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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107001.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107002.png" />-dimensional [[Affine space|affine space]] over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107003.png" />. An arrangement of hyperplanes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107004.png" />, is a finite collection of codimension-one affine subspaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107005.png" />, [[#References|[a5]]].
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Let  $  V $
 +
be an  $  {\mathcal l} $-
 +
dimensional [[Affine space|affine space]] over the field $  \mathbf K $.  
 +
An arrangement of hyperplanes, $  {\mathcal A} $,  
 +
is a finite collection of codimension-one affine subspaces in $  V $,  
 +
[[#References|[a5]]].
  
 
===Examples of arrangements of hyperplanes.===
 
===Examples of arrangements of hyperplanes.===
 
  
 
1) A subset of the coordinate hyperplanes is called a Boolean arrangement.
 
1) A subset of the coordinate hyperplanes is called a Boolean arrangement.
Line 8: Line 24:
 
2) An arrangement is in general position if at each point it is locally Boolean.
 
2) An arrangement is in general position if at each point it is locally Boolean.
  
3) The braid arrangement consists of the hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107006.png" />. It is the set of reflecting hyperplanes of the [[Symmetric group|symmetric group]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107007.png" /> letters.
+
3) The braid arrangement consists of the hyperplanes $  \{ {x _ {i} = x _ {j} } : {1 \leq  i < j \leq  {\mathcal l} } \} $.  
 +
It is the set of reflecting hyperplanes of the [[Symmetric group|symmetric group]] on $  {\mathcal l} $
 +
letters.
  
 
4) The reflecting hyperplanes of a finite reflection group.
 
4) The reflecting hyperplanes of a finite reflection group.
  
 
==Combinatorics.==
 
==Combinatorics.==
An edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107008.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a1107009.png" /> is a non-empty intersection of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070011.png" /> be the set of edges partially ordered by reverse inclusion. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070012.png" /> is a geometric semi-lattice with minimal element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070013.png" />, rank given by codimension, and maximal elements of the same rank, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070014.png" />. The [[Möbius function|Möbius function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070015.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070016.png" /> and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070017.png" />,
+
An edge $  X $
 +
of $  {\mathcal A} $
 +
is a non-empty intersection of elements of $  {\mathcal A} $.  
 +
Let $  L ( {\mathcal A} ) $
 +
be the set of edges partially ordered by reverse inclusion. Then $  L $
 +
is a geometric semi-lattice with minimal element $  V $,  
 +
rank given by codimension, and maximal elements of the same rank, $  r ( {\mathcal A} ) $.  
 +
The [[Möbius function|Möbius function]] on $  L $
 +
is defined by $  \mu ( V ) = 1 $
 +
and, for $  X > V $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070018.png" /></td> </tr></table>
+
$$
 +
\sum _ {V \leq  Y \leq  X } \mu ( Y ) = 0.
 +
$$
  
The characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070019.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070020.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070021.png" />. For a generic arrangement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070022.png" /> hyperplanes,
+
The characteristic polynomial of $  {\mathcal A} $
 +
is $  \chi ( {\mathcal A},t ) = \sum _ {X \in L }  \mu ( X ) t ^ { { \mathop{\rm codim} } X } $.  
 +
Let $  \beta ( {\mathcal A} ) = ( - 1 ) ^ {r ( {\mathcal A} ) } \chi ( {\mathcal A},1 ) $.  
 +
For a generic arrangement of $  n $
 +
hyperplanes,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070023.png" /></td> </tr></table>
+
$$
 +
\chi ( {\mathcal A},t ) = \sum _ {k = 0 } ^ { {r }  ( {\mathcal A} ) } ( - 1 )  ^ {k} \left ( \begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right ) t ^ { {\mathcal l} - k } .
 +
$$
  
 
For the braid arrangement,
 
For the braid arrangement,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070024.png" /></td> </tr></table>
+
$$
 +
\chi ( {\mathcal A},t ) = t ( t - 1 ) \dots ( t - ( {\mathcal l} - 1 ) ) .
 +
$$
  
Similar factorizations hold for all reflection arrangements involving the (co)exponents of the reflection group. Given a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070025.png" />-tuple of hyperplanes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070026.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070027.png" />; note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070028.png" /> may be empty. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070029.png" /> is dependent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070031.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070032.png" /> be the exterior algebra on symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070033.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070034.png" />, where the product is juxtaposition. Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070035.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070037.png" /> and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070038.png" />,
+
Similar factorizations hold for all reflection arrangements involving the (co)exponents of the reflection group. Given a $  p $-
 +
tuple of hyperplanes, $  S = ( H _ {1} \dots H _ {p} ) $,  
 +
let $  \cap S = H _ {1} \cap \dots \cap H _ {p} $;  
 +
note that $  \cap S $
 +
may be empty. One says that $  S $
 +
is dependent if $  \cap S \neq \emptyset $
 +
and $  { \mathop{\rm codim} } ( \cap S ) < | S | $.  
 +
Let $  E ( {\mathcal A} ) $
 +
be the exterior algebra on symbols $  ( H ) $
 +
for $  H \in {\mathcal A} $,  
 +
where the product is juxtaposition. Define $  \partial  : E \rightarrow E $
 +
by $  \partial  1 = 0 $,  
 +
$  \partial  ( H ) = 1 $
 +
and, for $  p \geq  2 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070039.png" /></td> </tr></table>
+
$$
 +
\partial  ( H _ {1} \dots H _ {p} ) = \sum _ {k = 1 } ^ { p }  ( - 1 ) ^ {k - 1 } ( H _ {1} \dots { {H _ {k} } hat } \dots H _ {p} ) .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070040.png" /> be the [[Ideal|ideal]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070041.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070042.png" />. The Orlik–Solomon algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070043.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070044.png" />. For connections with matroid theory, see [[#References|[a3]]].
+
Let $  I ( {\mathcal A} ) $
 +
be the [[Ideal|ideal]] of $  E ( {\mathcal A} ) $
 +
generated by $  \{ S : {\cap S = \emptyset } \} \cup \{ {\partial  S } : {S  \textrm{ dependent  } } \} $.  
 +
The Orlik–Solomon algebra of $  {\mathcal A} $
 +
is $  A ( {\mathcal A} ) = E ( {\mathcal A} ) /I ( {\mathcal A} ) $.  
 +
For connections with matroid theory, see [[#References|[a3]]].
  
 
==Divisor.==
 
==Divisor.==
The divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070045.png" /> is the union of the hyperplanes, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070046.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070047.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070049.png" /> has the homotopy type of a wedge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070050.png" /> spheres of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070051.png" />, [[#References|[a4]]]. The singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070052.png" /> are not isolated. The divisor of a general-position arrangement has normal crossings, but this is not true for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070053.png" />. Blowing-up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070054.png" /> along all edges where it is not locally a product of arrangements yields a normal crossing divisor. See also [[Divisor|Divisor]].
+
The divisor of $  {\mathcal A} $
 +
is the union of the hyperplanes, denoted by $  N ( {\mathcal A} ) $.  
 +
If $  \mathbf K = \mathbf R $
 +
or $  \mathbf K = \mathbf C $,  
 +
then $  N $
 +
has the homotopy type of a wedge of $  \beta ( {\mathcal A} ) $
 +
spheres of dimension $  r ( {\mathcal A} ) - 1 $,  
 +
[[#References|[a4]]]. The singularities of $  N $
 +
are not isolated. The divisor of a general-position arrangement has normal crossings, but this is not true for arbitrary $  {\mathcal A} $.  
 +
Blowing-up $  N $
 +
along all edges where it is not locally a product of arrangements yields a normal crossing divisor. See also [[Divisor|Divisor]].
  
 
==Complement.==
 
==Complement.==
The complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070055.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070056.png" />.
+
The complement of $  {\mathcal A} $
 +
is $  M ( {\mathcal A} ) = V \setminus  N ( {\mathcal A} ) $.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070057.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070058.png" /> is a finite set of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070059.png" />.
+
1) If $  \mathbf K = \mathbf F _ {q} $,  
 +
then $  M $
 +
is a finite set of cardinality $  | M | = \chi ( {\mathcal A},q ) $.
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070060.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070061.png" /> is a disjoint union of open convex sets (chambers) of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070062.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070064.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070065.png" /> chambers with compact closure, [[#References|[a7]]].
+
2) If $  \mathbf K = \mathbf R $,  
 +
then $  M $
 +
is a disjoint union of open convex sets (chambers) of cardinality $  ( - 1 )  ^  {\mathcal l}  \chi ( {\mathcal A}, - 1 ) $.  
 +
If $  r ( {\mathcal A} ) = {\mathcal l} $,  
 +
$  M $
 +
contains $  \beta ( {\mathcal A} ) $
 +
chambers with compact closure, [[#References|[a7]]].
  
3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070067.png" /> is an open complex (Stein) manifold of the homotopy type of a finite [[CW-complex|CW-complex]] (cf. also [[Stein manifold|Stein manifold]]). Its [[Cohomology|cohomology]] is torsion-free and its Poincaré polynomial (cf. [[Künneth formula|Künneth formula]]) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070068.png" />. The product structure is determined by the isomorphism of graded algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070069.png" />. The [[Fundamental group|fundamental group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070070.png" /> has an effective presentation, but the higher homotopy groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070071.png" /> are not known in general.
+
3) If $  \mathbf K = \mathbf C $,  
 +
then $  M $
 +
is an open complex (Stein) manifold of the homotopy type of a finite [[CW-complex|CW-complex]] (cf. also [[Stein manifold|Stein manifold]]). Its [[Cohomology|cohomology]] is torsion-free and its Poincaré polynomial (cf. [[Künneth formula|Künneth formula]]) is $  { \mathop{\rm Poin} } ( M,t ) = ( - t )  ^  {\mathcal l}  \chi ( {\mathcal A}, - t ^ {- 1 } ) $.  
 +
The product structure is determined by the isomorphism of graded algebras $  H  ^ {*} ( M ) \simeq A ( {\mathcal A} ) $.  
 +
The [[Fundamental group|fundamental group]] of $  M $
 +
has an effective presentation, but the higher homotopy groups of $  M $
 +
are not known in general.
  
The complement of a Boolean arrangement is a complex torus. In a general-position arrangement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070072.png" /> hyperplanes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070073.png" /> has non-trivial higher homotopy groups. For the braid arrangement, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070074.png" /> is called the pure braid space and its higher homotopy groups are trivial. The symmetric group acts freely on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070075.png" /> with as orbit space the braid space whose fundamental group is the braid group. The quotient of the divisor by the symmetric group is called the discriminant, which has connections with singularity theory.
+
The complement of a Boolean arrangement is a complex torus. In a general-position arrangement of $  n > {\mathcal l} $
 +
hyperplanes, $  M $
 +
has non-trivial higher homotopy groups. For the braid arrangement, $  M $
 +
is called the pure braid space and its higher homotopy groups are trivial. The symmetric group acts freely on $  M $
 +
with as orbit space the braid space whose fundamental group is the braid group. The quotient of the divisor by the symmetric group is called the discriminant, which has connections with singularity theory.
  
 
==Ball quotients.==
 
==Ball quotients.==
Line 49: Line 139:
  
 
==Logarithmic forms.==
 
==Logarithmic forms.==
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070076.png" />, choose a linear polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070077.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070078.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070079.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070080.png" /> denote all global regular (i.e., polynomial) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070081.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070082.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070083.png" /> denote the space of all global rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070084.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070085.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070086.png" /> of logarithmic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070087.png" />-forms with poles along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070088.png" /> is
+
For $  H \in {\mathcal A} $,  
 +
choose a linear polynomial $  \alpha _ {H} $
 +
with $  H = { \mathop{\rm ker} } \alpha _ {H} $
 +
and let $  Q ( {\mathcal A} ) = \prod _ {H \in {\mathcal A} }  \alpha _ {H} $.  
 +
Let $  \Omega  ^ {p} [ V ] $
 +
denote all global regular (i.e., polynomial) $  p $-
 +
forms on $  V $.  
 +
Let $  \Omega  ^ {p} ( V ) $
 +
denote the space of all global rational $  p $-
 +
forms on $  V $.  
 +
The space $  \Omega  ^ {p} ( {\mathcal A} ) $
 +
of logarithmic $  p $-
 +
forms with poles along $  {\mathcal A} $
 +
is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070089.png" /></td> </tr></table>
+
$$
 +
\Omega  ^ {p} ( {\mathcal A} ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070090.png" /></td> </tr></table>
+
$$
 +
=  
 +
\left \{ {\omega \in \Omega  ^ {p} ( V ) } : {Q \omega \in \Omega  ^ {p} [ V ] ,  Q ( d \omega ) \in \Omega ^ {p + 1 } [ V ] } \right \} .
 +
$$
  
The arrangement is free if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070091.png" /> is a free module over the polynomial ring. A free arrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070092.png" /> has integer exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070093.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070094.png" />. Reflection arrangements are free. This explains the factorization of their characteristic polynomials.
+
The arrangement is free if $  \Omega  ^ {1} ( {\mathcal A} ) $
 +
is a free module over the polynomial ring. A free arrangement $  {\mathcal A} $
 +
has integer exponents $  \{ b _ {1} \dots b _  {\mathcal l}  \} $,  
 +
so that $  \chi ( {\mathcal A},t ) = \prod _ {k = 1 }  ^  {\mathcal l}  ( t - b _ {k} ) $.  
 +
Reflection arrangements are free. This explains the factorization of their characteristic polynomials.
  
 
==Hypergeometric integrals.==
 
==Hypergeometric integrals.==
Certain rank-one local system cohomology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070095.png" /> may be identified with spaces of hypergeometric integrals, [[#References|[a1]]]. If the local system is suitably generic, these cohomology groups may be computed using the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070096.png" />. Only the top cohomology group is non-zero, and it has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110700/a11070097.png" />. See [[#References|[a6]]] for connections with the representation theory of Lie algebras and [[Quantum groups|quantum groups]], and with the Knizhnik–Zamolodchikov differential equations of physics.
+
Certain rank-one local system cohomology groups of $  M $
 +
may be identified with spaces of hypergeometric integrals, [[#References|[a1]]]. If the local system is suitably generic, these cohomology groups may be computed using the algebra $  A ( {\mathcal A} ) $.  
 +
Only the top cohomology group is non-zero, and it has dimension $  \beta ( {\mathcal A} ) $.  
 +
See [[#References|[a6]]] for connections with the representation theory of Lie algebras and [[Quantum groups|quantum groups]], and with the Knizhnik–Zamolodchikov differential equations of physics.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Aomoto, M. Kita, "Hypergeometric functions" , Springer (1994) (Translated from Japanese) {{MR|2799182}} {{MR|2482635}} {{MR|1866164}} {{MR|1805969}} {{MR|1792063}} {{MR|1768923}} {{MR|1803881}} {{MR|1749398}} {{MR|1614401}} {{MR|1401610}} {{MR|1256465}} {{MR|0988314}} {{ZBL|1229.33001}} {{ZBL|1169.33307}} {{ZBL|1174.33301}} {{ZBL|1189.33028}} {{ZBL|0972.33009}} {{ZBL|0927.33014}} {{ZBL|0943.32010}} {{ZBL|0787.33001}} {{ZBL|0859.33001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Barthel, F. Hirzebruch, T. Höfer, "Geradenkonfigurationen und Algebraische Flächen" , Vieweg (1987) {{MR|0912097}} {{ZBL|0645.14016}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G.M. Ziegler, "Oriented matroids" , Cambridge Univ. Press (1993) {{MR|1226888}} {{ZBL|0773.52001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Goresky, R. MacPherson, "Stratified Morse theory" , Springer (1988) {{MR|0932724}} {{ZBL|0639.14012}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Orlik, H. Terao, "Arrangements of hyperplanes" , Springer (1992) {{MR|1217488}} {{ZBL|0757.55001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Varchenko, "Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups" , World Sci. (1995) {{MR|1384760}} {{ZBL|0951.33001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> T. Zaslavsky, "Facing up to arrangements: face-count formulas for partitions of space by hyperplanes" , ''Memoirs'' , '''154''' , Amer. Math. Soc. (1975) {{MR|0357135}} {{ZBL|0296.50010}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Aomoto, M. Kita, "Hypergeometric functions" , Springer (1994) (Translated from Japanese) {{MR|2799182}} {{MR|2482635}} {{MR|1866164}} {{MR|1805969}} {{MR|1792063}} {{MR|1768923}} {{MR|1803881}} {{MR|1749398}} {{MR|1614401}} {{MR|1401610}} {{MR|1256465}} {{MR|0988314}} {{ZBL|1229.33001}} {{ZBL|1169.33307}} {{ZBL|1174.33301}} {{ZBL|1189.33028}} {{ZBL|0972.33009}} {{ZBL|0927.33014}} {{ZBL|0943.32010}} {{ZBL|0787.33001}} {{ZBL|0859.33001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Barthel, F. Hirzebruch, T. Höfer, "Geradenkonfigurationen und Algebraische Flächen" , Vieweg (1987) {{MR|0912097}} {{ZBL|0645.14016}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G.M. Ziegler, "Oriented matroids" , Cambridge Univ. Press (1993) {{MR|1226888}} {{ZBL|0773.52001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Goresky, R. MacPherson, "Stratified Morse theory" , Springer (1988) {{MR|0932724}} {{ZBL|0639.14012}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Orlik, H. Terao, "Arrangements of hyperplanes" , Springer (1992) {{MR|1217488}} {{ZBL|0757.55001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Varchenko, "Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups" , World Sci. (1995) {{MR|1384760}} {{ZBL|0951.33001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> T. Zaslavsky, "Facing up to arrangements: face-count formulas for partitions of space by hyperplanes" , ''Memoirs'' , '''154''' , Amer. Math. Soc. (1975) {{MR|0357135}} {{ZBL|0296.50010}} </TD></TR></table>

Latest revision as of 18:48, 5 April 2020


Let $ V $ be an $ {\mathcal l} $- dimensional affine space over the field $ \mathbf K $. An arrangement of hyperplanes, $ {\mathcal A} $, is a finite collection of codimension-one affine subspaces in $ V $, [a5].

Examples of arrangements of hyperplanes.

1) A subset of the coordinate hyperplanes is called a Boolean arrangement.

2) An arrangement is in general position if at each point it is locally Boolean.

3) The braid arrangement consists of the hyperplanes $ \{ {x _ {i} = x _ {j} } : {1 \leq i < j \leq {\mathcal l} } \} $. It is the set of reflecting hyperplanes of the symmetric group on $ {\mathcal l} $ letters.

4) The reflecting hyperplanes of a finite reflection group.

Combinatorics.

An edge $ X $ of $ {\mathcal A} $ is a non-empty intersection of elements of $ {\mathcal A} $. Let $ L ( {\mathcal A} ) $ be the set of edges partially ordered by reverse inclusion. Then $ L $ is a geometric semi-lattice with minimal element $ V $, rank given by codimension, and maximal elements of the same rank, $ r ( {\mathcal A} ) $. The Möbius function on $ L $ is defined by $ \mu ( V ) = 1 $ and, for $ X > V $,

$$ \sum _ {V \leq Y \leq X } \mu ( Y ) = 0. $$

The characteristic polynomial of $ {\mathcal A} $ is $ \chi ( {\mathcal A},t ) = \sum _ {X \in L } \mu ( X ) t ^ { { \mathop{\rm codim} } X } $. Let $ \beta ( {\mathcal A} ) = ( - 1 ) ^ {r ( {\mathcal A} ) } \chi ( {\mathcal A},1 ) $. For a generic arrangement of $ n $ hyperplanes,

$$ \chi ( {\mathcal A},t ) = \sum _ {k = 0 } ^ { {r } ( {\mathcal A} ) } ( - 1 ) ^ {k} \left ( \begin{array}{c} n \\ k \end{array} \right ) t ^ { {\mathcal l} - k } . $$

For the braid arrangement,

$$ \chi ( {\mathcal A},t ) = t ( t - 1 ) \dots ( t - ( {\mathcal l} - 1 ) ) . $$

Similar factorizations hold for all reflection arrangements involving the (co)exponents of the reflection group. Given a $ p $- tuple of hyperplanes, $ S = ( H _ {1} \dots H _ {p} ) $, let $ \cap S = H _ {1} \cap \dots \cap H _ {p} $; note that $ \cap S $ may be empty. One says that $ S $ is dependent if $ \cap S \neq \emptyset $ and $ { \mathop{\rm codim} } ( \cap S ) < | S | $. Let $ E ( {\mathcal A} ) $ be the exterior algebra on symbols $ ( H ) $ for $ H \in {\mathcal A} $, where the product is juxtaposition. Define $ \partial : E \rightarrow E $ by $ \partial 1 = 0 $, $ \partial ( H ) = 1 $ and, for $ p \geq 2 $,

$$ \partial ( H _ {1} \dots H _ {p} ) = \sum _ {k = 1 } ^ { p } ( - 1 ) ^ {k - 1 } ( H _ {1} \dots { {H _ {k} } hat } \dots H _ {p} ) . $$

Let $ I ( {\mathcal A} ) $ be the ideal of $ E ( {\mathcal A} ) $ generated by $ \{ S : {\cap S = \emptyset } \} \cup \{ {\partial S } : {S \textrm{ dependent } } \} $. The Orlik–Solomon algebra of $ {\mathcal A} $ is $ A ( {\mathcal A} ) = E ( {\mathcal A} ) /I ( {\mathcal A} ) $. For connections with matroid theory, see [a3].

Divisor.

The divisor of $ {\mathcal A} $ is the union of the hyperplanes, denoted by $ N ( {\mathcal A} ) $. If $ \mathbf K = \mathbf R $ or $ \mathbf K = \mathbf C $, then $ N $ has the homotopy type of a wedge of $ \beta ( {\mathcal A} ) $ spheres of dimension $ r ( {\mathcal A} ) - 1 $, [a4]. The singularities of $ N $ are not isolated. The divisor of a general-position arrangement has normal crossings, but this is not true for arbitrary $ {\mathcal A} $. Blowing-up $ N $ along all edges where it is not locally a product of arrangements yields a normal crossing divisor. See also Divisor.

Complement.

The complement of $ {\mathcal A} $ is $ M ( {\mathcal A} ) = V \setminus N ( {\mathcal A} ) $.

1) If $ \mathbf K = \mathbf F _ {q} $, then $ M $ is a finite set of cardinality $ | M | = \chi ( {\mathcal A},q ) $.

2) If $ \mathbf K = \mathbf R $, then $ M $ is a disjoint union of open convex sets (chambers) of cardinality $ ( - 1 ) ^ {\mathcal l} \chi ( {\mathcal A}, - 1 ) $. If $ r ( {\mathcal A} ) = {\mathcal l} $, $ M $ contains $ \beta ( {\mathcal A} ) $ chambers with compact closure, [a7].

3) If $ \mathbf K = \mathbf C $, then $ M $ is an open complex (Stein) manifold of the homotopy type of a finite CW-complex (cf. also Stein manifold). Its cohomology is torsion-free and its Poincaré polynomial (cf. Künneth formula) is $ { \mathop{\rm Poin} } ( M,t ) = ( - t ) ^ {\mathcal l} \chi ( {\mathcal A}, - t ^ {- 1 } ) $. The product structure is determined by the isomorphism of graded algebras $ H ^ {*} ( M ) \simeq A ( {\mathcal A} ) $. The fundamental group of $ M $ has an effective presentation, but the higher homotopy groups of $ M $ are not known in general.

The complement of a Boolean arrangement is a complex torus. In a general-position arrangement of $ n > {\mathcal l} $ hyperplanes, $ M $ has non-trivial higher homotopy groups. For the braid arrangement, $ M $ is called the pure braid space and its higher homotopy groups are trivial. The symmetric group acts freely on $ M $ with as orbit space the braid space whose fundamental group is the braid group. The quotient of the divisor by the symmetric group is called the discriminant, which has connections with singularity theory.

Ball quotients.

Examples of algebraic surfaces whose universal cover is the complex ball were constructed as "Kummer" covers of the projective plane branched along certain arrangements of projective lines, [a2].

Logarithmic forms.

For $ H \in {\mathcal A} $, choose a linear polynomial $ \alpha _ {H} $ with $ H = { \mathop{\rm ker} } \alpha _ {H} $ and let $ Q ( {\mathcal A} ) = \prod _ {H \in {\mathcal A} } \alpha _ {H} $. Let $ \Omega ^ {p} [ V ] $ denote all global regular (i.e., polynomial) $ p $- forms on $ V $. Let $ \Omega ^ {p} ( V ) $ denote the space of all global rational $ p $- forms on $ V $. The space $ \Omega ^ {p} ( {\mathcal A} ) $ of logarithmic $ p $- forms with poles along $ {\mathcal A} $ is

$$ \Omega ^ {p} ( {\mathcal A} ) = $$

$$ = \left \{ {\omega \in \Omega ^ {p} ( V ) } : {Q \omega \in \Omega ^ {p} [ V ] , Q ( d \omega ) \in \Omega ^ {p + 1 } [ V ] } \right \} . $$

The arrangement is free if $ \Omega ^ {1} ( {\mathcal A} ) $ is a free module over the polynomial ring. A free arrangement $ {\mathcal A} $ has integer exponents $ \{ b _ {1} \dots b _ {\mathcal l} \} $, so that $ \chi ( {\mathcal A},t ) = \prod _ {k = 1 } ^ {\mathcal l} ( t - b _ {k} ) $. Reflection arrangements are free. This explains the factorization of their characteristic polynomials.

Hypergeometric integrals.

Certain rank-one local system cohomology groups of $ M $ may be identified with spaces of hypergeometric integrals, [a1]. If the local system is suitably generic, these cohomology groups may be computed using the algebra $ A ( {\mathcal A} ) $. Only the top cohomology group is non-zero, and it has dimension $ \beta ( {\mathcal A} ) $. See [a6] for connections with the representation theory of Lie algebras and quantum groups, and with the Knizhnik–Zamolodchikov differential equations of physics.

References

[a1] K. Aomoto, M. Kita, "Hypergeometric functions" , Springer (1994) (Translated from Japanese) MR2799182 MR2482635 MR1866164 MR1805969 MR1792063 MR1768923 MR1803881 MR1749398 MR1614401 MR1401610 MR1256465 MR0988314 Zbl 1229.33001 Zbl 1169.33307 Zbl 1174.33301 Zbl 1189.33028 Zbl 0972.33009 Zbl 0927.33014 Zbl 0943.32010 Zbl 0787.33001 Zbl 0859.33001
[a2] G. Barthel, F. Hirzebruch, T. Höfer, "Geradenkonfigurationen und Algebraische Flächen" , Vieweg (1987) MR0912097 Zbl 0645.14016
[a3] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G.M. Ziegler, "Oriented matroids" , Cambridge Univ. Press (1993) MR1226888 Zbl 0773.52001
[a4] M. Goresky, R. MacPherson, "Stratified Morse theory" , Springer (1988) MR0932724 Zbl 0639.14012
[a5] P. Orlik, H. Terao, "Arrangements of hyperplanes" , Springer (1992) MR1217488 Zbl 0757.55001
[a6] A. Varchenko, "Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups" , World Sci. (1995) MR1384760 Zbl 0951.33001
[a7] T. Zaslavsky, "Facing up to arrangements: face-count formulas for partitions of space by hyperplanes" , Memoirs , 154 , Amer. Math. Soc. (1975) MR0357135 Zbl 0296.50010
How to Cite This Entry:
Arrangement of hyperplanes. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arrangement_of_hyperplanes&oldid=24369
This article was adapted from an original article by P. Orlik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article