The Alexander polynomial of certain classes of non-symmetric line arrangements

The Alexander polynomial of a projective hypersurface V ϲ Pᶰ is the characteristic polynomial of the monodromy operator acting on Hᶰ¯¹(F, C), where F is the Milnor fibre of V; unless V is smooth, the problem of its computation is open. The singular hypersurfaces that have drawn the most attention are projectivisations Ᾱ of central hyperplane arrangements A C Cᶰ⁺ ¹, as one can hope to take advantage of the combinatorial nature of such objects; one can assume without loss of generality that n=2. In this Thesis we prove that the Alexander polynomials of line arrangements Ᾱ C P² belonging to some particular non-symmetric classes are trivial: this constitutes evidence in favour of the validity of a conjecture due to Papadima and Suciu. The Thesis is organised as follows. In Chapter 1 we gather some known results on which we will build upon: the discussion of mixed Hodge structures on cohomology groups of algebraic varieties and the comparison between the polar and Hodge filtration are of particular importance; the construction of cubical hyperresolutions and their use in the definition of algebraic de Rham cohomology for singular algebraic varieties will be very useful too. Chapter 2 is divided in two parts. The first one is mainly devoted to defining the Alexander polynomial and presenting a formula by Libgober for its computation in case V is a curve. The second part is a survey of known results around the problem of determining the Alexander polynomial of a line arrangement, and closes with a discussion of some interesting examples; we try to highlight how the symmetry of the arrangement affects its Alexander polynomial. In Chapter 3 we introduce some classes of non-symmetric line arrangements Ᾱ and prove that their Alexander polynomials are trivial. The methods we use are essentially two: one is the combination of Libgober's formula with an easy deformation theory argument, thanks to which we can restrict ourselves to considering a finite number of “representative arrangements”; the other relies on associating to Ᾱ a threefold T fibred in surfaces over P¹ and on studying the monodromy around a special fibre of the latter. A key step of the second method is the proof of the existence of a Gysin morphism that connects the cohomology of T to that of a hyperplane section S: this result is of independent interest, as T and S do not satisfy the hypotheses usually required in order to obtain Lefschetz-type results.