On the diffeomorphic type of the complement to a line arrangement in a projective plane

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 three 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].