We show that the diffeomorphic type of the complement to a line arrangement in a complex projective plane P 2 depends only on the graph of line intersections if no line in the arrangement contains more than two points in which at least two lines intersect. This result also holds for some special arrangements which do not satisfy this property. However it is not true in general, see [Rybnikov G., On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal. Appl., 2011, 45(2), 137–148].
@article{bwmeta1.element.doi-10_2478_s11533-011-0116-x,
author = {Fedor Bogomolov and Viktor Kulikov},
title = {On the diffeomorphic type of the complement to a line arrangement in a projective plane},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {521-529},
zbl = {1243.14047},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0116-x}
}
Fedor Bogomolov; Viktor Kulikov. On the diffeomorphic type of the complement to a line arrangement in a projective plane. Open Mathematics, Tome 10 (2012) pp. 521-529. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0116-x/
[1] Eliyahu M., Garber D., Teicher M., A conjugation-free geometric presentation of fundamental groups of arrangements, Manuscripta Math., 2010, 133(1–2), 247–271 http://dx.doi.org/10.1007/s00229-010-0380-2 | Zbl 1205.14034
[2] Fan K.-M., Direct product of free groups as the fundamental group of the complement of a union of lines, Michigan Math. J., 1997, 44(2), 283–291 http://dx.doi.org/10.1307/mmj/1029005704 | Zbl 0911.14007
[3] Hirzebruch F., Arrangements of lines and algebraic surfaces, In: Arithmetic and Geometry II, Progr. Math., 36, Birkhäuser, Boston, 1983, 113–140
[4] Jiang T., Yau S.S.-T., Diffeomorphic types of the complements of arrangements of hyperplanes, Compositio Math., 1994, 92(2), 133–155 | Zbl 0828.57018
[5] Rybnikov G., On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal. Appl., 2011, 45(2), 137–148 http://dx.doi.org/10.1007/s10688-011-0015-8 | Zbl 1271.14085
[6] Wang S., Yau S.S.-T., Rigidity of differentiable structure for new class of line arrangements, Comm. Anal. Geom., 2005, 13(5), 1057–1075 | Zbl 1115.52010