Some contributions to the theory of singularities and their characteristic classes
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In this Ph.D. thesis, we give some contributions to the theory of singularities, as well as to the theory of characteristic classes of singular spaces. The first part of this thesis is devoted to the theory of singularities of mappings. We obtain formulas for an important analytical invariant, the image Milnor number of a map-germ. The first contribution is a version of the Lê-Greuel formula for the image Milnor number for corank 1 map-germs, which gives a recursive method to compute it. This work is in collaboration with my Ph.D. supervisor J. J. Nuño. In the second work of this thesis, we obtain two formulas for the image Milnor number for weighted-homogeneous map-germs for dimensions four and five. These formulas are obtained by combining the theory of characteristic classes of singular spaces with 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 projective varieties of the Brasselet-Schürmann-Yokura conjecture. This conjecture is within the theory of characteristic classes of singular spaces. Characteristic classes of singular varieties are homology classes generalizing the classical cohomological characteristic classes of manifolds. The conjecture states that two different characteristic classes coincide for compact complex algebraic varieties that are rational homology 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 Hodge Index Theorem which computes the signature of a compact complex manifold in terms of Hodge numbers.