Massey Products of Complex Hypersurface Complements

The study of the topology of hypersurface complements is a classical subject in algebraic geometry. Most of what is known about these spaces is related to invariants of their rational homotopy type. In this paper, we attempt to show that their Fp-homotopy type captures in general more information than the Q-homotopy type, where Fp is the prime field of p elements. Let X be the complement to a hypersurface S in CP . Then we have the following results due to Kohno [12, 13]: Massey products in H(X,Q) of length ≥ 3 vanish. Moreover, the Malcev Lie algebra of π1(X) and the completed holonomy Lie algebra of H≤2(X,Q) are isomorphic. Thus, the Q-completion of π1(X) is completely determined by the Q-cohomology algebra of X. In the case when S is a hyperplane arrangement X is Q-formal by Morgan [17], that is the entire Q-homotopy type of X is determined by the algebra H(X,Q). In this context, it seems natural to pose the following questions: Are there non-vanishing Massey products in H(X,Fp) for all primes p? Is X a Fp-formal space, particularly when X is a hyperplane arrangement complement? Massey products are known to be obstructions to formality, see [5, 7]. So, if the answer to the first question was yes, then the space X would not be Fp-formal. For compact Kähler manifolds the above questions were answered by Ekedhal in [6] by constructing such manifolds M with non-vanishing triple products in H(M,Fp). Thus, a compact Kähler manifold although is Q-formal by [5], in general it may not be Fp-formal. The case of non-compact complex algebraic varieties is already different over Q from the compact case. As pointed out by Morgan in [17] such varieties may not be Q-formal. The main result of this paper settles in affirmative the existence of non-vanishing Massey products in the Fp-cohomology of a hyperplane arrangement complement for all odd primes p, thus showing that arrangement complements are not Fp-formal in general.