Deformations of legendrian curves

Tese de doutoramento, Matemática (Geometria e Topologia), Universidade de Lisboa, Faculdade de Ciências, 2018In chapters 1 and 2 we study deformations of Legendrian curves in P*C². In chapter 1 we construct versal and semiuniversal objects in the category of deformations of the parametrization of a germ of a Legendrian curve as well as in the subcategory of equimultiple deformations. We show that these objects are given by the conormal or fake conormal of an hypersurface in C² x Cʳ. In chapter 2 we prove the existence of equisingular versal and semiuniversal deformations of a Legendrian curve, on this instance making use of deformations of the equation. By equisingular we mean that the plane projection of the fibres have fixed topological type. We prove in particular that the base space of such an equisingular versal deformation is smooth and construct it explicitly when the special fibre has semiquasihomogeneous or Newton non-degenerate plane projection. Chapter 3 concerns the construction of a moduli space for Legendrian curves singularities which are contactomorphic-equivalent and equisingular through a contact analogue of the Kodaira-Spencer map for curve singularities. We focus on the specific case of Legendrian curves which are the conormal of a plane curve with one Puiseux pair. To do so, it is fundamental to understand how deformations of such singularities behave, which was done in the previous chapter. The equisingular semiuniversal microlocal deformations constructed in chapter 2 already contain in their base space all the relevant fibres in the construction of such a moduli space. This is so because all deformations are isomorphic through a contact transformation to the pull-back of a semiuniversal deformation

Bounding the degree of solutions to Pfaff equations

We study hypersurfaces of complex projective manifolds which are invariant by a foliation, or more generally which are solutions to a Pfaff equation. We bound their degree using classical results on logarithmic forms