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61
Higher homotopy groups of complements of complex hyperplane arrangements
 Advances in Mathematics
, 2002
"... We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, ..."
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Cited by 31 (2 self)
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We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, and we give an explicit Z\pi_1module presentation of \pi_p, the first nonvanishing higher homotopy group. We also give a combinatorial formula for the \pi_1coinvariants of \pi_p.
For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of \pi_2, and study some of the properties of this module. For graphic arrangements associated to graphs with no 3cycles, we obtain information on \pi_2, directly from the graph. The \pi_1coinvariants of \pi_2 may distinguish the homotopy 2types of arrangement complements with the same \pi_1, and the same Betti numbers in low degrees.
Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian
, 2008
"... We introduce various families of irreducible homaloidal hypersurfaces in projective space P r, for all r ≥ 3. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves ..."
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Cited by 25 (8 self)
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We introduce various families of irreducible homaloidal hypersurfaces in projective space P r, for all r ≥ 3. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of hypersurfaces that are dual to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. The latter family fit non–classical versions of de Jonquières transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan–Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision into the present knowledge.
Morse theory, Milnor fibers and minimality of hyperplane arrangements
 Proc. Amer. Math. Soc
, 2002
"... Abstract. Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CWcomplex in which the number of pcells equals the pth betti nu ..."
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Cited by 22 (1 self)
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Abstract. Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CWcomplex in which the number of pcells equals the pth betti number. Combining this result with recent work of Papadima and Suciu, one obtains a characterization of when arrangement complements are EilenbergMac Lane spaces. 1.
Intersection homology and Alexander modules of hypersurface complements
, 2004
"... Let V be a degree d, reduced hypersurface in CP n+1, n ≥ 1, and fix a generic hyperplane, H. Denote by U the (affine) hypersurface complement, CP n+1 −V ∪H, and let U c be the infinite cyclic covering of U corresponding to the kernel of the linking number homomorphism. Using intersection homology ..."
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Cited by 21 (13 self)
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Let V be a degree d, reduced hypersurface in CP n+1, n ≥ 1, and fix a generic hyperplane, H. Denote by U the (affine) hypersurface complement, CP n+1 −V ∪H, and let U c be the infinite cyclic covering of U corresponding to the kernel of the linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules Hi(U c; Q) of the hypersurface complement and show that, if i ≤ n, these are torsion over the ring of rational Laurent polynomials. We also obtain obstructions on the associated global polynomials. Their zeros are roots of unity of order d and are entirely determined by the local topological information encoded by the link pairs of singular strata of a stratification of the pair (CP n+1, V). As an application, we give obstructions on the eigenvalues of monodromy operators associated to the Milnor fibre of a projective hypersurface arrangement.
Hyperplane arrangements and lefschetz’s hyperplane section theorem
 KODAI MATHEMATICAL JOURNAL
, 2005
"... The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangemen ..."
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Cited by 20 (9 self)
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The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real numbers. The cells and attaching maps are described in combinatorial terms of chambers. We also discuss the cellular chain complex with coefficients in a local system and a presentation for the fundamental group associated to the minimal CWdecomposition for the complement.
MULTIVARIABLE ALEXANDER INVARIANTS OF HYPERSURFACE COMPLEMENTS
, 2007
"... We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin’s vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomolog ..."
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Cited by 19 (12 self)
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We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin’s vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case, which combines the degeneration of the Hodge to de Rham spectral sequence with the purity of some cohomology groups.
Combinatorial Morse theory and minimality of hyperplane arrangements
 GEOMETRY AND TOPOLOGY
, 2007
"... Using combinatorial Morse theory on the CWcomplex S constructed in Salvetti [15] which gives the homotopy type of the complement to a complexified real arrangement of hyperplanes, we find an explicit combinatorial gradient vector field on S, such that S contracts over a minimal CWcomplex. The exis ..."
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Cited by 18 (1 self)
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Using combinatorial Morse theory on the CWcomplex S constructed in Salvetti [15] which gives the homotopy type of the complement to a complexified real arrangement of hyperplanes, we find an explicit combinatorial gradient vector field on S, such that S contracts over a minimal CWcomplex. The existence of such minimal complex was proved before Dimca and Padadima [5] and Randell [14] and there exists also some description of it by Yoshinaga [19]. Our description seems much more explicit and allows to find also an algebraic complex computing local system cohomology, where the boundary operator is effectively computable.
The spectral sequence of an equivariant chain complex and homology with local coefficients
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2010
"... We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CWcomplex X. In the process, we identify the d_1 differential in terms of the coalgebra structure of H^*(X, k), and the kπ ..."
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Cited by 15 (9 self)
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We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CWcomplex X. In the process, we identify the d_1 differential in terms of the coalgebra structure of H^*(X, k), and the kπ_1(X)module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov, on the mod p cohomology of cyclic pcovers of aspherical complexes. This approach provides information on the homology of all Galois covers of X. It also yields computable upper bounds on the ranks of the cohomology groups of X, with coefficients in a primepower order, rank one local system. When X admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cupproduct structure of H^*(X, k), thereby generalizing a result of Cohen and Orlik.