Topology of smoothings of non-isolated singularities of complex surfaces

Abstract : This thesis is dedicated to the study of the topology of smoothings of non-isolated singularities of complex surfaces. The question is to describe the topology of the manifold, called \textbf{Milnor fiber}, which appears during this process of smoothing. Considering the great difficulty of a description of the whole of this topology, many researches have focused on the study of the \textbf{boundary} of the Milnor fiber. In the case of isolated singularities, it is known since the work of Mumford (1961) that this boundary is a graph manifold, isomorphic to the link of the singularity. Different results (Michel \& Pichon 2003, 2014, Némethi \& Szil\'ard 2012) have then proved that, in the case of reduced non-isolated singularities \ajo[of surfaces], the boundary of the Milnor fiber is again a graph manifold, while restraining to the case of a smooth total space of smoothing. Fern\'andez de Bobadilla \& Menegon-Neto (2014) have widened the context, considering non-reduced surfaces, and allowing the total space to have an isolated singularity. In this work, we pursue the extension of this result to a larger context, allowing the total space to present non-isolated singularities, while restraining ourselves to the study of reduced surface singularities. Our proof is inspired by the one of Némethi and Szilard, and allows us furthermore to provide a method for the computation of \hsout{this manifold} \ajo[the boundary of the Milnor fiber]. This makes possible the actual computation of a large number of examples, representing a step forward in the quest for the comprehension of the manifolds that can actually appear as boundaries of Milnor fibers. We apply in particular the method to Newton non-degenerate singularities defined on $3$-dimensional toric germs. This is a generalization of a theorem of Oka (1986), expressing the boundary of the Milnor fiber in terms of the Newton polyhedron of the singularity.
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Octave Curmi. Topology of smoothings of non-isolated singularities of complex surfaces. Mathematics [math]. Université de Lille / Laboratoire Paul Painlevé, 2019. English. ⟨tel-02180967⟩

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