Alexandru Dimca

Singularities and topology of hypersurfaces

This book systematically presents a large number of basic results on the topology of complex algebraic varieties using the information on the local topology and geometry of a singularity. These concepts are then used in the computation of global topological invariants, such as homology groups, cohomology groups, fundamental groups, and Alexander polynomials. The reader will derive from this text a working knowledge of Whitney stratifications, Lefschetz-type theorems, knots and links, Milnor fibrations, vanishing cycles, weighted projective space, mixed Hodge structures and many other related notions and results. The book is intended for graduate work in algebraic and differential topology, and is an excellent source of natural examples and open-ended problems for the student working on a dissertation.

Sheaves in Topology

1 Derived Categories.- 1.1 Categories of Complexes C*(A).- 1.2 Homotopical Categories K*(A).- 1.3 The Derived Categories D*(A).- 1.4 The Derived Functors of Hom.- 2 Derived Categories in Topology.- 2.1 Generahties on Sheaves.- 2.2 Derived Tensor Products.- 2.3 Direct and Inverse Images.- 2.4 The Adjunction Triangle.- 2.5 Local Systems.- 3 Poincare-Verdier Duality.- 3.1 Cohomological Dimension of Rings and Spaces.- 3.2 The Functor f!.- 3.3 Poincare and Alexander Duality.- 3.4 Vanishing Results.- 4 Constructible Sheaves, Vanishing Cycles and Characteristic Varieties.- 4.1 Constructible Sheaves.- 4.2 Nearby and Vanishing Cycles.- 4.3 Characteristic Varieties and Characteristic Cycles.- 5 Perverse Sheaves.- 5.1 t-Structures and the Definition of Perverse.- 5.2 Properties of Perverse.- 5.3 D-Modules and Perverse.- 5.4 Intersection Cohomology.- 6 Applications to the Geometry of Singular Spaces.- Singularities, Milnor Fibers and Monodromy.- Topology of Deformations.- Topology of Polynomial Functions.- Hyperplane and Hypersurface Arrangements.- References.

Topology and geometry of cohomology jump loci

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to V_k at 1 equals R_k. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.

Sur la topologie des polynômes complexes

Soit f ∊ ℂ[x i,... ,x n ] un polynome de degre d. On sait [P], [ST] qu’il existe un ensemble fini A C C tel que

CHARACTERISTIC VARIETIES AND CONSTRUCTIBLE SHEAVES

f:{C^n}\backslash {f^{ - 1}}(\Lambda ) \to C\backslash \Lambda

Koszul complexes and hypersurface singularities

est une fibration localement triviale. Dorenavant A designe le plus petit ensemble qui possede cette propriete. Si t ∊ Λ, la fibre F t = f −1(t) est appellee fibre irreguliere de f, sinon elle est dite reguliere ou generique. Soit δ un nombre reel positif assez petit, δ ∉ Λ. On note

Multivariable Alexander invariants of hypersurface complements

{D_\delta } = \{ t \in C|\left| t \right| < \delta \} ,{S_\delta } = \partial {{\bar D}_\delta }et{T_\delta } = {f^{ - 1}}({D_\delta })