Combinatorial construction of logarithmic differential forms

Abstract

H. Terao has shown that the structure of the module of (rational) differential forms with at most logarithmic poles at an arrangement of hyperplanes (as defined by K. Saito) is very strongly controlled by the combinatorial structure of the arrangement. In this paper we demonstrate how the existence of rational logarithmic forms with poles of high order depends on the existence of highly degenerate (“special position”) subarrangements. The associated combinatorial structures are studied. First a strong version of K. Saito's “Preparation Lemma” for logarithmic forms leads to a new, simple proof of Terao's celebrated “Addition-Deletion Theorem” for hyperplane arrangements with free module of logarithmic 1-forms (“free arrangements”). Then structural characterizations are developed in two extreme cases for the generation of the module of logarithmic differential 1-forms: this module has a triangular basis iff the arrangement is supersolvable (strictly linearly fibered), and it is generated by forms of degree —1 iff the arrangement is generic in codimension 3. Both conditions on the geometry of the arrangement are combinatorial in a very strong sense (determined by restricted data on the lattice of intersections of the hyperplanes.) However, examples show that the cardinality and degree sequence of a minimal set of generators for the module of lógarithmic 1-forms are not in general determined by this intersection lattice.