### A note on the Lawrence-Krammer-Bigelow representation.

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We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik–Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the ${\mathrm{\U0001d530\U0001d529}}_{2}$ Bernstein–Gelfand–Gelfand resolution as an Aomoto complex.

The cohomological structure of hypersphere arragnements is given. The Gauss-Manin connections for related hypergeometrtic integrals are given in terms of invariant forms. They are used to get the explicit differential formula for the volume of a simplex whose faces are hyperspheres.

There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank $1$ local systems on the complement of the arrangement to gain information on the homology of the other three spaces, and on the monodromy operators of the various fibrations.

Let $Y$ be a divisor on a smooth algebraic variety $X$. We investigate the geometry of the Jacobian scheme of $Y$, homological invariants derived from logarithmic differential forms along $Y$, and their relationship with the property that $Y$ be a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.

We prove a vanishing theorem for the cohomology of the complement of a complex hyperplane arrangement with coefficients in a complex local system. This result is compared with other vanishing theorems, and used to study Milnor fibers of line arrangements, and hypersurface arrangements.

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,...

In this paper, we define the supersolvable order of hyperplanes in a supersolvable arrangement, and obtain a class of inductively free arrangements according to this order. Our main results improve the conclusion that every supersolvable arrangement is inductively free. In addition, we assert that the inductively free arrangement with the required induction table is supersolvable.

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers–Tymoczko. In particular, when an ideal subarrangement is equal to the entireWeyl arrangement, our...

This work contains an extended version of a course given in Arrangements in Pyrénées. School on hyperplane arrangements and related topics held at Pau (France) in June 2012. In the first part, we recall the computation of the fundamental group of the complement of a line arrangement. In the second part, we deal with characteristic varieties of line arrangements focusing on two aspects: the relationship with the position of the singular points (relative to projective curves of some prescribed degrees)...