##
**Arrangements and hypergeometric integrals.**
*(English)*
Zbl 0980.32010

MSJ Memoirs. 9. Tokyo: Mathematical Society of Japan. ix, 112 p. (2001).

The book under review is based on a series of lectures given by P. Orlik at a conference at Tokyo Metropolitan University, and on a series of lectures given by H. Terao at the University of Wisconsin, Hokkaido University, Tohoku University, and Kyushu University. Its aim is to give an elementary introduction to hypergeometric functions from the point of view of the theory of arrangements of hyperplanes. Let \(V\) be an affine complex space of dimension \(l\). An arrangement in \(V\) is a finite family \({\mathcal{A}}\) of affine hyperplanes in \(V\). Let \(N=N(\mathcal{A})=\bigcup_{H \in \mathcal{A}} H\) denote the divisor of \(\mathcal{A}\), and let \(M=M(\mathcal{A}) =V \setminus N(\mathcal{A})\) denote the complement of \(\mathcal{A}\). For each hyperplane \(H \in {\mathcal{A}}\) choose an affine form \(\alpha_H: V \to \mathbf C\) with \(\alpha_H^{-1} (0) =H\), and choose a complex weight \(\lambda_H \in \mathbf C\). Write \(\lambda=\{ \lambda_H \}_{ H\in {\mathcal{A}}}\), and define
\[
\Phi_\lambda= \prod_{H \in \mathcal{A}} \alpha_H^{\lambda_H}.
\]
A generalized hypergeometric integral is of the form
\[
\int_\sigma \Phi_\sigma \eta,
\]
where \(\sigma\) is a singular simplex on \(M\), \(\Phi_\sigma\) is a branch of \(\Phi_\lambda\) on the image of \(\sigma\), and \(\eta\) is a holomorphic form on \(M\).

The goal of Chapter 2 is to interpret hypergeometric integrals as the result of the hypergeometric pairing. Consider the 1-form \(\omega_\lambda=\sum_{H\in{\mathcal{A}}\lambda_H} {d\alpha_H \over \alpha_H}\), and the differential \(\nabla_\lambda=d+\omega_\lambda \wedge\), where \(d\) denotes the ordinary exterior differential. Define the presheaf \({\mathcal{L}}_\lambda\) on \(M\) by \({\mathcal{L}}_\lambda (U)=\{f:U\to \mathbf C;\;\nabla_\lambda(f)=0 \}\), where \(U \subset M\) and \(f\) is a holomorphic function. Then \({\mathcal{L}}_\lambda\) is a locally constant sheaf.

The cohomology \(H^\ast(M,\mathcal{L}_\lambda)\) can be regarded as the cohomology of \(M\) with coefficients in a certain local system. It is also equal to the cohomology of the twisted holomorphic de Rham complex \((\Omega^\cdot (M), \nabla_\lambda)\). Let \({\mathcal{L}}_\lambda^\vee\) denote the local system dual to \({\mathcal{L}}_\lambda\). Then \(H_\ast (M, {\mathcal{L}}_\lambda^\vee)\) is the homology of the chain complex \(C_\ast(M,{\mathcal{L}}_\lambda^\vee)\), where \(C_p(M,{\mathcal{L}}_\lambda^\vee)\) is the complex vector space with basis \(\sigma \otimes \Phi_\sigma\), where \(\sigma\) is a singular \(p\)-simplex on \(M\), and \(\Phi_\sigma\) is a branch of \(\Phi_\lambda\) on the image of \(\sigma\). The differential is induced by the standard one applied to \(\sigma\). Then hypergeometric integrals provide a pairing \(\langle\;,\;\rangle: H^p(M, {\mathcal{L}}_\lambda) \times H_p(M, {\mathcal{L}}_\lambda^\vee) \to \mathbf C\) called hypergeometric pairing.

Chapter 3 is a short survey on the classical theory of arrangements. It starts with some definitions related with the poset \(L({\mathcal{A}})\) of intersections of elements of \({\mathcal{A}}\), and with some basic constructions: coning, projective closure, projective quotient, deconing, and delation-restriction triple. The remainder of the chapter is dedicated to the study of two objects that will be used later in the book. The first object are the dense edges. Define an edge to be an element of \(L({\mathcal{A}})\), and call an edge \(X\) dense if \({\mathcal{A}}_X=\{H\in {\mathcal{A}}; H \supset X\}\) is not the direct sum of two nonempty arrangements. Let \({\mathcal{A}}\) be an arrangement of rank \(r\). The second object is the \(\beta\)-invariant which is defined by \(\beta= \beta({\mathcal{A}})= (-1)^r \chi ({\mathcal{A}},1)\), where \(\chi({\mathcal{A}},t)\) denotes the characteristic polynomial of \({\mathcal{A}}\). If \({\mathcal{A}}\) is the complexification of a real arrangement \({\mathcal{A}}_\mathbf R\), the \(\beta(\mathcal{A})\) is equal to the number of bounded chambers of \({\mathcal{A}}_\mathbf R\). Most of the results stated and proved in the next four chapters concern the so-called non-resonance case. Briefly speaking, non-resonance means that some sums of \(\pm \lambda_H\) are not positive integers. In particular, if \(\sum_{H \in {\mathcal{B}}} \lambda_H \not \in \mathbf Z\) for all \(\mathcal{B} \subset {\mathcal{A}}\), then the non-resonance hypothesis is satisfied. Define the Brieskorn algebra of \(\mathcal{A}\) to be the subalgebra \(B^\cdot(\mathcal{A})\) of \(\Omega^\cdot (M)\) generated by \(1\) and by the forms \({d\alpha_H \over \alpha_H}\), \(H \in \mathcal{A}\).

The goal of Chapter 4 is to prove that, under the non-resonance hypothesis, \(H^\ast(M, {\mathcal{L}}_\lambda)\) is isomorphic to \(H^\ast(B^\cdot (\mathcal{A}),\omega_\lambda \wedge)\).

Chapter 5 starts with the definition of the Orlik-Solomon algebra \(A^\cdot (\mathcal{A})\) of an arrangement \(\mathcal{A}\). This is a graded algebra defined by generators and relations. The set \(\{ a_H; H\in \mathcal{A}\}\) of generators of \(A^\cdot (\mathcal{A})\) is in one-to-one correspondence with \(\mathcal{A}\). The mapping \(a_H \mapsto {d\alpha_H \over \alpha_H}\) induces an isomorphism \(A^\cdot(\mathcal{A}) \to B^\cdot (\mathcal{A})\). In particular, by the results of Chapter 4, one has \(H^\ast(M,{\mathcal{L}}_\lambda) \simeq H^\ast( A^\cdot(\mathcal{A}), a_\lambda \wedge)\), where \(a_\lambda= \sum_{H \in \mathcal{A}} \lambda_H a_H\). The remainder of the chapter is dedicated to the definition and the study of the complex \(\mathtt{NBC}(\mathcal{A})\) and the set \(\beta{\mathbf{nbc}}\). \(\mathtt{NBC}(\mathcal{A})\) is a simplicial complex whose vertices are the hyperplanes of \(\mathcal{A}\), and \(\beta{\mathbf{nbc}}\) is a set of simplices of \(\mathtt{NBC}(\mathcal{A})\) whose dual forms a basis for \(H^\ast( \mathtt{NBC}(\mathcal{A}), \mathbf C)\).

The goal of Chapter 6 is to produce basis for \(H^\ast(M(\mathcal{A}), {\mathcal{L}}_\lambda)\) and for \(H_\ast(M(\mathcal{A}), {\mathcal{L}}_\lambda^\vee)\). Let \(\mathcal{A}\) be an arrangement of rank \(r\). Recall that we are under the non-resonance hypothesis. Then one has \(H^p(M(\mathcal{A}), {\mathcal{L}}_\lambda)= H_p(M(\mathcal{A}), {\mathcal{L}}_\lambda^\vee)=0\), for \(p\neq r\), and \(\dim H^r(M(\mathcal{A}), {\mathcal{L}}_\lambda) = \dim H_r (M(\mathcal{A}), {\mathcal{L}}_\lambda^\vee) =\beta(\mathcal{A})\). The group \(H^r(\mathtt{NBC}(\mathcal{A}), \mathbf C)\) is isomorphic to \(H^r(M(\mathcal{A}), {\mathcal{L}}_\lambda)\), thus the set \(\beta{\mathbf{nbc}}\) furnishes a basis for \(H^r (M(\mathcal{A}), {\mathcal{L}}_\lambda)\) through this isomorphism. Assume now that \(\mathcal{A}\) is the complexification of a real arrangement \({\mathcal{A}}_\mathbf R\). Let \(\text{bch} (\mathcal{A})\) denote the set of bounded chambers of \({\mathcal{A}}_\mathbf R\). As pointed out before, the cardinality of \(\text{bch} (\mathcal{A})\) is \(\beta(\mathcal{A})\). Then \(\text{bch} (\mathcal{A})\) determines a basis for \(H_r(M(\mathcal{A}), {\mathcal{L}}_\lambda^\vee)\). One also has a natural bijection \(\tau: \text{bch}(\mathcal{A}) \to \beta{\mathbf{nbc}}\). Assume that \(\mathcal{A}\) is the complexification of an essential real arrangement \({\mathcal{A}}_\mathbf R\). We order \(\beta{\mathbf{nbc}}\) with the lexicographic ordering, \(\beta{\mathbf{nbc}}= \{ B_j\}_{j=1}^\beta\), and order \(\text{bch} (\mathcal{A})= \{ \Delta_j \}_{j=1}^\beta\) setting \(\Delta_j= \tau^{-1} (B_j)\). Define the hypergeometric period matrix \(PM(\mathcal{A},\lambda)\) by \[ PM(\mathcal{A},\lambda)_{i,j} = \int_{\Delta_j} \Phi_j \psi_i, \] where \(\psi_i\) is a holomorphic form corresponding to \(B_i \in H^l( \mathtt{NBC}(\mathcal{A}), \mathbf C) \simeq H^l (M(\mathcal{A}), {\mathcal{L}}_\lambda)\).

In Chapter 7 the authors assume that \(\operatorname{Re} \lambda_H >0\) for all \(H \in \mathcal{A}\), and compute the determinant of \(PM(\mathcal{A}, \lambda)\) in terms of gamma functions.

Chapters 8, 9, and 10 form the second part of the book whose aim is to work in the dynamical setup. Let \(B\) be the configuration space of ordered \(n\)-tuples in \(\mathbf C\), and, for \(\mathbf t=(t_1, \dots, t_n)\) in \(B\), let \(M_\mathbf t= \mathbf C \setminus \{ t_1, \dots, t_n\}\) be the complement of \({\mathcal{A}}_\mathbf t =\{t_1, \dots, t_n\}\). Choose non-resonant weights \(\lambda_1, \dots, \lambda_n\), and let \(\mathcal{L}\) be the corresponding local system. \({\mathcal{H}}_1= \bigcup_{\mathbf t \in B} H_1( M_\mathbf t, {\mathcal{L}}^\vee)\) is a locally trivial bundel over \(B\), and \({\mathcal{H}}_1 \otimes {\mathcal{O}}_B\) is isomorphic to \(({\mathcal{O}}_B)^{n-1}\) via the map which sends \(\gamma\) to \((\widehat \varphi_1, \dots, \widehat \varphi_{n-1})^T\), where \(\widehat \varphi_j = \int_\gamma \Phi \varphi_j\), and \(\varphi_j= {\lambda_{j+1} du \over u-t_{j+1}}\).

It is shown in Chapter 8 that the vector \((\widehat \varphi_1, \dots, \widehat \varphi_{n-1})^T\) satisfies the KZ-equation \[ d' \left( \begin{matrix} \widehat \varphi_1\\ \vdots\\ \widehat \varphi_{n-1}\end{matrix}\right) = \Omega \wedge \left( \begin{matrix} \widehat \varphi_1\\ \vdots\\ \widehat \varphi_{n-1}\end{matrix}\right), \] where \(d'\) is the exterior differential in \(B\), and \(\Omega\) is explicitly given.

In Chapter 9 the authors describe the space \(B\) of arrangements with a given combinatorial type. Choose such a space. One has a locally trivial fibration \(\pi: M \to B\) whose fiber at \(\mathbf t \in B\) is the complement \(M_\mathbf t\) of the arrangement \({\mathcal{A}}_\mathbf t\) corresponding to \(\mathbf t\). Set \({\mathcal{H}}_l= \bigcup_{\mathbf t \in B} H_l( M_\mathbf t, {\mathcal{L}}^\vee)\). Then \({\mathcal{H}}_l\) is a locally trivial bundle over \(B\). Any local section \(\sigma\) of \({\mathcal{H}}_l \otimes {\mathcal{O}}_B\) determines a differential equation which generalizes the KZ-equation of Chapter 8. These equations are the subject of Chapter 10.

The goal of Chapter 2 is to interpret hypergeometric integrals as the result of the hypergeometric pairing. Consider the 1-form \(\omega_\lambda=\sum_{H\in{\mathcal{A}}\lambda_H} {d\alpha_H \over \alpha_H}\), and the differential \(\nabla_\lambda=d+\omega_\lambda \wedge\), where \(d\) denotes the ordinary exterior differential. Define the presheaf \({\mathcal{L}}_\lambda\) on \(M\) by \({\mathcal{L}}_\lambda (U)=\{f:U\to \mathbf C;\;\nabla_\lambda(f)=0 \}\), where \(U \subset M\) and \(f\) is a holomorphic function. Then \({\mathcal{L}}_\lambda\) is a locally constant sheaf.

The cohomology \(H^\ast(M,\mathcal{L}_\lambda)\) can be regarded as the cohomology of \(M\) with coefficients in a certain local system. It is also equal to the cohomology of the twisted holomorphic de Rham complex \((\Omega^\cdot (M), \nabla_\lambda)\). Let \({\mathcal{L}}_\lambda^\vee\) denote the local system dual to \({\mathcal{L}}_\lambda\). Then \(H_\ast (M, {\mathcal{L}}_\lambda^\vee)\) is the homology of the chain complex \(C_\ast(M,{\mathcal{L}}_\lambda^\vee)\), where \(C_p(M,{\mathcal{L}}_\lambda^\vee)\) is the complex vector space with basis \(\sigma \otimes \Phi_\sigma\), where \(\sigma\) is a singular \(p\)-simplex on \(M\), and \(\Phi_\sigma\) is a branch of \(\Phi_\lambda\) on the image of \(\sigma\). The differential is induced by the standard one applied to \(\sigma\). Then hypergeometric integrals provide a pairing \(\langle\;,\;\rangle: H^p(M, {\mathcal{L}}_\lambda) \times H_p(M, {\mathcal{L}}_\lambda^\vee) \to \mathbf C\) called hypergeometric pairing.

Chapter 3 is a short survey on the classical theory of arrangements. It starts with some definitions related with the poset \(L({\mathcal{A}})\) of intersections of elements of \({\mathcal{A}}\), and with some basic constructions: coning, projective closure, projective quotient, deconing, and delation-restriction triple. The remainder of the chapter is dedicated to the study of two objects that will be used later in the book. The first object are the dense edges. Define an edge to be an element of \(L({\mathcal{A}})\), and call an edge \(X\) dense if \({\mathcal{A}}_X=\{H\in {\mathcal{A}}; H \supset X\}\) is not the direct sum of two nonempty arrangements. Let \({\mathcal{A}}\) be an arrangement of rank \(r\). The second object is the \(\beta\)-invariant which is defined by \(\beta= \beta({\mathcal{A}})= (-1)^r \chi ({\mathcal{A}},1)\), where \(\chi({\mathcal{A}},t)\) denotes the characteristic polynomial of \({\mathcal{A}}\). If \({\mathcal{A}}\) is the complexification of a real arrangement \({\mathcal{A}}_\mathbf R\), the \(\beta(\mathcal{A})\) is equal to the number of bounded chambers of \({\mathcal{A}}_\mathbf R\). Most of the results stated and proved in the next four chapters concern the so-called non-resonance case. Briefly speaking, non-resonance means that some sums of \(\pm \lambda_H\) are not positive integers. In particular, if \(\sum_{H \in {\mathcal{B}}} \lambda_H \not \in \mathbf Z\) for all \(\mathcal{B} \subset {\mathcal{A}}\), then the non-resonance hypothesis is satisfied. Define the Brieskorn algebra of \(\mathcal{A}\) to be the subalgebra \(B^\cdot(\mathcal{A})\) of \(\Omega^\cdot (M)\) generated by \(1\) and by the forms \({d\alpha_H \over \alpha_H}\), \(H \in \mathcal{A}\).

The goal of Chapter 4 is to prove that, under the non-resonance hypothesis, \(H^\ast(M, {\mathcal{L}}_\lambda)\) is isomorphic to \(H^\ast(B^\cdot (\mathcal{A}),\omega_\lambda \wedge)\).

Chapter 5 starts with the definition of the Orlik-Solomon algebra \(A^\cdot (\mathcal{A})\) of an arrangement \(\mathcal{A}\). This is a graded algebra defined by generators and relations. The set \(\{ a_H; H\in \mathcal{A}\}\) of generators of \(A^\cdot (\mathcal{A})\) is in one-to-one correspondence with \(\mathcal{A}\). The mapping \(a_H \mapsto {d\alpha_H \over \alpha_H}\) induces an isomorphism \(A^\cdot(\mathcal{A}) \to B^\cdot (\mathcal{A})\). In particular, by the results of Chapter 4, one has \(H^\ast(M,{\mathcal{L}}_\lambda) \simeq H^\ast( A^\cdot(\mathcal{A}), a_\lambda \wedge)\), where \(a_\lambda= \sum_{H \in \mathcal{A}} \lambda_H a_H\). The remainder of the chapter is dedicated to the definition and the study of the complex \(\mathtt{NBC}(\mathcal{A})\) and the set \(\beta{\mathbf{nbc}}\). \(\mathtt{NBC}(\mathcal{A})\) is a simplicial complex whose vertices are the hyperplanes of \(\mathcal{A}\), and \(\beta{\mathbf{nbc}}\) is a set of simplices of \(\mathtt{NBC}(\mathcal{A})\) whose dual forms a basis for \(H^\ast( \mathtt{NBC}(\mathcal{A}), \mathbf C)\).

The goal of Chapter 6 is to produce basis for \(H^\ast(M(\mathcal{A}), {\mathcal{L}}_\lambda)\) and for \(H_\ast(M(\mathcal{A}), {\mathcal{L}}_\lambda^\vee)\). Let \(\mathcal{A}\) be an arrangement of rank \(r\). Recall that we are under the non-resonance hypothesis. Then one has \(H^p(M(\mathcal{A}), {\mathcal{L}}_\lambda)= H_p(M(\mathcal{A}), {\mathcal{L}}_\lambda^\vee)=0\), for \(p\neq r\), and \(\dim H^r(M(\mathcal{A}), {\mathcal{L}}_\lambda) = \dim H_r (M(\mathcal{A}), {\mathcal{L}}_\lambda^\vee) =\beta(\mathcal{A})\). The group \(H^r(\mathtt{NBC}(\mathcal{A}), \mathbf C)\) is isomorphic to \(H^r(M(\mathcal{A}), {\mathcal{L}}_\lambda)\), thus the set \(\beta{\mathbf{nbc}}\) furnishes a basis for \(H^r (M(\mathcal{A}), {\mathcal{L}}_\lambda)\) through this isomorphism. Assume now that \(\mathcal{A}\) is the complexification of a real arrangement \({\mathcal{A}}_\mathbf R\). Let \(\text{bch} (\mathcal{A})\) denote the set of bounded chambers of \({\mathcal{A}}_\mathbf R\). As pointed out before, the cardinality of \(\text{bch} (\mathcal{A})\) is \(\beta(\mathcal{A})\). Then \(\text{bch} (\mathcal{A})\) determines a basis for \(H_r(M(\mathcal{A}), {\mathcal{L}}_\lambda^\vee)\). One also has a natural bijection \(\tau: \text{bch}(\mathcal{A}) \to \beta{\mathbf{nbc}}\). Assume that \(\mathcal{A}\) is the complexification of an essential real arrangement \({\mathcal{A}}_\mathbf R\). We order \(\beta{\mathbf{nbc}}\) with the lexicographic ordering, \(\beta{\mathbf{nbc}}= \{ B_j\}_{j=1}^\beta\), and order \(\text{bch} (\mathcal{A})= \{ \Delta_j \}_{j=1}^\beta\) setting \(\Delta_j= \tau^{-1} (B_j)\). Define the hypergeometric period matrix \(PM(\mathcal{A},\lambda)\) by \[ PM(\mathcal{A},\lambda)_{i,j} = \int_{\Delta_j} \Phi_j \psi_i, \] where \(\psi_i\) is a holomorphic form corresponding to \(B_i \in H^l( \mathtt{NBC}(\mathcal{A}), \mathbf C) \simeq H^l (M(\mathcal{A}), {\mathcal{L}}_\lambda)\).

In Chapter 7 the authors assume that \(\operatorname{Re} \lambda_H >0\) for all \(H \in \mathcal{A}\), and compute the determinant of \(PM(\mathcal{A}, \lambda)\) in terms of gamma functions.

Chapters 8, 9, and 10 form the second part of the book whose aim is to work in the dynamical setup. Let \(B\) be the configuration space of ordered \(n\)-tuples in \(\mathbf C\), and, for \(\mathbf t=(t_1, \dots, t_n)\) in \(B\), let \(M_\mathbf t= \mathbf C \setminus \{ t_1, \dots, t_n\}\) be the complement of \({\mathcal{A}}_\mathbf t =\{t_1, \dots, t_n\}\). Choose non-resonant weights \(\lambda_1, \dots, \lambda_n\), and let \(\mathcal{L}\) be the corresponding local system. \({\mathcal{H}}_1= \bigcup_{\mathbf t \in B} H_1( M_\mathbf t, {\mathcal{L}}^\vee)\) is a locally trivial bundel over \(B\), and \({\mathcal{H}}_1 \otimes {\mathcal{O}}_B\) is isomorphic to \(({\mathcal{O}}_B)^{n-1}\) via the map which sends \(\gamma\) to \((\widehat \varphi_1, \dots, \widehat \varphi_{n-1})^T\), where \(\widehat \varphi_j = \int_\gamma \Phi \varphi_j\), and \(\varphi_j= {\lambda_{j+1} du \over u-t_{j+1}}\).

It is shown in Chapter 8 that the vector \((\widehat \varphi_1, \dots, \widehat \varphi_{n-1})^T\) satisfies the KZ-equation \[ d' \left( \begin{matrix} \widehat \varphi_1\\ \vdots\\ \widehat \varphi_{n-1}\end{matrix}\right) = \Omega \wedge \left( \begin{matrix} \widehat \varphi_1\\ \vdots\\ \widehat \varphi_{n-1}\end{matrix}\right), \] where \(d'\) is the exterior differential in \(B\), and \(\Omega\) is explicitly given.

In Chapter 9 the authors describe the space \(B\) of arrangements with a given combinatorial type. Choose such a space. One has a locally trivial fibration \(\pi: M \to B\) whose fiber at \(\mathbf t \in B\) is the complement \(M_\mathbf t\) of the arrangement \({\mathcal{A}}_\mathbf t\) corresponding to \(\mathbf t\). Set \({\mathcal{H}}_l= \bigcup_{\mathbf t \in B} H_l( M_\mathbf t, {\mathcal{L}}^\vee)\). Then \({\mathcal{H}}_l\) is a locally trivial bundle over \(B\). Any local section \(\sigma\) of \({\mathcal{H}}_l \otimes {\mathcal{O}}_B\) determines a differential equation which generalizes the KZ-equation of Chapter 8. These equations are the subject of Chapter 10.

Reviewer: Luis Paris (Dijon)

### MSC:

32S22 | Relations with arrangements of hyperplanes |

33C70 | Other hypergeometric functions and integrals in several variables |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |