Arrangement of hyperplanes

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