the $\unicode[STIX]{x1D6FE}$ -part $P^{\unicode[STIX]{x1D6FE}}$ of $P$ as a function on $T_{\unicode[STIX]{x1D6FE}}=\operatorname{Spec}(\mathbb{C}[L_{\unicode[STIX]{x1D6FE}}\cap \mathbb{Z}^{n-1}])\simeq (\mathbb{C}^{\ast })^{\operatorname{dim}\unicode[STIX]{x1D6FE}}$ .

Theorem 3.8. Assume that $\operatorname{dim}\unicode[STIX]{x1D6E5}=n-1$ , and for any face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ , the hypersurface $(P^{\unicode[STIX]{x1D6FE}})^{-1}(0)\subset T_{\unicode[STIX]{x1D6FE}}$ of $T_{\unicode[STIX]{x1D6FE}}$ has only isolated singular points. Then, for generic parameter vectors $c\in \mathbb{C}^{n}$ , we have the concentration

(3.27) $$\begin{eqnarray}H^{j}(W;{\mathcal{L}})\simeq 0\quad (j\not =n-1).\end{eqnarray}$$

From now on, let us generalize Theorem 3.3 to the following more general situation. For $0<k<n$ , let $B_{i}=\{b_{i}(1),b_{i}(2),\ldots ,b_{i}(N_{i})\}\subset \mathbb{Z}^{n-k}$ ( $1\leqslant i\leqslant k$ ) be $k$ finite subsets of the lattice $\mathbb{Z}^{n-k}$ , and set $N=N_{1}+N_{2}+\cdots +N_{k}$ . For $1\leqslant i\leqslant k$ and $(z_{i1},\ldots ,z_{iN_{i}})\in \mathbb{C}^{N_{i}}$ , we define a Laurent polynomial $P_{i}(x)$ on $T_{0}=(\mathbb{C}^{\ast })^{n-k}$ by $P_{i}(x)=\sum _{j=1}^{N_{i}}z_{ij}x^{b_{i}(j)}$ ( $x=(x_{1},\ldots ,x_{n-k})\in T_{0}=(\mathbb{C}^{\ast })^{n-k}$ ). Let us set $W=T_{0}\setminus \bigcup _{i=1}^{k}P_{i}^{-1}(0)$ . Then, for $c=(c_{1},\ldots ,c_{n-k},\widetilde{c_{1}},\ldots ,\widetilde{c_{k}})\in \mathbb{C}^{n}$ , the possibly multivalued function

(3.28) $$\begin{eqnarray}P_{1}(x)^{-\widetilde{c_{1}}}\cdots P_{k}(x)^{-\widetilde{c_{k}}}x_{1}^{c_{1}-1}\cdots x_{n-k}^{c_{n-k}-1}\end{eqnarray}$$

on $W$ generates the local system

(3.29) $$\begin{eqnarray}{\mathcal{L}}=\mathbb{C}_{W}P_{1}(x)^{-\widetilde{c_{1}}}\cdots P_{k}(x)^{-\widetilde{c_{k}}}x_{1}^{c_{1}-1}\cdots x_{n-k}^{c_{n-k}-1}.\end{eqnarray}$$

Let $e_{i}=(0,0,\ldots ,0,1,0,\ldots ,0)\in \mathbb{Z}^{k}$ ( $1\leqslant i\leqslant k$ ) be the standard basis of $\mathbb{Z}^{k}$ , and set $a_{i}(j)=(b_{i}(j),e_{i})\in \mathbb{Z}^{n-k}\times \mathbb{Z}^{k}=\mathbb{Z}^{n}$ ( $1\leqslant i\leqslant k$ , $1\leqslant j\leqslant N_{i}$ ) and

(3.30) $$\begin{eqnarray}A=\{a_{1}(1),\ldots ,a_{1}(N_{1}),\ldots \ldots ,a_{k}(1),\ldots ,a_{k}(N_{k})\}\subset \mathbb{Z}^{n}.\end{eqnarray}$$

For $1\leqslant i\leqslant k$ , let $\unicode[STIX]{x1D6E5}_{i}\subset \mathbb{R}^{n-k}$ be the convex hull of $B_{i}$ in $\mathbb{R}^{n-k}$ . Denote by $\unicode[STIX]{x1D6E5}\subset \mathbb{R}^{n-k}$ their Minkowski sum $\unicode[STIX]{x1D6E5}_{1}+\cdots +\unicode[STIX]{x1D6E5}_{k}$ . Assume that $\operatorname{dim}\unicode[STIX]{x1D6E5}=n-k$ . Then, by using the $n$ -dimensional closed convex polyhedral cone $K=\mathbb{R}_{+}A\subset \mathbb{R}^{n}$ generated by $A$ in $\mathbb{R}^{n}$ , we can define the nonresonance of the parameter $c\in \mathbb{C}^{n}$ as in Definition 3.1. For a face $\unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E5}$ of $\unicode[STIX]{x1D6E5}$ , let $\unicode[STIX]{x1D6FE}_{i}\prec \unicode[STIX]{x1D6E5}_{i}$ be the faces of $\unicode[STIX]{x1D6E5}_{i}$ ( $1\leqslant i\leqslant k$ ) canonically associated to $\unicode[STIX]{x1D6FE}$ such that $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{1}+\cdots +\unicode[STIX]{x1D6FE}_{k}$ .

Definition 3.9. We say that the $k$ -tuple of the Laurent polynomials $(P_{1},\ldots ,P_{k})$ is “weakly” (resp. “strongly”) nondegenerate if for any face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ such that $\operatorname{dim}\unicode[STIX]{x1D6FE}<\operatorname{dim}\unicode[STIX]{x1D6E5}=n-k$ (resp. $\operatorname{dim}\unicode[STIX]{x1D6FE}\leqslant \operatorname{dim}\unicode[STIX]{x1D6E5}=n-k$ ) and nonempty subset $J\subset \{1,2,\ldots ,k\}$ , the subvariety

(3.31) $$\begin{eqnarray}\{x\in T_{0}=(\mathbb{C}^{\ast })^{n-k}\mid P_{i}^{\unicode[STIX]{x1D6FE}_{i}}(x)=0~(i\in J)\}\subset T_{0}\end{eqnarray}$$

is a nondegenerate complete intersection.

Remark 3.10. Denote the convex hull of $\bigcup _{i=1}^{k}(\unicode[STIX]{x1D6E5}_{i}\times \{e_{i}\})\subset \mathbb{R}^{n-k}\times \mathbb{R}^{k}=\mathbb{R}^{n}$ in $\mathbb{R}^{n}$ by $\unicode[STIX]{x1D6E5}_{1}\ast \cdots \ast \unicode[STIX]{x1D6E5}_{k}$ . Then, $\unicode[STIX]{x1D6E5}_{1}\ast \cdots \ast \unicode[STIX]{x1D6E5}_{k}$ is naturally identified with the Newton polytope of the Laurent polynomial $R(x,t)=\sum _{i=1}^{k}P_{i}(x)t_{i}$ on $\widetilde{T_{0}}:=T_{0}\times (\mathbb{C}^{\ast })_{t}^{k}\simeq (\mathbb{C}^{\ast })_{x,t}^{n}$ . In [10], the authors considered the condition that for any face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}_{1}\ast \cdots \ast \unicode[STIX]{x1D6E5}_{k}$ , the hypersurface $\{(x,t)\in \widetilde{T_{0}}\mid R^{\unicode[STIX]{x1D6FE}}(x,t)=0\}\subset \widetilde{T_{0}}$ of $\widetilde{T_{0}}$ is smooth and reduced. It is easy to see that our strong nondegeneracy of the $k$ -tuple $(P_{1},\ldots ,P_{k})$ in Definition 3.9 is equivalent to their condition.

Let $\unicode[STIX]{x1D704}:W=T_{0}\setminus \bigcup _{i=1}^{k}P_{i}^{-1}(0)\longrightarrow T_{0}$ be the inclusion map, and set ${\mathcal{M}}=R\unicode[STIX]{x1D704}_{\ast }{\mathcal{L}}\in \mathbf{D}_{c}^{b}(T_{0})$ .

Theorem 3.11. Assume that $\operatorname{dim}\unicode[STIX]{x1D6E5}=n-k$ , the parameter vector $c\in \mathbb{C}^{n}$ is nonresonant and $(P_{1},\ldots ,P_{k})$ is weakly nondegenerate. Then, there exists an isomorphism

(3.32) $$\begin{eqnarray}H_{c}^{j}(T_{0};{\mathcal{M}})\simeq H^{j}(T_{0};{\mathcal{M}})\simeq H^{j}(W;{\mathcal{L}})\end{eqnarray}$$

for any $j\in \mathbb{Z}$ . Moreover, we have the concentration

(3.33) $$\begin{eqnarray}H^{j}(W;{\mathcal{L}})\simeq 0\quad (j\not =n-k).\end{eqnarray}$$

Proof. The proof is similar to that of Theorem 3.3. Let $\unicode[STIX]{x1D6F4}_{0}$ be the dual fan of $\unicode[STIX]{x1D6E5}$ in $\mathbb{R}^{n-k}$ , and let $X$ be the (possibly singular) toric variety associated to it. For a face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ , we denote by $X_{\unicode[STIX]{x1D6FE}}\simeq (\mathbb{C}^{\ast })^{\operatorname{dim}\unicode[STIX]{x1D6FE}}$ the $T_{0}$ -orbit associated to $\unicode[STIX]{x1D6FE}$ . Let $i:X_{\unicode[STIX]{x1D6E5}}\simeq T_{0}{\hookrightarrow}X$ be the inclusion map. Then, by the weak nondegeneracy of the $k$ -tuple $(P_{1},\ldots ,P_{k})$ , for any $T_{0}$ -orbits $X_{\unicode[STIX]{x1D6FE}}$ in $X\setminus X_{\unicode[STIX]{x1D6E5}}$ and the closure $S=\overline{i(\bigcup _{i=1}^{k}P_{i}^{-1}(0))}\subset X$ of the hypersurface $i(\bigcup _{i=1}^{k}P_{i}^{-1}(0))\subset i(T_{0})$ in $X$ , their intersection $S\cap X_{\unicode[STIX]{x1D6FE}}\subset X_{\unicode[STIX]{x1D6FE}}$ is a normal crossing divisor in $X_{\unicode[STIX]{x1D6FE}}$ . In fact, $S$ itself is normal crossing on a neighborhood of such $X_{\unicode[STIX]{x1D6FE}}$ , and any irreducible component of it intersects $X_{\unicode[STIX]{x1D6FE}}$ transversely. Moreover, by the nonresonance of $c\in \mathbb{C}^{n}$ , for any $\unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E5}$ such that $\operatorname{dim}\unicode[STIX]{x1D6FE}=n-k-1$ , the monodromy of the local system ${\mathcal{L}}$ around the codimension-one $T_{0}$ -orbit $X_{\unicode[STIX]{x1D6FE}}\subset X$ in $X$ is nontrivial. Indeed, let $\unicode[STIX]{x1D6FE}\prec \unicode[STIX]{x1D6E5}$ be such a facet of $\unicode[STIX]{x1D6E5}$ , and let $\unicode[STIX]{x1D6FE}_{i}\prec \unicode[STIX]{x1D6E5}_{i}$ be the faces of $\unicode[STIX]{x1D6E5}_{i}$ ( $1\leqslant i\leqslant k$ ) associated to $\unicode[STIX]{x1D6FE}$ such that $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{1}+\cdots +\unicode[STIX]{x1D6FE}_{k}$ . We denote the convex hull of $\bigcup _{i=1}^{k}(\unicode[STIX]{x1D6E5}_{i}\times \{e_{i}\})$ (resp. $\bigcup _{i=1}^{k}(\unicode[STIX]{x1D6FE}_{i}\times \{e_{i}\})$ ) $\subset \mathbb{R}^{n-k}\times \mathbb{R}^{k}=\mathbb{R}^{n}$ in $\mathbb{R}^{n}$ by $\unicode[STIX]{x1D6E5}_{1}\ast \cdots \ast \unicode[STIX]{x1D6E5}_{k}$ (resp. $\unicode[STIX]{x1D6FE}_{1}\ast \cdots \ast \unicode[STIX]{x1D6FE}_{k}$ ). Then, $\unicode[STIX]{x1D6E5}_{1}\ast \cdots \ast \unicode[STIX]{x1D6E5}_{k}$ is the join of $\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{k}$ and $\unicode[STIX]{x1D6FE}_{1}\ast \cdots \ast \unicode[STIX]{x1D6FE}_{k}$ is its facet. We denote by $\unicode[STIX]{x1D6E4}$ the facet of the cone $K=\mathbb{R}_{+}A$ generated by $\unicode[STIX]{x1D6FE}_{1}\ast \cdots \ast \unicode[STIX]{x1D6FE}_{k}\subset K$ . Let $\unicode[STIX]{x1D708}\in \mathbb{Z}^{n-k}\setminus \{0\}$ be the primitive inner conormal vector of the facet $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}\subset \mathbb{R}^{n-k}$ , and for $1\leqslant i\leqslant k$ set

(3.34) $$\begin{eqnarray}m_{i}=\min _{v\in \unicode[STIX]{x1D6E5}_{i}}\langle \unicode[STIX]{x1D708},v\rangle =\min _{v\in \unicode[STIX]{x1D6FE}_{i}}\langle \unicode[STIX]{x1D708},v\rangle \in \mathbb{Z}.\end{eqnarray}$$

Then, the primitive inner conormal vector $\widetilde{\unicode[STIX]{x1D708}}\in \mathbb{Z}^{n}\setminus \{0\}$ of the facet $\unicode[STIX]{x1D6E4}$ of $K\subset \mathbb{R}^{n}$ is explicitly given by the formula

(3.35) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D708}}=\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D708}\\ -m_{1}\\ \vdots \\ -m_{k}\end{array}\right)\in \mathbb{Z}^{n}\setminus \{0\},\end{eqnarray}$$

and the condition $c=(c_{1},\ldots ,c_{n-k},\widetilde{c_{1}},\ldots ,\widetilde{c_{k}})\notin \{\mathbb{Z}^{n}+\operatorname{Lin}(\unicode[STIX]{x1D6E4})\}$ is equivalent to the one

(3.36) $$\begin{eqnarray}m(\unicode[STIX]{x1D6FE}):=\left\langle \unicode[STIX]{x1D708},\left(\begin{array}{@{}c@{}}c_{1}-1\\ \vdots \\ c_{n-k}-1\end{array}\right)\right\rangle -\mathop{\sum }_{i=1}^{k}m_{i}\cdot \widetilde{c_{i}}\quad \notin \mathbb{Z}.\end{eqnarray}$$

Moreover, we can easily see that the order of the (multivalued) function

(3.37) $$\begin{eqnarray}P_{1}(x)^{-\widetilde{c_{1}}}\cdots P_{k}(x)^{-\widetilde{c_{k}}}x_{1}^{c_{1}-1}\cdots x_{n-k}^{c_{n-k}-1}\end{eqnarray}$$

along the codimension-one $T_{0}$ -orbit $X_{\unicode[STIX]{x1D6FE}}\subset X$ in $X$ is equal to $m(\unicode[STIX]{x1D6FE})\notin \mathbb{Z}$ . Finally, by constructing suitable distance functions as in the proof of [7, Lemma 4.2], we can show that

(3.38) $$\begin{eqnarray}(Ri_{\ast }{\mathcal{M}})_{p}\simeq 0\quad \text{for any}~p\in X\setminus T_{0}.\end{eqnarray}$$

Namely, there exists an isomorphism $i_{!}{\mathcal{M}}\simeq Ri_{\ast }{\mathcal{M}}$ in $\mathbf{D}_{c}^{b}(X)$ . Applying the functor $R\unicode[STIX]{x1D6E4}_{c}(X;\cdot )=R\unicode[STIX]{x1D6E4}(X;\cdot )$ to it, we obtain the desired isomorphisms

(3.39) $$\begin{eqnarray}H_{c}^{j}(T_{0};{\mathcal{M}})\simeq H^{j}(T_{0};{\mathcal{M}})\simeq H^{j}(W;{\mathcal{L}})\end{eqnarray}$$

for $j\in \mathbb{Z}$ . Then, the remaining assertion can be proved as in the proof of Theorem 3.3. This completes the proof.◻

In the situation of Theorem 3.11, for any $1\leqslant i\leqslant k$ , the hypersurface $P_{i}^{-1}(0)\subset T_{0}$ has only isolated singular points. Assume moreover that the hypersurface $\bigcup _{i=1}^{k}P_{i}^{-1}(0)\subset T_{0}$ is normal crossing outside them. Then, as in Corollary 3.4, by Theorem 2.4, we can also express the dimension of $H^{n-k}(W;{\mathcal{L}})$ in terms of some mixed volumes of the polytopes $\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{k}$ and the Milnor numbers of the isolated singular points. Since the statement of this result is involved, we leave its precise formulation to the reader.

As in the case where $k=1$ , we have the following result. For a face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ , let $L_{\unicode[STIX]{x1D6FE}}\simeq \mathbb{R}^{\operatorname{dim}\unicode[STIX]{x1D6FE}}$ be the linear subspace of $\mathbb{R}^{n-k}$ parallel to the affine span of $\unicode[STIX]{x1D6FE}$ in $\mathbb{R}^{n-k}$ , and for $1\leqslant i\leqslant k$ , consider the $\unicode[STIX]{x1D6FE}_{i}$ -part $P_{i}^{\unicode[STIX]{x1D6FE}_{i}}$ of $P_{i}$ as a function on $T_{\unicode[STIX]{x1D6FE}}=\operatorname{Spec}(\mathbb{C}[L_{\unicode[STIX]{x1D6FE}}\cap \mathbb{Z}^{n-k}])\simeq (\mathbb{C}^{\ast })^{\operatorname{dim}\unicode[STIX]{x1D6FE}}$ .

Theorem 3.12. Assume that $\operatorname{dim}\unicode[STIX]{x1D6E5}=n-k$ , and for any $1\leqslant i\leqslant k$ , the hypersurface $P_{i}^{-1}(0)\subset T_{0}$ of $T_{0}$ has only isolated singular points. Assume moreover that for any face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ such that $\operatorname{dim}\unicode[STIX]{x1D6FE}<\operatorname{dim}\unicode[STIX]{x1D6E5}=n-k$ and nonempty subset $J\subset \{1,2,\ldots ,k\}$ , the $k$ -tuple of the Laurent polynomials $(P_{1},\ldots ,P_{k})$ satisfies the following condition.

If $J=\{i\}$ for some $1\leqslant i\leqslant k$ and $\operatorname{dim}\unicode[STIX]{x1D6FE}_{i}=\operatorname{dim}\unicode[STIX]{x1D6FE}=\operatorname{dim}\unicode[STIX]{x1D6E5}-1=n-k-1$ , the hypersurface $(P_{i}^{\unicode[STIX]{x1D6FE}_{i}})^{-1}(0)\subset T_{\unicode[STIX]{x1D6FE}}$ of $T_{\unicode[STIX]{x1D6FE}}$ has only isolated singular points. Otherwise, the subvariety

(3.40) $$\begin{eqnarray}\{x\in T_{0}=(\mathbb{C}^{\ast })^{n-k}\mid P_{i}^{\unicode[STIX]{x1D6FE}_{i}}(x)=0~(i\in J)\}\subset T_{0}\end{eqnarray}$$

of $T_{0}$ is a nondegenerate complete intersection.

Then, for generic parameter vectors $c\in \mathbb{C}^{n}$ , we have the concentration

(3.41) $$\begin{eqnarray}H^{j}(W;{\mathcal{L}})\simeq 0\quad (j\not =n-k).\end{eqnarray}$$

Proof. Let $\unicode[STIX]{x1D6F4}_{0}$ be the dual fan of $\unicode[STIX]{x1D6E5}$ in $\mathbb{R}^{n-k}$ , and let $X$ be the (possibly singular) toric variety associated to it. Then, our assumptions imply that for any $1\leqslant i\leqslant k$ , the hypersurface $S_{i}=\overline{i(P_{i}^{-1}(0))}\subset X$ has only stratified isolated singular points in $X$ , and we can prove the assertion following the proofs of Theorems 3.8 and 3.11.◻

For a face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ and $1\leqslant i\leqslant k$ such that $\operatorname{dim}\unicode[STIX]{x1D6FE}_{i}<\operatorname{dim}\unicode[STIX]{x1D6FE}\leqslant n-k-1$ , the hypersurface $(P_{i}^{\unicode[STIX]{x1D6FE}_{i}})^{-1}(0)\subset T_{\unicode[STIX]{x1D6FE}}$ of $T_{\unicode[STIX]{x1D6FE}}$ is smooth or has nonisolated singularities. In the latter case, we cannot prove the concentration in Theorem 3.12 by our methods. This is the reason why we do not allow such cases in our assumptions of Theorem 3.12. However, in the very special case where the Newton polytopes $\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{k}$ are similar to each other, we do not have this problem and obtain the following simpler result.

Theorem 3.13. Assume that $\operatorname{dim}\unicode[STIX]{x1D6E5}=n-k$ , the Newton polytopes $\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{k}$ are similar to each other, and for any face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ and $1\leqslant i\leqslant k$ , the hypersurface $(P_{i}^{\unicode[STIX]{x1D6FE}_{i}})^{-1}(0)\subset T_{\unicode[STIX]{x1D6FE}}$ of $T_{\unicode[STIX]{x1D6FE}}$ has only isolated singular points. Assume moreover that for any face $\unicode[STIX]{x1D6FE}$ of $\unicode[STIX]{x1D6E5}$ such that $\operatorname{dim}\unicode[STIX]{x1D6FE}<\operatorname{dim}\unicode[STIX]{x1D6E5}=n-k$ and any subset $J\subset \{1,2,\ldots ,k\}$ such that $\sharp J\geqslant 2$ , the subvariety

(3.42) $$\begin{eqnarray}\{x\in T_{0}=(\mathbb{C}^{\ast })^{n-k}\mid P_{i}^{\unicode[STIX]{x1D6FE}_{i}}(x)=0~(i\in J)\}\subset T_{0}\end{eqnarray}$$

of $T_{0}$ is a nondegenerate complete intersection. Then, for generic parameter vectors $c\in \mathbb{C}^{n}$ , we have the concentration

(3.43) $$\begin{eqnarray}H^{j}(W;{\mathcal{L}})\simeq 0\quad (j\not =n-k).\end{eqnarray}$$

4 Some results on the twisted Morse theory

In this section, we prepare some auxiliary results on the twisted Morse theory which will be used in Section 5. The following proposition is a refinement of the results in [5, page 10]. See also [7, Proposition 7.1].

Proposition 4.1. Let $T$ be an algebraic torus $(\mathbb{C}^{\ast })_{x}^{n}$ , and let $T=\sqcup _{\unicode[STIX]{x1D6FC}}Z_{\unicode[STIX]{x1D6FC}}$ be its algebraic stratification. In particular, we assume that each stratum $Z_{\unicode[STIX]{x1D6FC}}$ in it is smooth. Let $h(x)$ be a Laurent polynomial on $T=(\mathbb{C}^{\ast })_{x}^{n}$ such that the hypersurface $\{h=0\}\subset T$ intersects $Z_{\unicode[STIX]{x1D6FC}}$ transversely for any $\unicode[STIX]{x1D6FC}$ . For $a\in \mathbb{C}^{n}$ , consider the (possibly multivalued) function $g_{a}(x):=h(x)x^{-a}$ on $T$ . Then, there exists a nonempty Zariski open subset $\unicode[STIX]{x1D6FA}\subset \mathbb{C}^{n}$ of $\mathbb{C}^{n}$ such that the restriction $g_{a}|_{Z_{\unicode[STIX]{x1D6FC}}}:Z_{\unicode[STIX]{x1D6FC}}\longrightarrow \mathbb{C}$ of $g_{a}$ to $Z_{\unicode[STIX]{x1D6FC}}$ has only isolated nondegenerate (i.e., Morse type) critical points for any $a\in \unicode[STIX]{x1D6FA}\subset \mathbb{C}^{n}$ and $\unicode[STIX]{x1D6FC}$ .

Proof. We may assume that each stratum $Z_{\unicode[STIX]{x1D6FC}}$ is connected. We fix a stratum $Z_{\unicode[STIX]{x1D6FC}}$ and set $k=\operatorname{dim}Z_{\unicode[STIX]{x1D6FC}}$ . For a subset $I\subset \{1,2,\ldots ,n\}$ such that $|I|=k=\operatorname{dim}Z_{\unicode[STIX]{x1D6FC}}$ , denote by $\unicode[STIX]{x1D70B}_{I}:T=(\mathbb{C}^{\ast })_{x}^{n}\longrightarrow (\mathbb{C}^{\ast })^{k}$ the projection associated to $I$ . We also denote by $Z_{\unicode[STIX]{x1D6FC},I}\subset Z_{\unicode[STIX]{x1D6FC}}$ the maximal Zariski open subset of $Z_{\unicode[STIX]{x1D6FC}}$ such that the restriction of $\unicode[STIX]{x1D70B}_{I}$ to it is locally biholomorphic. By the implicit function theorem, the variety $Z_{\unicode[STIX]{x1D6FC}}$ is covered by such open subsets $Z_{\unicode[STIX]{x1D6FC},I}$ . For simplicity, let us consider the case where $I=\{1,2,\ldots ,k\}\subset \{1,2,\ldots ,n\}$ . Then, we may regard $g_{a}|_{Z_{\unicode[STIX]{x1D6FC}}}$ locally as a function $g_{a,\unicode[STIX]{x1D6FC},I}(x_{1},\ldots ,x_{k})$ on the Zariski open subset $\unicode[STIX]{x1D70B}_{I}(Z_{\unicode[STIX]{x1D6FC},I})\subset (\mathbb{C}^{\ast })^{k}$ of the form

(4.1) $$\begin{eqnarray}g_{a,\unicode[STIX]{x1D6FC},I}(x_{1},\ldots ,x_{k})=\frac{h_{a,\unicode[STIX]{x1D6FC},I}(x_{1},\ldots ,x_{k})}{x_{1}^{a_{1}}\cdots x_{k}^{a_{k}}}.\end{eqnarray}$$

By our assumption, the hypersurface $\{h_{a,\unicode[STIX]{x1D6FC},I}=0\}\subset \unicode[STIX]{x1D70B}_{I}(Z_{\unicode[STIX]{x1D6FC},I})\subset (\mathbb{C}^{\ast })^{k}$ is smooth. Then, as in the proof of [7, Proposition 7.1], we can show that there exists a nonempty Zariski open subset $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},I}\subset \mathbb{C}^{n}$ such that the (possibly multivalued) function $g_{a,\unicode[STIX]{x1D6FC},I}(x_{1},\ldots ,x_{k})$ on $\unicode[STIX]{x1D70B}_{I}(Z_{\unicode[STIX]{x1D6FC},I})\subset (\mathbb{C}^{\ast })^{k}$ has only isolated nondegenerate (i.e., Morse type) critical points for any $a\in \unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D6FC},I}\subset \mathbb{C}^{n}$ . This completes the proof.◻

Corollary 4.2. In the situation of Proposition 4.1, assume moreover that for the Newton polytope $NP(h)\subset \mathbb{R}^{n}$ of $h$ , we have $\operatorname{dim}NP(h)=n$ . Then, there exists $a\in \operatorname{Int}NP(h)$ such that the restriction $g_{a}|_{Z_{\unicode[STIX]{x1D6FC}}}:Z_{\unicode[STIX]{x1D6FC}}\longrightarrow \mathbb{C}$ of $g_{a}$ to $Z_{\unicode[STIX]{x1D6FC}}$ has only isolated nondegenerate (i.e., Morse type) critical points for any $\unicode[STIX]{x1D6FC}$ .

Now, let $Q_{1},\ldots ,Q_{l}$ be Laurent polynomials on $T=(\mathbb{C}^{\ast })^{n}$ , and for $1\leqslant i\leqslant l$ , denote by $\unicode[STIX]{x1D6E5}_{i}\subset \mathbb{R}^{n}$ the Newton polytope $NP(Q_{i})$ of $Q_{i}$ . Set $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6E5}_{1}+\cdots +\unicode[STIX]{x1D6E5}_{l}$ . Then, by Corollary 4.2, we obtain the following result which might be of independent interest.

Theorem 4.3. Let ${\mathcal{L}}$ be a nontrivial local system of rank one on $T=(\mathbb{C}^{\ast })^{n}$ . Assume that for any $1\leqslant i\leqslant l$ , we have $\operatorname{dim}\unicode[STIX]{x1D6E5}_{i}=n$ , and the subvariety

(4.2) $$\begin{eqnarray}Z_{i}=\{x\in T\mid Q_{1}(x)=\cdots =Q_{i}(x)=0\}\subset T\end{eqnarray}$$

of $T$ is a nondegenerate complete intersection. Then, for any $1\leqslant i\leqslant l$ , we have the concentration

(4.3) $$\begin{eqnarray}H^{j}(Z_{i};{\mathcal{L}})\simeq 0\quad (j\not =n-i).\end{eqnarray}$$

Moreover, we have

(4.4) $$\begin{eqnarray}\operatorname{dim}H^{n-i}(Z_{i};{\mathcal{L}})=\mathop{\sum }_{\substack{ m_{1},\ldots ,m_{i}\geqslant 1 \\ m_{1}+\cdots +m_{i}=n}}\operatorname{Vol}_{\mathbb{Z}}(\underbrace{\unicode[STIX]{x1D6E5}_{1},\ldots ,\unicode[STIX]{x1D6E5}_{1}}_{m_{1}\text{-times}},\ldots ,\underbrace{\unicode[STIX]{x1D6E5}_{i},\ldots ,\unicode[STIX]{x1D6E5}_{i}}_{m_{i}\text{-times}}).\end{eqnarray}$$

Proof. We prove the assertion by induction on $i$ . For $i=0$ , we have $Z_{i}=T$ and the assertion is obvious. Since $Z_{i}\subset T$ is affine, by Artin’s vanishing theorem, we have the concentration

(4.5) $$\begin{eqnarray}H^{j}(Z_{i};{\mathcal{L}})\simeq 0\quad (j>n-i=\operatorname{dim}Z_{i}).\end{eqnarray}$$

On the other hand, by Corollary 4.2, there exists $a_{i}\in \operatorname{Int}NP(Q_{i})\subset \mathbb{R}^{n}$ such that the real-valued function

(4.6) $$\begin{eqnarray}g_{i}:Z_{i-1}\longrightarrow \mathbb{R},\qquad x\longmapsto |Q_{i}(x)x^{-a_{i}}|\end{eqnarray}$$

has only isolated nondegenerate (Morse type) critical points. Note that the Morse index of $g_{i}$ at each critical point is $\operatorname{dim}Z_{i-1}=n-i+1$ . Let $\unicode[STIX]{x1D6F4}_{0}$ be the dual fan of the $n$ -dimensional polytope $\unicode[STIX]{x1D6E5}$ in $\mathbb{R}^{n}$ , and let $\unicode[STIX]{x1D6F4}$ be its smooth subdivision. We denote by $X_{\unicode[STIX]{x1D6F4}}$ the toric variety associated to $\unicode[STIX]{x1D6F4}$ . Then, $X_{\unicode[STIX]{x1D6F4}}$ is a smooth compactification of $T$ such that $D=X_{\unicode[STIX]{x1D6F4}}\setminus T$ is a normal crossing divisor in it. By our assumption, the closures $\overline{Z_{i-1}},\overline{Z_{i}}\subset X_{\unicode[STIX]{x1D6F4}}$ of $Z_{i-1},Z_{i}$ in $X_{\unicode[STIX]{x1D6F4}}$ are smooth. Moreover, they intersect $D$ , etc. transversely. Let $U$ be a sufficiently small tubular neighborhood of $\overline{Z_{i}}\cap D$ in $\overline{Z_{i-1}}$ . Then, by [28, Section 3.5] (see also [16]), for any $t\in \mathbb{R}_{+}$ , there exist isomorphisms

(4.7) $$\begin{eqnarray}H^{j}(\{g_{i}<t\};{\mathcal{L}})\simeq H^{j}(\{g_{i}<t\}\setminus U;{\mathcal{L}})\quad (j\in \mathbb{Z}).\end{eqnarray}$$

Moreover, the level set $g_{i}^{-1}(t)\cap (Z_{i-1}\setminus U)$ of $g_{i}$ in $Z_{i-1}\setminus U$ is compact in $Z_{i-1}$ and intersects $\unicode[STIX]{x2202}U$ transversely for any $t\in \mathbb{R}_{+}$ . Hence, for $t\gg 0$ , we have isomorphisms

(4.8) $$\begin{eqnarray}H^{j}(\{g_{i}<t\};{\mathcal{L}})\simeq H^{j}(Z_{i-1};{\mathcal{L}})\quad (j\in \mathbb{Z}).\end{eqnarray}$$

Moreover, for $0<t\ll 1$ , we have isomorphisms

(4.9) $$\begin{eqnarray}H^{j}(\{g_{i}<t\};{\mathcal{L}})\simeq H^{j}(Z_{i};{\mathcal{L}})\quad (j\in \mathbb{Z}).\end{eqnarray}$$

When $t\in \mathbb{R}$ decreases passing through one of the critical values of $g_{i}$ , only the dimensions of $H^{n-i+1}(\{g_{i}<t\};{\mathcal{L}})$ and $H^{n-i}(\{g_{i}<t\};{\mathcal{L}})$ may change, and the other cohomology groups $H^{j}(\{g_{i}<t\};{\mathcal{L}})$ $(j\not =n-i+1,n-i)$ remain the same. Then, by our induction hypothesis for $i-1$ and Equation (4.5), we obtain the desired concentration

(4.10) $$\begin{eqnarray}H^{j}(Z_{i};{\mathcal{L}})\simeq 0\quad (j\not =n-i).\end{eqnarray}$$

Moreover, the last assertion follows from Theorem 2.4. This completes the proof. ◻

From now on, assume also that the $l$ -tuple $(Q_{1},\ldots ,Q_{l})$ is strongly nondegenerate, and $\operatorname{dim}\unicode[STIX]{x1D6E5}_{l}=n$ . Let $T=\bigsqcup _{\unicode[STIX]{x1D6FC}}Z_{\unicode[STIX]{x1D6FC}}$ be the algebraic stratification of $T$ associated to the hypersurface $S=\bigcup _{i=1}^{l-1}Q_{i}^{-1}(0)\subset T$ , and set $M=T\setminus S$ . Then, by Corollary 4.2, there exists $a\in \operatorname{Int}(\unicode[STIX]{x1D6E5}_{l})$ such that the restriction of the (possibly multivalued) function $Q_{l}(x)x^{-a}$ to $Z_{\unicode[STIX]{x1D6FC}}$ has only isolated nondegenerate (i.e., Morse type) critical points for any $\unicode[STIX]{x1D6FC}$ . In particular, it has only stratified isolated singular points. We fix such $a\in \operatorname{Int}(\unicode[STIX]{x1D6E5}_{l})$ and define a real-valued function $g:T\longrightarrow \mathbb{R}_{+}$ by $g(x)=|Q_{l}(x)x^{-a}|$ . For $t\in \mathbb{R}_{+}$ , we set also

(4.11) $$\begin{eqnarray}M_{t}=\{x\in M=T\setminus S\mid g(x)<t\}\subset M.\end{eqnarray}$$

Then, we have the following result.

Lemma 4.4. Let ${\mathcal{L}}$ be a local system on $M=T\setminus S$ . Then, for any $c>0$ , there exists a sufficiently small $0<\unicode[STIX]{x1D700}\ll 1$ such that we have the concentration

(4.12) $$\begin{eqnarray}H^{j}(M_{c+\unicode[STIX]{x1D700}},M_{c-\unicode[STIX]{x1D700}};{\mathcal{L}})\simeq 0\quad (j\not =n).\end{eqnarray}$$

Proof. Let $\unicode[STIX]{x1D6F4}_{0}$ be the dual fan of the $n$ -dimensional polytope $\unicode[STIX]{x1D6E5}$ in $\mathbb{R}^{n}$ , and let $\unicode[STIX]{x1D6F4}$ be its smooth subdivision. We denote by $X_{\unicode[STIX]{x1D6F4}}$ the toric variety associated to $\unicode[STIX]{x1D6F4}$ . Then, $X_{\unicode[STIX]{x1D6F4}}$ is a smooth compactification of $T$ such that $D=X_{\unicode[STIX]{x1D6F4}}\setminus T$ is a normal crossing divisor in it. By the strong nondegeneracy of $(Q_{1},\ldots ,Q_{l})$ , the hypersurface $\overline{Q_{l}^{-1}(0)}\subset X_{\unicode[STIX]{x1D6F4}}$ intersects $D$ , etc. transversely. Let $U$ be a sufficiently small tubular neighborhood of $\overline{Q_{l}^{-1}(0)}\cap D$ in $X_{\unicode[STIX]{x1D6F4}}$ , and for $t\in \mathbb{R}_{+}$ , set $M_{t}^{\prime }=M_{t}\setminus U$ . Then, by [28, Section 3.5], for any $t\in \mathbb{R}_{+}$ , there exist isomorphisms

(4.13) $$\begin{eqnarray}H^{j}(M_{t};{\mathcal{L}})\simeq H^{j}(M_{t}^{\prime };{\mathcal{L}})\quad (j\in \mathbb{Z}).\end{eqnarray}$$

Moreover, the level set $g^{-1}(t)\cap (T\setminus U)$ of $g$ in $T\setminus U$ is compact in $T$ and intersects $\unicode[STIX]{x2202}U$ transversely for any $t\in \mathbb{R}_{+}$ . For $c>0$ , let $p_{1},\ldots ,p_{r}\in T\setminus g^{-1}(0)=T\setminus Q_{l}^{-1}(0)$ be the stratified isolated singular points of the function $h(x)=Q_{l}(x)x^{-a}$ in $T$ such that $g(p_{i})=|h(p_{i})|=c$ . Note that we have

(4.14) $$\begin{eqnarray}g(x)=|h(x)|=\exp [\operatorname{Re}\{\log h(x)\}].\end{eqnarray}$$

Then, there exist small open balls $B_{i}$ centered at $p_{i}$ in $T$ and $0<\unicode[STIX]{x1D700}\ll 1$ such that we have isomorphisms

(4.15) $$\begin{eqnarray}H^{j}(M_{c+\unicode[STIX]{x1D700}}^{\prime },M_{c-\unicode[STIX]{x1D700}}^{\prime };{\mathcal{L}})\simeq \bigoplus _{i=1}^{r}H^{j}(B_{i}\cap M_{c+\unicode[STIX]{x1D700}},B_{i}\cap M_{c-\unicode[STIX]{x1D700}};{\mathcal{L}})\quad (j\in \mathbb{Z}).\end{eqnarray}$$

For $1\leqslant i\leqslant r$ , by taking a local branch $\log h$ of the logarithm of the function $h\not =0$ on a neighborhood of $p_{i}\in T\setminus h^{-1}(0)$ , we set $f_{i}=\log h-\log h(p_{i})$ . Then, $f_{i}$ has also a stratified isolated singular point at $p_{i}$ . Let $F_{i}\subset B_{i}$ be the Milnor fiber of $f_{i}$ at $p_{i}\in f_{i}^{-1}(0)$ . Then, for any $1\leqslant i\leqslant r$ , by shrinking $B_{i}$ if necessary, we can easily prove the isomorphisms

(4.16) $$\begin{eqnarray}H^{j}(B_{i}\cap M_{c+\unicode[STIX]{x1D700}},B_{i}\cap M_{c-\unicode[STIX]{x1D700}};{\mathcal{L}})\simeq H^{j}(B_{i}\setminus S,F_{i}\setminus S;{\mathcal{L}})\quad (j\in \mathbb{Z}).\end{eqnarray}$$

Let $j:M=T\setminus S{\hookrightarrow}T$ be the inclusion. Since the Milnor fibers $F_{i}\subset B_{i}$ intersect each stratum $Z_{\unicode[STIX]{x1D6FC}}$ transversely, we have also isomorphisms

(4.17) $$\begin{eqnarray}H^{j}(B_{i}\setminus S,F_{i}\setminus S;{\mathcal{L}})\simeq H^{j-1}\unicode[STIX]{x1D719}_{f_{i}}(Rj_{\ast }{\mathcal{L}})_{p_{i}}\quad (j\in \mathbb{Z}),\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{f_{i}}$ are Deligne’s vanishing cycle functors. Hence, by (the proof of) [4, Proposition 6.1.1], the assertion follows from

(4.18) $$\begin{eqnarray}\operatorname{supp}\unicode[STIX]{x1D719}_{f_{i}}(Rj_{\ast }{\mathcal{L}})\subset \{p_{i}\}\quad (1\leqslant i\leqslant r)\end{eqnarray}$$

and the fact that $Rj_{\ast }{\mathcal{L}}$ and $\unicode[STIX]{x1D719}_{f_{i}}(Rj_{\ast }{\mathcal{L}})$ are perverse sheaves (up to some shifts). This completes the proof.◻

5 A new vanishing theorem

Now, let $P_{1},\ldots ,P_{k}$ be Laurent polynomials on $T_{0}=(\mathbb{C}^{\ast })^{n-k}$ , and for $1\leqslant i\leqslant k$ , denote by $\unicode[STIX]{x1D6E5}_{i}\subset \mathbb{R}^{n-k}$ the Newton polytope $NP(P_{i})$ of $P_{i}$ . Set $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6E5}_{1}+\cdots +\unicode[STIX]{x1D6E5}_{k}$ . Let us set $W=T_{0}\setminus \bigcup _{i=1}^{k}P_{i}^{-1}(0)$ , and for $(c,\widetilde{c})=(c_{1},\ldots ,c_{n-k},\widetilde{c_{1}},\ldots ,\widetilde{c_{k}})\in \mathbb{C}^{n}$ consider the local system

(5.1) $$\begin{eqnarray}{\mathcal{L}}=\mathbb{C}_{W}P_{1}(x)^{\widetilde{c_{1}}}\cdots P_{k}(x)^{\widetilde{c_{k}}}x_{1}^{c_{1}}\cdots x_{n-k}^{c_{n-k}}\end{eqnarray}$$

on $W$ .

Theorem 5.1. Assume that the $k$ -tuple of the Laurent polynomials $(P_{1},\ldots ,P_{k})$ is strongly nondegenerate, $(c,\widetilde{c})=(c_{1},\ldots ,c_{n-k},\widetilde{c_{1}},\ldots ,\widetilde{c_{k}})\notin \mathbb{Z}^{n}$ , and for any $1\leqslant i\leqslant k$ , we have $\operatorname{dim}\unicode[STIX]{x1D6E5}_{i}=n-k$ . Then, we have the concentration

(5.2) $$\begin{eqnarray}H^{j}(W;{\mathcal{L}})\simeq 0\quad (j\not =n-k).\end{eqnarray}$$

Proof. Set $T=T_{0}\times (\mathbb{C}^{\ast })_{t_{1},\ldots ,t_{k}}^{k}\simeq (\mathbb{C}^{\ast })_{x,t}^{n}$ , and consider the Laurent polynomials

(5.3) $$\begin{eqnarray}\widetilde{P_{i}}(x,t)=t_{i}-P_{i}(x)\quad (1\leqslant i\leqslant k)\end{eqnarray}$$

on $T$ . For $1\leqslant i\leqslant k$ , we set also

(5.4) $$\begin{eqnarray}Z_{i}=\{(x,t)\in T\mid \widetilde{P_{1}}(x,t)=\cdots =\widetilde{P_{i}}(x,t)=0\}.\end{eqnarray}$$

We define a local system $\widetilde{{\mathcal{L}}}$ on $T$ by

(5.5) $$\begin{eqnarray}\widetilde{{\mathcal{L}}}=\mathbb{C}_{T}x_{1}^{c_{1}}\cdots x_{n-k}^{c_{n-k}}t_{1}^{\widetilde{c_{1}}}\cdots t_{k}^{\widetilde{c_{k}}}.\end{eqnarray}$$

Then, $Z_{k}\simeq W$ , and we have isomorphisms

(5.6) $$\begin{eqnarray}H^{j}(W;{\mathcal{L}})\simeq H^{j}(Z_{k};\widetilde{{\mathcal{L}}})\quad (j\in \mathbb{Z}).\end{eqnarray}$$

First, let us consider the case where $\widetilde{c}=(\widetilde{c_{1}},\ldots ,\widetilde{c_{k}})\notin \mathbb{Z}^{k}$ . In this case, without loss of generality, we may assume that $\widetilde{c_{k}}\notin \mathbb{Z}$ . Then, by the Künneth formula, for $i=1,2,\ldots ,k-1$ , we have the vanishings

(5.7) $$\begin{eqnarray}H^{j}(Z_{i};\widetilde{{\mathcal{L}}})\simeq 0\quad (j\in \mathbb{Z}).\end{eqnarray}$$

Moreover, we can naturally identify $Z_{k-1}\subset T$ with $(T_{0}\setminus \bigcup _{i=1}^{k-1}P_{i}^{-1}(0))$ $\times \,\mathbb{C}_{t_{k}}^{\ast }$ . Consider $\widetilde{P_{k}}$ as a Laurent polynomial on $T_{1}=T_{0}\times \mathbb{C}_{t_{k}}^{\ast }\simeq (\mathbb{C}^{\ast })^{n-k+1}$ . Note that we have $\operatorname{dim}NP(\widetilde{P_{k}})=n-k+1=\operatorname{dim}T_{1}$ . By taking a sufficiently generic

(5.8) $$\begin{eqnarray}(a_{1},\ldots ,a_{n-k},a_{n-k+1})\in \operatorname{Int}NP(\widetilde{P_{k}})\subset \mathbb{R}^{n-k+1},\end{eqnarray}$$

we define a real-valued function $g$ on $T_{1}=T_{0}\times \mathbb{C}_{t_{k}}^{\ast }$ by

(5.9) $$\begin{eqnarray}g(x,t_{k})=\left|\widetilde{P_{k}}(x,t_{k})\times x_{1}^{-a_{1}}\cdots x_{n-k}^{-a_{n-k}}t_{k}^{-a_{n-k+1}}\right|.\end{eqnarray}$$

Then, by applying Lemma 4.4 to the Morse function $g:T_{1}=T_{0}\times \mathbb{C}^{\ast }\longrightarrow \mathbb{R}$ and arguing as in the proof of Theorem 4.3, we obtain the desired concentration

(5.10) $$\begin{eqnarray}H^{j}(Z_{k};\widetilde{{\mathcal{L}}})\simeq 0\quad (j\not =n-k).\end{eqnarray}$$

The proof for the remaining case where $(c,\widetilde{c})=(c_{1},\ldots ,c_{n-k},\widetilde{c_{1}},\ldots ,\widetilde{c_{k}})\notin \mathbb{Z}^{n}$ and $\widetilde{c}=(\widetilde{c_{1}},\ldots ,\widetilde{c_{k}})\in \mathbb{Z}^{k}$ is similar. In this case, $Z_{1}\subset T$ is isomorphic to the product $Z_{1}^{\prime }\times (\mathbb{C}^{\ast })^{k-1}$ for a hypersurface $Z_{1}^{\prime }$ in $T_{0}\times \mathbb{C}_{t_{1}}^{\ast }$ , and $\widetilde{{\mathcal{L}}}$ is isomoprhic to the pullback of a local system on $T_{0}\times \mathbb{C}_{t_{1}}^{\ast }$ . Hence, by the Künneth formula and the proof of Theorem 4.3, we obtain the concentration

(5.11) $$\begin{eqnarray}H^{j}(Z_{1};\widetilde{{\mathcal{L}}})\simeq 0\quad (j\not =n-k,\ldots ,n-1).\end{eqnarray}$$

Repeating this argument with the help of Lemma 4.4 and the proof of Theorem 4.3, we obtain also

(5.12) $$\begin{eqnarray}H^{j}(W;{\mathcal{L}})\simeq H^{j}(Z_{k};\widetilde{{\mathcal{L}}})\simeq 0\quad (j\not =n-k,\ldots ,n-1).\end{eqnarray}$$

Then, the assertion is obtained by applying Artin’s vanishing theorem to the $(n-k)$ -dimensional affine variety $Z_{k}\subset T$ . This completes the proof.◻