Abstract
We use a new method to study arrangement in CP l, define a class of nice point arrangements and show that if two nice point arrangements have the same combinatorics, then their complements are diffeomorphic to each other. In particular, the moduli space of nice point arrangements with same combinatorics in CP l is connected. It generalizes the result on point arrangements in CP 3 to point arrangements in CP l for any l.
Similar content being viewed by others
References
Orlik P, Terao H. Arrangements of Hyperplanes. In: Grundlehren der Mathematischen Wissenschaften, Vol. 300. Berlin: Springer-Verlag, 1992
Orlik P, Solomon L. Combinatorics and topology of complements of hyperplanes. Invent Math, 56: 167–189 (1980)
Falk M. The cohomology and fundamental group of a hyperplane complement. In: Proceedings of the IMA Participating Institutions Conference, Vol. 90. Providence, RI: Amer Math Soc, 1989, 55–72
Falk M. Homotopy types of line arrangements. Invent Math, 111(1): 139–150 (1993)
Falk M. On the algebra associated with a geometric lattice. Adv Math, 80(2): 152–163 (1990)
Jiang T, Yau S S T. Topological invariance of intersection lattices of arrangements in CP 2. Bull Amer Math Soc, 29(1): 88–93 (1993)
Jiang T, Yau S S T. Intersection lattices and topological structures of complements of arrangements in CP 2. Ann Scuola Norm Sup Pisa Cl Sci, 26(2): 357–381 (1998)
Rybnikov G. On the fundamental group of the complement of a complex hyperplane arrangement. Arxiv.org /abs/math/9805056, 1998
Randell R. Lattice-isotopic arrangements are topologically isomorphic. Proc Amer Math Soc, 107(2): 555–559 (1989)
Jiang T, Yau S S T. Diffeomorphic types of the complements of arrangements of hyperplanes. Compos Math, 92(2): 133–155 (1994)
Jiang T, Yau S S T. Complement of arrangement of hyperplanes. In: SinGularities and complex geometry, AMS/IP Stud Adv Math, Vol. 5. Providence, RI: Amer Math Soc, 1997, 93–104
Wang S, Yau S S T. Rigidity of differentiable structure for new class of line arrangements. Comm Anal Geom, 13(5): 1057–1075 (2005)
Wang S, Yau S S T. Diffeomorphic types of the complements of arrangements in CP 3 I: point arrangements. J Math Soc Japan, 59(2): 423–447 (2007)
Wang B, Yau S S T. Diffeomorphic types of the complements of arrangements in CP 3 II. Sci China Ser A, 51: 785–802 (2008)
Hartshorne R. Algebraic Geometry. New York: Springer-Verlag, 1977
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Prof. ZHONG TongDe on the occasion of his 80th birthday
This work was partially supported by National Natural Science Foundation of China (Grant No. 10731030) and Program of Shanghai Subject Chief Scientist (PSSCS) of Shanghai
Rights and permissions
About this article
Cite this article
Yau, S.ST., Ye, F. Diffeomorphic types of complements of nice point arrangements in CPl . Sci. China Ser. A-Math. 52, 2774–2791 (2009). https://doi.org/10.1007/s11425-009-0202-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-009-0202-8