Critical Set of the Master Function and Characteristic Variety of the Associated Gauss–Manin Differential Equations

Let $k,n$ be positive integers, $k<n$. Denote $J=\{1,\ldots,n\}$.

Consider the complex affine space ${\mathbb{C}}^{k}$ with coordinates $t_{1},\ldots,t_{k}$. Let ${\mathcal{C}}=(H_{j})_{j\in J}$, be an arrangement of $n$ affine hyperplanes in ${\mathbb{C}}^{k}$. Denote $U({\mathcal{C}})={\mathbb{C}}^{k}-\cup_{j\in J}H_{j}$, the complement. An edge $X_{\alpha}\subset{\mathbb{C}}^{k}$ of ${\mathcal{C}}$ is a nonempty intersection of some hyperplanes of ${\mathcal{C}}$. Denote by $J_{\alpha}\subset J$ the subset of indices of all hyperplanes containing $X_{\alpha}$. Denote $l_{\alpha}=\mathrm{codim}_{{\mathbb{C}}^{k}}X_{\alpha}$.

We assume that ${\mathcal{C}}$ is essential, that is, ${\mathcal{C}}$ has a vertex, an edge which is a point.

An edge is called dense if the subarrangement of all hyperplanes containing it is irreducible: the hyperplanes cannot be partitioned into nonempty sets so that, after a change of coordinates, hyperplanes in different sets are in different coordinates. In particular, each hyperplane of ${\mathcal{C}}$ is a dense edge.

Define complex vector spaces $\mathcal{A}^{p}({\mathcal{C}})$, $p=0,\ldots,k$. For $p=0$, we set $\mathcal{A}^{p}({\mathcal{C}})={\mathbb{C}}$. For $p\geq 1$, $\mathcal{A}^{p}({\mathcal{C}})$ is generated by symbols $(H_{j_{1}},\ldots,H_{j_{p}})$ with ${j_{i}}\in J$, such that

1. (i)

   $(H_{j_{1}},\ldots,H_{j_{p}})=0$ if $H_{j_{1}}$,…,$H_{j_{p}}$ are not in general position, that is, if the intersection $H_{j_{1}}\cap\cdots\cap H_{j_{p}}$ is empty or has codimension less than $p$;

2. (ii)

   $(H_{j_{\sigma(1)}},\ldots,H_{j_{\sigma(p)}})=(-1)^{|\sigma|}(H_{j_{1}},\ldots,% H_{j_{p}})$ for any element $\sigma$ of the symmetric group $\Sigma_{p}$;

3. (iii)

   $\sum_{i=1}^{p+1}(-1)^{i}(H_{j_{1}},\ldots,\widehat{H}_{j_{i}},\ldots,H_{j_{p+1% }})=0$ for any $(p+1)$-tuple $H_{j_{1}},\ldots,H_{j_{p+1}}$ of hyperplanes in ${\mathcal{C}}$ which are not in general position and such that $H_{j_{1}}\cap\cdots\cap H_{j_{p+1}}\not=\emptyset$.

The direct sum $\mathcal{A}({\mathcal{C}})=\oplus_{p=1}^{N}\mathcal{A}^{p}({\mathcal{C}})$ is the Orlik–Solomon algebra with respect to multiplication $(H_{j_{1}},\ldots,H_{j_{p}})\cdot(H_{j_{p+1}},\ldots,H_{j_{p+q}})=(H_{j_{1}},% \ldots,H_{j_{p}},H_{j_{p+1}},\ldots,H_{j_{p+q}})$.

Fix a point $a=(a_{1},\ldots,a_{n})\in({\mathbb{C}}^{\times})^{n}$ called the weight. Then the arrangement ${\mathcal{C}}$ is weighted: for $j\in J$, we assign weight $a_{j}$ to hyperplane $H_{j}$. For an edge $X_{\alpha}$, define its weight $a_{\alpha}=\sum_{j\in J_{\alpha}}a_{j}$. We define $\omega^{(a)}=\sum_{j\in J}a_{j}\cdot(H_{j})\ \in\mathcal{A}^{1}({\mathcal{C}})$. Multiplication by $\omega^{(a)}$ defines the differential $d^{(a)}:\mathcal{A}^{p}({\mathcal{C}})\to\mathcal{A}^{p+1}({\mathcal{C}})$, $x\mapsto\omega^{(a)}\cdot x$, on $\mathcal{A}({\mathcal{C}})$,  $(d^{(a)})^{2}=0$. The complex $(\mathcal{A}({\mathcal{C}}),d^{(a)})$ is called the Aomoto complex.

For an edge $X_{\alpha}$,  $l_{\alpha}=p$, a flag starting at $X_{\alpha}$ is a sequence $X_{\alpha_{0}}\supset X_{\alpha_{1}}\supset\ldots\supset X_{\alpha_{p}}=X_{\alpha}$ of edges such that $l_{\alpha_{j}}=j$ for $j=0,\ldots,p$. For an edge $X_{\alpha}$, we define $(\overline{{\mathcal{F}}}_{\alpha})_{\mathbb{Z}}$ as the free ${\mathbb{Z}}$-module generated by the elements $\overline{F}_{{\alpha_{0}},\ldots,{\alpha_{p}}=\alpha}$ labeled by the elements of the set of all flags starting at $X_{\alpha}$. We define $({\mathcal{F}}_{\alpha})_{\mathbb{Z}}$ as the quotient of $(\overline{{\mathcal{F}}}_{\alpha})_{\mathbb{Z}}$ by the submodule generated by all the elements of the form

Such an element is determined by $j\in\{1,\ldots,p-1\}$ and an incomplete flag $X_{\alpha_{0}}\supset\cdots\supset X_{\alpha_{j-1}}\supset X_{\alpha_{j+1}}% \supset\cdots\supset X_{\alpha_{p}}=X_{\alpha}$ with $l_{\alpha_{i}}$ $=$ $i$.

We denote by ${F}_{{\alpha_{0}},\ldots,{\alpha_{p}}}$ the image in $({\mathcal{F}}_{\alpha})_{\mathbb{Z}}$ of the element $\overline{F}_{{\alpha_{0}},\ldots,{\alpha_{p}}}$. For $p=0,\ldots,k$, we set $({{\mathcal{F}}}^{p}({\mathcal{C}}))_{\mathbb{Z}}=\oplus_{X_{\alpha},\,l_{% \alpha}=p}\,({{\mathcal{F}}}_{\alpha})_{\mathbb{Z}}$, ${{\mathcal{F}}}^{p}({\mathcal{C}})=({{\mathcal{F}}}^{p}({\mathcal{C}}))_{% \mathbb{Z}}\otimes{\mathbb{C}}$, ${\mathcal{F}}({\mathcal{C}})=\oplus_{p=1}^{N}{\mathcal{F}}^{p}({\mathcal{C}})$. We define the differential $d_{\mathbb{Z}}:({\mathcal{F}}^{p}({\mathcal{C}}))_{\mathbb{Z}}\to({\mathcal{F}% }^{p+1}({\mathcal{C}}))_{\mathbb{Z}}$ by

$d^{2}_{\mathbb{Z}}=0$. Tensoring $d_{\mathbb{Z}}$ with ${\mathbb{C}}$, we obtain the differential $d:{\mathcal{F}}^{p}({\mathcal{C}})\to{\mathcal{F}}^{p+1}({\mathcal{C}})$. In particular, we have (see [Schechtman and Varchenko1991])

A formula for the Euler characteristic $\chi(U({\mathcal{C}}))$ in terms of the matroid associated with ${\mathcal{C}}$ is given in [Schechtman et al.1995] ([Schechtman et al.1995], Proposition 2.3). The condition $\chi(U({\mathcal{C}}))\neq 0$ is discussed in [Crapo1967] ([Crapo1967], Theorem 2) cited as Theorem 2.4 in [Schechtman et al.1995]. On the equality of the absolute value $|\chi(U({\mathcal{C}}))|$ and the number of bounded components of the real part of $U({\mathcal{C}})$ see, for example, [Varchenko1995].

The vector spaces $\mathcal{A}^{p}({\mathcal{C}})$ and ${\mathcal{F}}^{p}({\mathcal{C}})$ are dual, see [Schechtman and Varchenko1991]. The pairing $\mathcal{A}^{p}({\mathcal{C}})\otimes{\mathcal{F}}^{p}({\mathcal{C}})\to{% \mathbb{C}}$ is defined as follows. For $H_{j_{1}},\ldots,H_{j_{p}}$ in general position, set $F(H_{j_{1}},\ldots,H_{j_{p}})=F_{{\alpha_{0}},\ldots,{\alpha_{p}}}$, where $X_{\alpha_{0}}={\mathbb{C}}^{k}$, $X_{\alpha_{1}}=H_{j_{1}}$, …, $X_{\alpha_{p}}=H_{j_{1}}\cap\cdots\cap H_{j_{p}}$. Then we define $\langle(H_{j_{1}},\ldots,H_{j_{p}}),F_{{\alpha_{0}},\ldots,{\alpha_{p}}}% \rangle=(-1)^{|\sigma|},$ if $F_{{\alpha_{0}},\ldots,{\alpha_{p}}}=F(H_{j_{\sigma(1)}},\ldots,H_{j_{\sigma(p% )}})$ for some $\sigma\in S_{p}$, and $\langle(H_{j_{1}},\ldots,H_{j_{p}}),F_{{\alpha_{0}},\ldots,{\alpha_{p}}}% \rangle=0$ otherwise.

An element $F\in{\mathcal{F}}^{k}({\mathcal{C}})$ is called singular if $F$ annihilates the image of the map $d^{(a)}:\mathcal{A}^{k-1}({\mathcal{C}})\to\mathcal{A}^{k}({\mathcal{C}})$, see [Varchenko2006]. Denote by $\operatorname{Sing}_{a}{\mathcal{F}}^{k}({\mathcal{C}})\subset{\mathcal{F}}^{k% }({\mathcal{C}})$ the subspace of all singular vectors.

The weight $a$ determines the contravariant map

where the sum is taken over all $p$-tuples $(H_{j_{1}},\ldots,H_{j_{p}})$ such that $H_{j_{1}}\supset X_{\alpha_{1}}$,…, $H_{j_{p}}\supset X_{\alpha_{p}}$. Identifying $\mathcal{A}^{p}({\mathcal{C}})$ with ${\mathcal{F}}^{p}({\mathcal{C}})^{*}$, we consider the map as a bilinear form, $S^{(a)}:{\mathcal{F}}^{p}({\mathcal{C}})\otimes{\mathcal{F}}^{p}({\mathcal{C}}% )\to{\mathbb{C}}$. The bilinear form is called the contravariant form. The contravariant form is symmetric. For $F_{1},F_{2}\in{\mathcal{F}}^{p}({\mathcal{C}})$,

where the sum is over all unordered $p$-element subsets (see [Schechtman and Varchenko1991]).

Theorem 2.4 is a corollary of Lemma 2.2 and Theorems 2.1, 2.3.

Notice that none of the dense edges has weight zero if all weights are positive.

For $j\in J$, fix defining equations $f_{j}=0$ for the hyperplanes $H_{j}$, where $f_{j}=b^{1}_{j}t_{1}+\cdots+b^{k}_{j}t_{k}+z_{j}$ with $b^{i}_{j},z_{j}\in{\mathbb{C}}$. Consider the logarithmic differential 1-form $\omega_{j}=df_{j}/f_{j}$ on ${\mathbb{C}}^{k}$. Let $\bar{\mathcal{A}}({\mathcal{C}})$ be the exterior ${\mathbb{C}}$-algebra of differential forms generated by 1 and $\omega_{j}$, $j\in J$. The map ${\mathcal{A}}({\mathcal{C}})\to\bar{\mathcal{A}}({\mathcal{C}}),\ (H_{j})% \mapsto\omega_{j}$, is an isomorphism. We identify ${\mathcal{A}}({\mathcal{C}})$ and $\bar{\mathcal{A}}({\mathcal{C}})$.

For $I=\{i_{1},\ldots,i_{k}\}\subset J$, denote $d_{I}=d_{i_{1},\ldots,i_{k}}=\det_{i,l=1}^{k}(b^{i}_{i_{l}})$. Then

The master function of the weighted arrangement $({\mathcal{C}},a)$ is

a multivalued function on $U({\mathcal{C}})$. We have $d\Phi_{{\mathcal{C}},a}=\sum_{j\in J}a_{j}\omega_{j}=\omega^{(a)}\in\mathcal{A% }^{1}({\mathcal{C}})$. Let $C_{{\mathcal{C}},a}=\{u\in U({\mathcal{C}})\ |\ \frac{\partial\Phi_{{\mathcal{% C}},a}}{\partial t_{i}}(u)=0\ \operatorname{for}\ i=1,\ldots,k\}$ be the critical set of $\Phi_{{\mathcal{C}},a}$. The critical point equations can be reformulated as the equation $\omega^{(a)}|_{u}=0$. Notice that

Define the Hessian of the master function, $\operatorname{Hess}_{{\mathcal{C}},a}=\operatorname{det}_{i,j=1}^{k}\Big(\frac% {\partial^{2}\Phi_{{\mathcal{C}},a}}{\partial t_{i}\partial t_{j}}\Big)$. A critical point $u\in C_{{\mathcal{C}},a}$ is nondegenerate if $\operatorname{Hess}_{{\mathcal{C}},a}(u)\neq 0$. We have the formula in [Varchenko2006]:

Consider the projective space ${\mathbb{P}}^{k}$ compactifying ${\mathbb{C}}^{k}$. Assign the weight $a_{\infty}=-\sum_{j\in J}a_{j}$ to the hyperplane $H_{\infty}={\mathbb{P}}^{k}-{\mathbb{C}}^{k}$. Denote by ${\mathcal{C}}^{\vee}$ the arrangement $(H_{j})_{j\in J\cup\infty}$ in ${\mathbb{P}}^{k}$. The weighted arrangement $({\mathcal{C}},a)$ is called unbalanced if the weight of any dense edge of ${\mathcal{C}}^{\vee}$ is nonzero, see [Varchenko2011]. For example, $({\mathcal{C}},a)$ is unbalanced if all weights $(a_{j})_{j\in J}$ are positive. The unbalanced weights form a Zariski open subset in the space of all weights $a\in({\mathbb{C}}^{\times})^{n}$.

Let $\mathcal{O}(U({\mathcal{C}}))$ be the algebra of regular functions on $U({\mathcal{C}})$ and $I_{{\mathcal{C}},a}=\Big\langle\frac{\partial\Phi_{{\mathcal{C}},a}}{\partial t% _{i}}\ |\ i=1,\ldots,k\Big\rangle\subset\mathcal{O}(U({\mathcal{C}}))$ the ideal generated by the first derivatives of $\Phi_{{\mathcal{C}},a}$. Let ${\mathcal{O}(C_{{\mathcal{C}},a})}=\mathcal{O}(U({\mathcal{C}}))/I_{{\mathcal{% C}},a}$ be the algebra of functions on the critical set and $[\,]:\mathcal{O}(U({\mathcal{C}}))\to{\mathcal{O}(C_{{\mathcal{C}},a})}$, $f\mapsto[f]$, the projection. The algebra ${\mathcal{O}(C_{{\mathcal{C}},a})}$ is finite-dimensional, if all critical points are isolated. In that case, ${\mathcal{O}(C_{{\mathcal{C}},a})}=\oplus_{u\in C_{{\mathcal{C}},a}}{\mathcal{% O}(C_{{\mathcal{C}},a})}_{u}$, where ${\mathcal{O}(C_{{\mathcal{C}},a})}_{u}$ is the local algebra corresponding to the point $u$.

Let $\mathcal{R}_{u}:{\mathcal{O}(C_{{\mathcal{C}},a})}_{u}\to{\mathbb{C}}$ be the Grothendieck residue,

Here $\Gamma_{u}$ is the real $k$-cycle located in a small neighborhood of $u$ and defined by the equations $|\frac{\partial\Phi_{{\mathcal{C}},a}}{\partial t_{i}}|=\epsilon_{i},\ i=1,% \ldots,k$, where $\epsilon_{s}$ are sufficiently small positive numbers. The cycle is oriented by the condition $d\arg\frac{\partial\Phi_{{\mathcal{C}},a}}{\partial t_{1}}\wedge\cdots\wedge d% \arg\frac{\partial\Phi_{{\mathcal{C}},a}}{\partial t_{k}}>0$, see [Griffiths and Harris1994].

Denote by $[{\operatorname{Hess}}_{{\mathcal{C}},a}]_{u}$ the image of the Hessian in ${\mathcal{O}(C_{{\mathcal{C}},a})}_{u}$. We have

where $\mu_{u}=\dim_{\mathbb{C}}{\mathcal{O}(C_{{\mathcal{C}},a})}_{u}$, the Milnor number, see [Arnol’d et al.1985]. Define the bilinear form on ${\mathcal{O}(C_{{\mathcal{C}},a})}_{u}$,

If ${\mathcal{O}(C_{{\mathcal{C}},a})}$ is finite-dimensional, we define the residue bilinear form $(\,,\,)_{C_{{\mathcal{C}},a}}$ on ${\mathcal{O}(C_{{\mathcal{C}},a})}$ as

This form is nondegenerate, see [Arnol’d et al.1985], and $([f][g],[h])_{C_{{\mathcal{C}},a}}=([f],[g][h])_{C_{{\mathcal{C}},a}}$ for all $[f],[g],[h]\in{\mathcal{O}(C_{{\mathcal{C}},a})}$. In other words, the pair $({\mathcal{O}(C_{{\mathcal{C}},a})},{\ }(\,,\,)_{C_{{\mathcal{C}},a}})$ is a Frobenius algebra.

A differential $k$-form $H\in{\mathcal{A}}^{k}({\mathcal{C}})$ can be written as $H=f_{H}dt_{1}\wedge\cdots\wedge dt_{k}$, where $f_{H}\in\mathcal{O}(U({\mathcal{C}}))$. Define a map $\operatorname{F}:U({\mathcal{C}})\to{\mathcal{F}}^{k}({\mathcal{C}})$ which sends $u\in U({\mathcal{C}})$ to the element $\operatorname{F}(u)\in{\mathcal{F}}^{k}({\mathcal{C}})$ such that $\langle H,\operatorname{F}(u)\rangle=f_{H}(u)$ for any $H\in\mathcal{A}^{k}({\mathcal{C}})$. The map $\operatorname{F}$ is called the specialization map, the vector $\operatorname{F}(u)$ is called the special vector at $u$, see [Varchenko2006]. In the theory of quantum integrable systems special vectors are called the Bethe vectors.

Let $(F_{m})_{m\in M}$ be a basis of ${\mathcal{F}}^{k}({\mathcal{C}})$ and $(H^{m})_{m\in M}\subset\mathcal{A}^{k}({\mathcal{C}})$ the dual basis. We have $H^{m}=f_{H^{m}}dt_{1}\wedge\cdots\wedge dt_{k}$ for some $f_{H^{m}}\in\mathcal{O}(U({\mathcal{C}}))$. The element

is called the canonical element. For $u\in U({\mathcal{C}})$, we have

Let $[E]$ be the image of the canonical element in ${\mathcal{O}(C_{{\mathcal{C}},a})}\otimes{\mathcal{F}}^{k}({\mathcal{C}})$.

Assume that the algebra ${\mathcal{O}(C_{{\mathcal{C}},a})}$ is finite-dimensional. Define the linear map

The map $\mathcal{E}$ is called the canonical isomorphism. We provide a proof of Theorem 2.12 in Sect. 2.16.

On the restriction of $S^{(a)}$ to the subspace ${\operatorname{Sing}\,}{\mathcal{F}}^{k}({\mathcal{C}})$ see also [Falk and Varchenko2012].

If all critical points are nondegenerate, then

see (2.10).

First assume that the weight $a$ is generic and all critical points of $\Phi_{{\mathcal{C}},a}$ are nondegenerate. Then the special vectors $(\operatorname{F}(u))_{u\in C_{{\mathcal{C}},a}}$ form a basis of $\operatorname{Sing}_{a}{\mathcal{F}}^{k}({\mathcal{C}})$ by Theorems 2.11, 2.7 and Corollary 2.5. (In the theory of quantum integrable systems this fact is called the completeness of the Bethe ansatz method, see [Varchenko1995]; [Varchenko2006].)

Theorem 2.11 and formula (2.10) applied to the basis $(\operatorname{F}(u))_{u\in C_{{\mathcal{C}},a}}$ show that $S^{(a)}$ restricted to ${\operatorname{Sing}\,}_{a}{\mathcal{F}}^{k}({\mathcal{C}})$ is nondegenerate and $\mathcal{E}:{\mathcal{O}(C_{{\mathcal{C}},a})}\to{\operatorname{Sing}\,}{% \mathcal{F}}^{k}({\mathcal{C}})$ is an isomorphism of vector spaces that identifies the residue form on ${\mathcal{O}(C_{{\mathcal{C}},a})}$ and the contravariant form on ${\operatorname{Sing}\,}{\mathcal{F}}^{k}({\mathcal{C}})$ multiplied by $(-1)^{k}$. (More precisely, this follows from the following fact. Let $u\in C_{{\mathcal{C}},a}$ and $g_{u}\in{\mathcal{O}(C_{{\mathcal{C}},a})}$ be the function which equals 1 at $u$ and equals 0 at other points of $C_{{\mathcal{C}},a}$. Then $\mathcal{E}(g_{u})=\operatorname{F}(u)/{\operatorname{Hess}}_{{\mathcal{C}},a}% (u)$.)

The orthogonal projection ${\mathcal{F}}^{k}({\mathcal{C}})\to{\operatorname{Sing}\,}_{a}{\mathcal{F}}^{k% }({\mathcal{C}})$ is defined by the formula

Let $\mathcal{S}^{(a)}(F)=fdt_{1}\wedge\cdots\wedge dt_{k}$ and $u\in U({\mathcal{C}})$, then $f(u)=S^{(a)}(F,\operatorname{F}(u))$ by the definitions of $\operatorname{F}(u)$, $\mathcal{S}^{(a)}$, $S^{(a)}$. Hence the map $[\mathcal{S}^{(a)}]$ defined in (2.19) sends $F$ to the element of ${\mathcal{O}(C_{{\mathcal{C}},a})}$ which equals $S^{(a)}(F,\operatorname{F}(u))$ at every $u\in C_{{\mathcal{C}},a}$. Applying formula (2.18) to this element we obtain

These two formulas prove Theorem 2.16 if all critical points are nondegenerate.

Assume now that the weight $a$ is unbalanced. Then all critical points of $\Phi_{{\mathcal{C}},a}$ are isolated and the sum of the corresponding Milnor numbers equals $|\chi(U({\mathcal{C}}))|$. We deform the weight $a$ to make it generic and to make all critical points nondegenerate. Then ${\operatorname{Sing}\,}_{a}{\mathcal{F}}({\mathcal{C}})$ and $S^{(a)}\big|_{{\operatorname{Sing}\,}_{a}{\mathcal{F}}({\mathcal{C}})}$ continuously depend on the deformation as well as ${\mathcal{O}(C_{{\mathcal{C}},a})}$ and $(\,,\,)_{C_{{\mathcal{C}},a}}$. The maps $[\mathcal{S}^{(a)}]\big|_{{\operatorname{Sing}\,}_{a}{\mathcal{F}}({\mathcal{C% }})}$ and $\mathcal{E}$ also continuously depend on the deformation. This implies that for the initial unbalanced weight $a$, we have the identity $\mathcal{E}{\circ}[\mathcal{S}^{(a)}]\big|_{\operatorname{Sing}_{a}{\mathcal{F% }}^{k}({\mathcal{C}})}=(-1)^{k}$ and the fact that $\mathcal{E}$ identifies the residue form on ${\mathcal{O}(C_{{\mathcal{C}},a})}$ and the contravariant form on ${\operatorname{Sing}\,}{\mathcal{F}}^{k}({\mathcal{C}})$ multiplied by $(-1)^{k}$. This proves Theorems 2.16 and 2.12.

If the weight $a$ is unbalanced, the formula $H^{p}({\mathcal{F}}({\mathcal{C}}),d)=H^{p}(({\mathcal{F}}({\mathcal{C}}))_{% \mathbb{Z}},d_{\mathbb{Z}})\otimes{\mathbb{C}}$ and the isomorphism $[\mathcal{S}^{(a)}]\big|_{\operatorname{Sing}_{a}{\mathcal{F}}^{k}({\mathcal{C% }})}:H^{k}({\mathcal{F}},d)\to{\mathcal{O}(C_{{\mathcal{C}},a})}$ define an integral structure on ${\mathcal{O}(C_{{\mathcal{C}},a})}$. More precisely, for a $k$-flag of edges $X_{\alpha_{0}}\supset X_{\alpha_{1}}\supset\cdots\supset X_{\alpha_{k}}$, let $\mathcal{S}^{(a)}(F_{\alpha_{0},\ldots,\alpha_{k}})=f_{\alpha_{0},\ldots,% \alpha_{k}}dt_{1}\wedge\cdots\wedge dt_{k}$. Denote by $w_{\alpha_{0},\ldots,\alpha_{k}}$ the element $[f_{\alpha_{0},\ldots,\alpha_{k}}]\in{\mathcal{O}(C_{{\mathcal{C}},a})}$.

Similarly, for a $k$-flag of edges $X_{\alpha_{0}}\supset X_{\alpha_{1}}\supset\cdots\supset X_{\alpha_{k}}$, let $v_{\alpha_{0},\ldots,\alpha_{k}}$ be the orthogonal projection of $F_{\alpha_{0},\ldots,\alpha_{k}}$ to ${\operatorname{Sing}\,}_{a}{\mathcal{F}}^{k}({\mathcal{C}})$.

We have

The elements $\{w_{\alpha_{0},\ldots,\alpha_{k}}\}\subset{\mathcal{O}(C_{{\mathcal{C}},a})}$ and $\{v_{\alpha_{0},\ldots,\alpha_{k}}\}\subset{\operatorname{Sing}\,}_{a}{% \mathcal{F}}^{k}({\mathcal{C}})$ will be called the marked elements. The relations (2.18), (2.19) will be called the marked relations.

If $a$ is unbalanced, then

as vector spaces. The first space is a cohomology space of a skew-commutative graded algebra $\mathcal{A}({\mathcal{C}})$. The last space is the vector space of a commutative algebra. Isomorphisms (2.26) identify these skew-commutative and commutative objects. It is interesting to identify the multiplication operators on the last space with suitable operators on the first two spaces. It turns out that those operators appear in the associated Gauss–Manin (hypergeometric) differential equations, see Sect. 3.7 and [Mukhin et al.2011], [Varchenko2006], [Varchenko2011].

Another identification of skew-commutative and commutative objects of an arrangement see in [Gelfand and Varchenko1987], [Proudfoot2006].

Consider a deformation ${\mathcal{C}}(s)$ of the arrangement ${\mathcal{C}}$, which preserves the combinatorics of ${\mathcal{C}}$. Assume that the edges of ${\mathcal{C}}(s)$ can be identified with the edges of ${\mathcal{C}}$ so that the elements in formula (2.1) and the differential in formula (2.2) do not depend on $s$. Assume that the deformed arrangement ${\mathcal{C}}(s)$ has a deformed weight $a(s)$, which remains unbalanced. Then for every $s$, the elements $\{w_{\alpha_{0},\ldots,\alpha_{k}}(s)\}$ span $\mathcal{O}(C_{{\mathcal{C}}(s),a(s)})$ as a vector space with linear relations (2.18) not depending on $s$. This allows us to identify all the vector spaces $\mathcal{O}(C_{{\mathcal{C}}(s),a(s)})$. In particular, if an element $w(s)\in\mathcal{O}(C_{{\mathcal{C}}(s),a(s)})$ is given, then the derivative $\frac{dw}{ds}$ is well-defined. This construction is called the combinatorial connection on the family of algebras $\mathcal{O}(C_{{\mathcal{C}}(s),a(s)})$, see [Varchenko2015]. All the elements $\{w_{\alpha_{0},\ldots,\alpha_{k}}(s)\}$ are flat sections of the combinatorial connection.

Similarly we can define the combinatorial connection on the family of vector spaces ${\operatorname{Sing}\,}_{a(s)}{\mathcal{F}}^{k}({\mathcal{C}}(s))$.

An essential arrangement ${\mathcal{C}}$ is with normal crossings, if exactly $k$ hyperplanes meet at every vertex of ${\mathcal{C}}$. Assume that ${\mathcal{C}}$ is an essential arrangement with normal crossings. A subset $\{j_{1},\ldots,j_{p}\}\subset J$ is called independent if the hyperplanes $H_{j_{1}},\ldots,H_{j_{p}}$ intersect transversally.

A basis of $\mathcal{A}^{p}({\mathcal{C}})$ is formed by $(H_{j_{1}},\ldots,H_{j_{p}})$, where $\{{j_{1}}<\cdots<{j_{p}}\}$ are independent ordered $p$-element subsets of $J$. The dual basis of ${\mathcal{F}}^{p}({\mathcal{C}})$ is formed by the corresponding vectors $F(H_{j_{1}},\ldots,H_{j_{p}})$. These bases of $\mathcal{A}^{p}({\mathcal{C}})$ and ${\mathcal{F}}^{p}({\mathcal{C}})$ are called standard. We have

For an independent subset $\{j_{1},\ldots,j_{p}\}$, we have $S^{(a)}(F(H_{j_{1}},\ldots,H_{j_{p}}),F(H_{j_{1}},\ldots,{\ }H_{j_{p}}))=a% _{j_{1}}\cdots a_{j_{p}}$ and $S^{(a)}(F(H_{j_{1}},\ldots,H_{j_{p}}),F(H_{i_{1}},\ldots,H_{i_{k}}))=0$ for distinct elements of the standard basis. If $a$ is unbalanced, then the marked elements in ${\mathcal{O}(C_{{\mathcal{C}},a})}$ are

where $\{i_{1},\ldots,i_{k}\}$ runs through the set of all independent $k$-element subsets of $J$. We have

We put $w_{i_{1},\ldots,i_{k}}=0$ if the set $\{i_{1},\ldots,i_{k}\}$ is dependent. The marked relations are labeled by independent subsets $\{i_{2},\ldots,i_{k}\}$ and have the form

The marked elements $v_{i_{1},\ldots,i_{k}}$ in ${\operatorname{Sing}\,}_{a}{\mathcal{F}}^{k}({\mathcal{C}})$ are orthogonal projections to ${\operatorname{Sing}\,}_{a}{\mathcal{F}}^{k}({\mathcal{C}})$ of the elements $F(H_{i_{1}},\ldots,H_{i_{k}})$ with the skew-symmetry property

and the marked relations

labeled by independent subsets $\{i_{2},\ldots,i_{k}\}$.

Recall that $J=\{1,\ldots,n\}$. Consider ${\mathbb{C}}^{k}$ with coordinates $t_{1},\ldots,t_{k}$, ${\mathbb{C}}^{n}$ with coordinates $z_{1},\ldots,z_{n}$, the projection $\pi:{\mathbb{C}}^{n}\times{\mathbb{C}}^{k}\to{\mathbb{C}}^{n}$. Fix $n$ nonzero linear functions on ${\mathbb{C}}^{k}$, $g_{j}=b_{j}^{1}t_{1}+\cdots+b_{j}^{k}t_{k},$ $j\in J,$ where $b_{j}^{i}\in{\mathbb{C}}$. We assume that the functions $g_{j},j\in J$, span the dual space $({\mathbb{C}}^{k})^{*}$.

Define $n$ linear functions on ${\mathbb{C}}^{n}\times{\mathbb{C}}^{k}$, $f_{j}=g_{j}+z_{j}=b_{j}^{1}t_{1}+\cdots+b_{j}^{k}t_{k}+z_{j},$ $j\in J.$ We consider in ${\mathbb{C}}^{n}\times{\mathbb{C}}^{k}$ the arrangement of hyperplanes ${\mathcal{C}}=\{H_{j}\}_{j\in J}$, where $H_{j}$ is the zero set of $f_{j}$, and denote by $U({\mathcal{C}})={\mathbb{C}}^{n}\times{\mathbb{C}}^{k}-\cup_{j\in J}H_{j}$ the complement.

For every $x=(x_{1},\ldots,x_{n})\in{\mathbb{C}}^{n}$, the arrangement ${\mathcal{C}}$ induces an arrangement ${\mathcal{C}}(x)$ in the fiber $\pi^{-1}(x)$. We identify every fiber with ${\mathbb{C}}^{k}$. Then ${\mathcal{C}}(x)$ consists of hyperplanes $\{H_{j}(x)\}_{j\in J}$, defined in ${\mathbb{C}}^{k}$ by the equations $g_{j}+x_{j}=0$. Thus $\{{\mathcal{C}}(x)\}_{x\in{\mathbb{C}}^{n}}$ is a family of arrangements in ${\mathbb{C}}^{k}$, whose hyperplanes are transported parallelly as $x$ changes. We denote by $U({\mathcal{C}}(x))={\mathbb{C}}^{k}-\cup_{j\in J}H_{j}(x)$ the complement.

For almost all points $x\in{\mathbb{C}}^{n}$, the arrangement ${\mathcal{C}}(x)$ is with normal crossings. Such points form the complement in ${\mathbb{C}}^{n}$ to the union of suitable hyperplanes called the discriminant.

The collection $(g_{j})_{j\in J}$ induces a matroid structure on $J$. A subset $C=\{i_{1},\ldots,i_{r}\}\subset J$ is a circuit if $(g_{i})_{i\in C}$ are linearly dependent but any proper subset of $C$ gives linearly independent $g_{i}$’s. Denote by $\mathfrak{C}$ the set of all circuits in $J$.

For a circuit $C=\{i_{1},\ldots,i_{r}\}$, let $(\lambda^{C}_{i})_{i\in C}$ be a nonzero collection of complex numbers such that $\sum_{i\in C}\lambda^{C}_{i}g_{i}=0$. Such a collection is unique up to multiplication by a nonzero number. For every circuit $C$ we fix such a collection and denote $f_{C}=\sum_{i\in C}\lambda^{C}_{i}z_{i}$. The equation $f_{C}=0$ defines a hyperplane $H_{C}$ in ${\mathbb{C}}^{n}$. It is convenient to assume that $\lambda^{C}_{i}=0$ for $i\in J-C$ and write $f_{C}=\sum_{i\in J}\lambda^{C}_{i}z_{i}$.

For any $x\in{\mathbb{C}}^{n}$, the hyperplanes $\{H_{i}(x)\}_{i\in C}$ in ${\mathbb{C}}^{k}$ have nonempty intersection if and only if $x\in H_{C}$. If $x\in H_{C}$, then the intersection has codimension $r-1$ in ${\mathbb{C}}^{k}$. The union $\Delta=\cup_{C\in\mathfrak{C}}H_{C}$ is called the discriminant. The arrangement ${\mathcal{C}}(x)$ in ${\mathbb{C}}^{k}$ has normal crossings if and only if $x\in{\mathbb{C}}^{n}-\Delta$, see [Varchenko2011].

On the discriminant see also [Bayer and Brandt1997].

For any $x^{1},x^{2}\in{\mathbb{C}}^{n}-\Delta$, the spaces ${\mathcal{F}}^{p}({\mathcal{C}}(x^{1}))$, ${\mathcal{F}}^{p}({\mathcal{C}}(x^{2}))$ are canonically identified if a vector $F(H_{j_{1}}(x^{1}),\ldots,H_{j_{p}}(x^{1}))$ of the first space is identified with the vector $F(H_{j_{1}}(x^{2}),\ldots,H_{j_{p}}(x^{2}))$ of the second. In other words, we identify the standard bases of these spaces.

Assume that a weight $a\in({\mathbb{C}}^{\times})^{n}$ is given. Then each arrangement ${\mathcal{C}}(x)$ is weighted. The identification of spaces ${\mathcal{F}}^{p}({\mathcal{C}}(x^{1}))$, ${\mathcal{F}}^{p}({\mathcal{C}}(x^{2}))$ for $x^{1},x^{2}\in{\mathbb{C}}^{n}-\Delta$ identifies the corresponding subspaces $\operatorname{Sing}_{a}{\mathcal{F}}^{k}({\mathcal{C}}(x^{1}))$, $\operatorname{Sing}_{a}{\mathcal{F}}^{k}({\mathcal{C}}(x^{2}))$ and contravariant forms.

Assume that the weighted arrangement $({\mathcal{C}}(x),a)$ is unbalanced for some $x\in{\mathbb{C}}^{n}-\Delta$, then $({\mathcal{C}}(x),a)$ is unbalanced for all $x\in{\mathbb{C}}^{n}-\Delta$. The identification of $\operatorname{Sing}_{a}{\mathcal{F}}^{k}({\mathcal{C}}(x^{1}))$ and $\operatorname{Sing}_{a}{\mathcal{F}}^{k}({\mathcal{C}}(x^{2}))$ also identifies the marked elements $v_{j_{1},\ldots,j_{k}}(x^{1})$ and $v_{j_{1},\ldots,j_{k}}(x^{1})$, see Sect. 2.20. For $x\in{\mathbb{C}}^{n}-\Delta$, denote $V={\mathcal{F}}^{k}({\mathcal{C}}(x))$, $\operatorname{Sing}_{a}V=\operatorname{Sing}_{a}{\mathcal{F}}^{k}({\mathcal{C}% }(x))$, $v_{j_{1},\ldots,j_{k}}=v_{j_{1},\ldots,j_{k}}(x)$. The triple $(V,\operatorname{Sing}_{a}V,S^{(a)})$, with marked elements $v_{j_{1},\ldots,j_{k}}$, does not depend on $x$ under the identification.

As a result of this reasoning we obtain the canonically trivialized vector bundle

with the canonically trivialized subbundle $\sqcup_{x\in{\mathbb{C}}^{n}-\Delta}\,\operatorname{Sing}_{a}{\mathcal{F}}^{k}% ({\mathcal{C}}(x))\to{\mathbb{C}}^{n}-\Delta$ and the constant contravariant form on the fibers. This trivialization identifies the bundle in (3.2) with the bundle $({{\mathbb{C}}^{n}-\Delta})\times V\to{{\mathbb{C}}^{n}-\Delta}$ and the subbundle $\sqcup_{x\in{\mathbb{C}}^{n}-\Delta}\,\operatorname{Sing}_{a}{\mathcal{F}}^{k}% ({\mathcal{C}}(x))\to{\mathbb{C}}^{n}-\Delta$ with the subbbundle

The bundle in (3.3) will be called the combinatorial bundle, the flat connection on it will be called combinatorial, see [Varchenko2011]; [Varchenko2015], cf. Sect. 2.19.

For a circuit $C=\{i_{1},\ldots,i_{r}\}\subset J$, we define the linear operator $L_{C}:V\to V$ as follows. Let $C_{m}=C-\{i_{m}\}$. Let $F(H_{j_{1}},\ldots,H_{j_{k}})$ be an element of the standard basis. We set $L_{C}:F(H_{j_{1}},\ldots,H_{j_{k}})\mapsto 0$ if $|\{{j_{1}},\ldots,{j_{k}}\}\cap C|<r-1$. If $\{{j_{1}},\ldots,{j_{k}}\}\cap C=C_{m}$, then by (2.27) we have $F(H_{j_{1}},\ldots,H_{j_{k}})=\pm F(H_{i_{1}},H_{i_{2}},\ldots,\widehat{H_{i_{% m}}},\ldots,H_{i_{r-1}},H_{i_{r}},H_{s_{1}},\ldots,H_{s_{k-r+1}})$ with $\{{s_{1}},\ldots,{s_{k-r+1}}\}=\{{j_{1}},\ldots,{j_{k}}\}-C_{m}$. We set

Consider on ${\mathbb{C}}^{n}\times{\mathbb{C}}^{k}$ the logarithmic 1-forms $\omega_{C}=\frac{df_{C}}{f_{C}},\ C\in\mathfrak{C}$. Recall $f_{C}=\sum_{j\in J}\lambda^{C}_{j}z_{j}$. We set

We have

We obtain formulas for the action of $K_{j}$ on the marked elements $v_{j_{1},\ldots,j_{k}}\in{\operatorname{Sing}\,}_{a}V$ from formulas for the action of $L_{C}$.

Let $C=\{i_{1},\ldots,i_{r}\}$ be a circuit and $v_{j_{1},\ldots,j_{k}}\in{\operatorname{Sing}\,}_{a}V$ a marked element. If $|\{{j_{1}},\ldots,{j_{k}}\}\cap C|<r-1$, then $L_{C}(v_{j_{1},\ldots,j_{k}})=0$. If $\{{j_{1}},\ldots,{j_{k}}\}\cap C=C_{m}$, then by (2.31) we have $v_{j_{1},\ldots,j_{k}}=\pm v_{i_{1},i_{2},\ldots,\widehat{i_{m}},\ldots,i_{r-1% },i_{r},s_{1},\ldots,s_{k-r+1}}$ with $\{{s_{1}},\ldots,{s_{k-r+1}}\}=\{{j_{1}},\ldots,{j_{k}}\}-C_{m}$. We have

The master function of $({\mathcal{C}},a)$ is $\Phi_{{\mathcal{C}},a}={\sum}_{j\in J}\,a_{j}\log f_{j}$, a multivalued function on $U({{\mathcal{C}}})$. Let $\kappa\in{\mathbb{C}}^{\times}$. The function $e^{\Phi_{{\mathcal{C}},a}/\kappa}$ defines a rank one local system $\mathcal{L}_{\kappa}$ on $U({\mathcal{C}})$ whose horizontal sections over open subsets of $U({\mathcal{C}})$ are univalued branches of $e^{\Phi_{{\mathcal{C}},a}/\kappa}$ multiplied by complex numbers, see, for example, [Schechtman and Varchenko1991], [Varchenko1995]. The vector bundle

is called the homology bundle. The homology bundle has a canonical flat Gauss–Manin connection.

For a fixed $x$, choose any $\gamma\in H_{k}(U({\mathcal{C}}(x)),\mathcal{L}_{\kappa}|_{U({\mathcal{C}}(x))})$. The linear map

is an element of ${\operatorname{Sing}\,}{\mathcal{F}}^{k}({\mathcal{C}}(x))$ by Stokes’ theorem. It is known that for generic $\kappa$ any element of ${\operatorname{Sing}\,}{\mathcal{F}}^{k}({\mathcal{C}}(x))$ corresponds to a certain $\gamma$ and in that case this construction gives the integration isomorphism

see [Schechtman and Varchenko1991]. The precise values of $\kappa$, such that (3.9) is an isomorphism, can be deduced from the determinant formula in [Varchenko1989].

For generic $\kappa$ the fiber isomorphisms (3.9) define an isomorphism of the homology bundle and the combinatorial bundle (3.3). The Gauss–Manin connection induces a connection on the combinatorial bundle. That connection on the combinatorial bundle will also be called the Gauss–Manin connection.

Thus, there are two connections on the combinatorial bundle: the combinatorial connection and the Gauss–Manin connection depending on $\kappa$. In this situation we consider the differential equations for flat sections of the Gauss–Manin connection with respect to the combinatorially flat standard basis. Namely, let $\gamma(x)\in H_{k}(U({\mathcal{C}}(x)),\mathcal{L}_{\kappa}|_{U({\mathcal{C}}(% x))})$ be a flat section of the Gauss–Manin connection. Let us write the corresponding section $I_{\gamma}(x)$ of the bundle $({{\mathbb{C}}^{n}-\Delta})\times{\operatorname{Sing}\,}_{a}V\to{{\mathbb{C}}^% {n}-\Delta}$ in the combinatorially flat standard basis,

By Theorem 3.3, we can also write

For $I=\sum I^{j_{1},\ldots,j_{k}}v_{j_{1},\ldots,j_{k}}$ and $j\in J$, we denote $\frac{\partial I}{\partial z_{j}}=\sum\frac{\partial I^{j_{1},\ldots,j_{k}}}{% \partial z_{j}}v_{j_{1},\ldots,j_{k}}$. This formula defines the combinatorial connection on the combinatorial bundle.

On the Gauss–Manin connection and these differential equations see also [Cohen and Orlik2006].

Denote by $C_{{\mathcal{C}},a}$ the critical set of $\Phi_{{\mathcal{C}},a}$ in the ${\mathbb{C}}^{k}$-direction,

Let $\mathcal{O}(C_{{\mathcal{C}},a})$ be the algebra of regular functions on $C_{{\mathcal{C}},a}$ and $\mathcal{O}(C_{{\mathcal{C}}(x),a})$ the algebra of regular functions on $C_{{\mathcal{C}}(x),a}=C_{{\mathcal{C}},a}\cap\pi^{-1}(x)$. Namely, for $x=(x_{1},\ldots,x_{n})\in{\mathbb{C}}^{n}$, let $I_{{\mathcal{C}}(x),a}$ be the ideal in $\mathcal{O}(U({\mathcal{C}}(x)))$ generated by $\frac{\partial\Phi_{{\mathcal{C}},a}}{\partial t_{i}},i=1,\ldots,k$. We set

Assume that the weight $a$ is such that the pair $({\mathcal{C}}(x),a)$ is unbalanced for some $x\in{{\mathbb{C}}^{n}-\Delta}$. Then $\dim{\mathcal{O}(C_{{\mathcal{C}}(x),a})}=|\chi(U({\mathcal{C}}(x)))|$ for every $x\in{{\mathbb{C}}^{n}-\Delta}$ and we obtain the vector bundle of algebras

For $x\in{{\mathbb{C}}^{n}-\Delta}$, consider the canonical element $E(x)$ of the arrangement ${\mathcal{C}}(x)$ and its image $[E(x)]$ in ${\mathcal{O}(C_{{\mathcal{C}}(x),a})}\otimes{\operatorname{Sing}_{a}V}$, see Lemma 2.10. Recall the canonical isomorphism (2.16),

This fiber isomorphism establishes an isomorphism $\mathcal{E}$ of the bundles (3.14) and (3.3). The isomorphism $\mathcal{E}$ and the combinatorial and Gauss–Manin connections on the bundle (3.3) induce two connections on the bundle (3.14) which will also be called the combinatorial and Gauss–Manin connections, respectively.

For $x\in{{\mathbb{C}}^{n}-\Delta}$ and a circuit $C=\{i_{1},\ldots,i_{r}\}\subset J$, we define the linear operator $L_{C}:{\mathcal{O}(C_{{\mathcal{C}}(x),a})}\to{\mathcal{O}(C_{{\mathcal{C}}(x)% ,a})}$ as follows. Let $w_{j_{1},\ldots,j_{k}}\in{\mathcal{O}(C_{{\mathcal{C}}(x),a})}$ be a marked element. If $|\{{j_{1}},\ldots,{j_{k}}\}\cap C|<r-1$, then we set $L_{C}(w_{j_{1},\ldots,j_{k}})=0$. If $\{{j_{1}},\ldots,{j_{k}}\}\cap C=C_{m}$, then by (2.29) we have $w_{j_{1},\ldots,j_{k}}=$ $\pm\,w_{i_{1},i_{2},\ldots,\widehat{i_{m}},\ldots,i_{r-1},i_{r},s_{1},\ldots,s% _{k-r+1}}$ with $\{{s_{1}},\ldots,{s_{k-r+1}}\}=\{{j_{1}},\ldots,{j_{k}}\}-C_{m}$. We set

cf. formula (3.6). For $j\in J$, we define the operator $K_{j}(x):{\mathcal{O}(C_{{\mathcal{C}}(x),a})}\to{\mathcal{O}(C_{{\mathcal{C}}% (x),a})}$ by the formula

cf. formula (3.4).

For a section $I=\sum_{j_{1},\ldots,j_{k}}I^{j_{1},\ldots,j_{k}}w_{j_{1},\ldots,j_{k}}$ of the bundle of algebras (3.14) and $j\in J$, we define $\frac{\partial I}{\partial z_{j}}=\sum\frac{\partial I^{j_{1},\ldots,j_{k}}}{% \partial z_{j}}w_{j_{1},\ldots,j_{k}}$. This formula defines the combinatorial connection on the bundle of algebras (3.14).

Notice that solutions of these differential equations are given by the hypergeometric integrals

where $\gamma(x)\in H_{k}(U({\mathcal{C}}(x)),\mathcal{L}_{\kappa}|_{U({\mathcal{C}}(% x))})$ is a flat section of the Gauss–Manin connection on the homology bundle and $I_{\gamma}^{j_{1},\ldots,j_{k}}(x)=\int_{\gamma(x)}e^{\Phi_{{\mathcal{C}},a}/% \kappa}\omega_{j_{1}}\wedge\cdots\wedge\omega_{j_{k}}$.

Notice the similarities between the differential equations in (3.19) and the standard differential equations associated with Frobenius structures, see [Dubrovin1996], [Manin1999].

Consider ${\mathbb{C}}^{n}$ with coordinates $q_{1},\ldots,q_{n}$ and the dual space $({\mathbb{C}}^{n})^{*}$ with the dual coordinates $p_{1},\ldots,p_{n}$. The space ${\mathbb{C}}^{n}\times{({\mathbb{C}}^{n})^{*}}$ has symplectic form $\omega=\sum_{j=1}^{n}dp_{j}\wedge dq_{j}$. Two functions $M,N$ on ${\mathbb{C}}^{n}\times{({\mathbb{C}}^{n})^{*}}$ are in involution if $\{M,N\}=$ $\sum_{j=1}^{n}\Big(\frac{\partial M}{\partial q_{j}}\frac{\partial N}{\partial p% _{j}}-\frac{\partial M}{\partial p_{j}}\frac{\partial N}{\partial q_{j}}\Big)=0$.

For a $k$-dimensional vector subspace $Y\subset{\mathbb{C}}^{n}$, denote by $Y^{\perp}\subset{({\mathbb{C}}^{n})^{*}}$ the annihilator of $Y$. The $n$-dimensional vector space $Y\times Y^{\perp}$ is a Lagrangian subspace of ${\mathbb{C}}^{n}\times{({\mathbb{C}}^{n})^{*}}$ with defining equations

The set of all functions $\{F_{\alpha},G_{\beta}\}$ is in involution.

Fix a weight $a\in({\mathbb{C}}^{\times})^{n}$. Consider the invertible rational symplectic map $r_{a}:{\mathbb{C}}^{n}\times{({\mathbb{C}}^{n})^{*}}\to{\mathbb{C}}^{n}\times{% ({\mathbb{C}}^{n})^{*}}$,

Denote

Let $I=\{i_{1},\ldots,i_{k}\}\subset J$ be a $k$-element subset and $\bar{I}$ its complement.

We order the functions of the coordinate system $q_{I},p_{\bar{I}}$ according to the increase of the index. For example, if $k=3,n=6$, $I=\{1,3,6\}$, then the order is $q_{1},p_{2},q_{3},p_{4},p_{5},q_{6}$.

Fix a basis $b^{i}=(b^{i}_{1},\ldots,b^{i}_{n})$, $i=1,\ldots,k$, of $Y$. Let $t=(t_{1},\ldots,t_{k})$ be the associated coordinate system on $Y$. Then $q_{j}|_{Y}=\sum_{i=1}^{k}b^{i}_{j}t_{i}$ . For $I=\{i_{1},\ldots,i_{k}\}\subset J$, we denote $d_{I}=d_{i_{1},\ldots,i_{k}}=\det_{i,l=1}^{k}(b^{i}_{i_{l}})$, cf. Sect. 2.9.

Let $I=\{i_{1},\ldots,i_{k}\}\subset J$ satisfy the hypotheses of Lemma 4.3, and $q_{I},p_{\bar{I}}$ the corresponding ordered coordinate system on $L_{Y,a}$. The functions $q_{1},\ldots,q_{n}$ form an ordered coordinate system on ${\mathbb{C}}^{n}$. Let $\pi_{L_{Y,a}}:L_{Y,a}\to{\mathbb{C}}^{n}$ be the restriction to $L_{Y,a}$ of the natural projection ${\mathbb{C}}^{n}\times{({\mathbb{C}}^{n})^{*}}\to{\mathbb{C}}^{n}$. Let $\operatorname{Jac}_{I}(q_{I},p_{\bar{I}})$ be the Jacobian of $\pi_{L_{Y,a}}$ with respect to the chosen coordinate systems.

For $x\in{\mathbb{C}}^{n}$, denote $C_{Y,a}(x)=\pi^{-1}_{L_{Y,a}}(x)$, the fiber of the projection $\pi_{L_{Y,a}}$. The fiber is defined in $({\mathbb{C}}^{\times})^{n}$ with coordinates $p_{1},\ldots,p_{n}$ by the equations

cf. (4.1). Let $I_{L_{Y}(x),a}$ be the ideal in $\mathcal{O}(({\mathbb{C}}^{\times})^{n})$ generated by the left-hand sides of equations (4.2). Set

Return to the objects and notations of Sect. 3 and consider ${\mathbb{C}}^{k}$ with coordinates $t_{1},\ldots,t_{k}$, ${\mathbb{C}}^{n}$ with coordinates $z_{1},\ldots,z_{n}$, the projection $\pi:{\mathbb{C}}^{n}\times{\mathbb{C}}^{k}\to{\mathbb{C}}^{n}$ and $n$ nonzero linear functions on ${\mathbb{C}}^{k}$, $g_{j}=b_{j}^{1}t_{1}+\cdots+b_{j}^{k}t_{k},$ $j\in J$. As in Sect. 3 we assume that $g_{j},j\in J$, span the dual space $({\mathbb{C}}^{k})^{*}$. We consider the linear functions $f_{j}=g_{j}+z_{j}=b_{j}^{1}t_{1}+\cdots+b_{j}^{k}t_{k}+z_{j}$ on ${\mathbb{C}}^{n}\times{\mathbb{C}}^{k}$ and the arrangement of hyperplanes ${\mathcal{C}}=\{H_{j}\}_{j\in J}$ in ${\mathbb{C}}^{n}\times{\mathbb{C}}^{k}$, where $H_{j}$ is defined by the equation $f_{j}=0$. We assume that a weight $a\in({\mathbb{C}}^{n})^{\times}$ is given and consider the critical set $C_{{\mathcal{C}},a}$ defined by (3.12). In the rest of the paper we denote by

the $k$-dimensional subspace of ${\mathbb{C}}^{n}$ spanned by the vectors $b^{i}=(b^{i}_{1},\ldots,b^{i}_{n})$, $i=1,\ldots,k$.

Lemmas 4.6, 4.7, 4.8 and 4.9 prove Theorem 4.5.

Recall the Hessian of the master function $\Phi_{{\mathcal{C}},a}$, see Sect. 2.10. Under the diffeomorphism $C_{{\mathcal{C}},a}\to L_{Y,a}$ of Theorem 4.5, we may consider the Hessian as a function on $L_{Y,a}$. Then

Notice that the ratio of ${\operatorname{Hess}}_{{\mathcal{C}},a}$ and $d_{I}^{2}\operatorname{Jac}_{I}$ is never zero.

The map $\Psi_{{\mathcal{C}},a}$ of Theorem 4.5 establishes an isomorphism

for all $j\in J$, and for any $x\in{\mathbb{C}}^{n}$, the isomorphism

In particular, if the weight $a$ is such that $({\mathcal{C}}(x),a)$ is unbalanced, then the number of solutions of system (4.2), counted with multiplicities, equals $|U({\mathcal{C}}(x))|$ and can be calculated in terms of the matroid associated with the arrangement ${\mathcal{C}}(x)$, see Lemma 3.6.

The isomorphism $\Psi_{{\mathcal{C}}(x),a}^{*}$ allows us to compare objects associated with ${\mathcal{O}(C_{{\mathcal{C}}(x),a})}$ and objects associated with ${\mathcal{O}(L_{Y,a}(x))}$. For example, let $q_{I},p_{\bar{I}}$ be an ordered coordinate system on $L_{Y,a}$ like in Lemma 4.2. Assume that $x\in{\mathbb{C}}^{n}$ is such that $\dim{\mathcal{O}(L_{Y,a}(x))}$ is finite. Consider the Grothendieck residue $\mathcal{R}:{\mathcal{O}(L_{Y,a}(x))}\to{\mathbb{C}}$,

where the differentials are ordered as in the ordered coordinate system $q_{I},p_{\bar{I}}$, cf. (2.9). Define the nondegenerate bilinear form $(\,,\,)_{L_{Y,a}(x)}$ on ${\mathcal{O}(L_{Y,a}(x))}$ by the formula

If $({\mathcal{C}}(x),a)$ is unbalanced for some $x\in{{\mathbb{C}}^{n}-\Delta}$, then we have the vector bundle of algebras

The fiber isomorphism $\Psi_{{\mathcal{C}}(x),a}^{*}$ identifies this bundle of algebras with the bundle of algebras in (3.14). The combinatorial and Gauss–Manin connections on the bundle of algebras in (3.14) induce the corresponding connections on the bundle in (4.13).

For $x\in{{\mathbb{C}}^{n}-\Delta}$, we define the marked elements $p_{i_{1},\ldots,i_{k}}$ in ${\mathcal{O}(L_{Y,a}(x))}$ as the images under $\Psi_{{\mathcal{C}}(x),a}^{*}$ of the marked elements $w_{i_{1},\ldots,i_{k}}$ in ${\mathcal{O}(C_{{\mathcal{C}}(x),a})}$. By formula (2.28), we have

with the skew-symmetry property

and the marked relations

labeled by independent subsets $\{i_{2},\ldots,i_{k}\}$. By Corollary (2.18), the marked elements $p_{i_{1},\ldots,i_{k}}$ span ${\mathcal{O}(L_{Y,a}(x))}$ as a vector space and relations (4.15), (4.16) are the only linear relations between them. This fact defines an integral structure on ${\mathcal{O}(L_{Y,a}(x))}$.

For a section $I=\sum_{j_{1},\ldots,j_{k}}I^{j_{1},\ldots,j_{k}}p_{j_{1},\ldots,j_{k}}$ of the bundle of algebras (4.13) and $j\in J$, we define $\frac{\partial I}{\partial q_{j}}=\sum\frac{\partial I^{j_{1},\ldots,j_{k}}}{% \partial q_{j}}p_{j_{1},\ldots,j_{k}}$. This formula defines the combinatorial connection on (4.13).

Assume that for any $j\in J$, the subspace $Y^{\perp}$ does not lie in the hyperplane $p_{j}=0$. Assume that all coordinates of the weight $a\in({\mathbb{C}}^{\times})^{n}$ are positive. Assume that all entries of the matrix $(b^{i}_{j})_{\substack{i=1,\ldots,k\\ j=1,\ldots,n}}$, defining $Y\subset{\mathbb{C}}^{n}$ in (4.4), are real. Assume that the critical set $C_{{\mathcal{C}},a}$ is nonempty.