Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The Poincaré polynomial of an mp arrangement


Author: Chris Macmeikan
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1575-1580
MSC (2000): Primary 14F25; Secondary 14R20
Published electronically: January 20, 2004
MathSciNet review: 2051116
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Abstract: Let $\mathcal{A}=\{A_i\}_{i\in I}$ be an mp arrangement in a complex algebraic variety $X$ with corresponding complement $Q(\mathcal{A})=X\backslash\bigcup_{i\in I}A_{i}$ and intersection poset $L(\mathcal{A})$. Examples of such arrangements are hyperplane arrangements and toral arrangements, i.e., collections of codimension 1 subtori, in an algebraic torus. Suppose a finite group $\Gamma$ acts on $X$ as a group of automorphisms and stabilizes the arrangement $\{A_i\}_{i\in I}$ setwise. We give a formula for the graded character of $\Gamma$ on the cohomology of $Q(\mathcal{A})$ in terms of the graded character of $\Gamma$ on the cohomology of certain subvarieties in $L(\mathcal{A})$.


References [Enhancements On Off] (What's this?)

  • 1. Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 0498551
  • 2. Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 0498552
  • 3. A. Dimca and G. I. Lehrer, Purity and equivariant weight polynomials, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 161–181. MR 1635679
  • 4. Alan H. Durfee, Algebraic varieties which are a disjoint union of subvarieties, Geometry and topology (Athens, Ga., 1985) Lecture Notes in Pure and Appl. Math., vol. 105, Dekker, New York, 1987, pp. 99–102. MR 873286
  • 5. Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488
  • 6. Gian-Carlo Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368 (1964). MR 0174487

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Additional Information

Chris Macmeikan
Affiliation: Tokyo University of Science, Noda, Chiba 278-8510, Japan
Address at time of publication: Department of Mathematics, Keio University, Hiyoshi, Kohoku, Yokohama 223-8522, Japan
Email: chris_macmeikan@ma.noda.tus.ac.jp, chris@math.keio.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07398-8
Received by editor(s): May 28, 2002
Received by editor(s) in revised form: January 7, 2003
Published electronically: January 20, 2004
Additional Notes: This research was partially supported by an Australian Research Council grant for the project “Group Representation Theory and Cohomology of Algebraic Varieties”
Communicated by: Michael Stillman
Article copyright: © Copyright 2004 American Mathematical Society