MIT COMBINATORICS SEMINAR
Characteristic varieties of real and complex arrangements
Alex Suciu (Northeastern University)
Room 2-338
4:15 p.m., Wednesday, March 18, 1998
The k-th Fitting ideal of the Alexander invariant of
an arrangement A of n complex hyperplanes defines a
characteristic subvariety, V_k(A), of the complex algebraic
torus (C^*)^n. The characteristic varieties of an arrangement
provide subtle and effectively computable homotopy-type invariants
of its complement. In joint work with Daniel Cohen, we show that
the tangent cone at the identity of the top characteristic variety
V_1(A) coincides with R_1(A), the first-cohomology support locus
of the Orlik-Solomon algebra. We conclude that the variety V_1(A)
is combinatorially determined, and that Falk's variety R_1(A) is the
union of a subspace arrangement in C^n. We illustrate these techniques
by computing the top characteristic varieties of braid arrangements
and monomial arrangements.
If A is a real 2-arrangement (in the sense of Goresky and McPherson),
the characteristic varieties are no longer subtori through the origin.
The nature of these varieties vividly illustrates the difference
between real and complex arrangements. In joint work with Daniel Matei,
we study the homotopy types of complements of arrangements of n
transverse planes in R^4, obtaining a complete classification for n<=6,
and lower bounds for the number of homotopy types in general. Furthermore,
we show that the homotopy type of the complement of a 2-arrangement in R^4
is not determined by its cohomology ring, thereby answering a question
of Ziegler.
The papers on which the talk will be based can be found at
http://www.math.neu.edu/~suciu/publications.html.