Some contributions to the theory of singularities and their characteristic classes

Résumé

In this Ph.D. thesis, we give some contributions to the theory of singularities, as well as to the theory ofcharacteristic classes of singular spaces. The first part of this thesis is devoted to the theory ofsingularities of mappings. We obtain formulas for an important analytical invariant, the image Milnornumber of a map-germ. The first contribution is a version of the Lê-Greuel formula for the image Milnornumber for corank 1 map-germs, which gives a recursive method to compute it. This work is incollaboration with my Ph.D. supervisor J. J. Nuño. In the second work of this thesis, we obtain twoformulas for the image Milnor number for weighted-homogeneous map-germs for dimensions four andfive. These formulas are obtained by combining the theory of characteristic classes of singular spaceswith a recursive method through examples of singularities. This work is in collaboration with Prof. G.Peñafort. The last part which composes the main work of this Ph.D. thesis is the proof for projectivevarieties of the Brasselet-Schürmann-Yokura conjecture. This conjecture is within the theory ofcharacteristic classes of singular spaces. Characteristic classes of singular varieties are homology classesgeneralizing the classical cohomological characteristic classes of manifolds. The conjecture states thattwo different characteristic classes coincide for compact complex algebraic varieties that are rationalhomology manifolds. This work is in collaboration with my Ph.D. supervisor J. Fernández de Bobadilla.This conjecture is the characteristic class version for rational homology manifolds of the famous HodgeIndex Theorem which computes the signature of a compact complex manifold in terms of Hodgenumbers.