L'éclatement en géométrie algébrique, différentielle et symplectique
The blow-up is a transformation which plays an important role in geometry, because it can be used to resolve singularities,  relate birationally equivalent varieties, and construct varieties with new properties. This thesis first presents blowing-up as  developped in classical algebraic geometry. We will study it in the case of affine and (quasi-)projective varieties, on a point and  along an ideal and a subvariety. Then a discussion about its extension to the differential category will be carried out, over the real and complex  fields, on a point and along a submanifold. An example of a resolution of singularity will then follow. Subsequently we will discuss  blowing-up in the symplectic category, where we will do the same as for complex manifolds, paying careful  attention to the symplectic form. To conclude, we will study a theorem by François Lalonde, where the symplectic blow-up  plays a major part in proof. This theorem states that any 4-variety fibered by 2-spheres over a Riemann surface, and  different than the Cartesian product of two 2-spheres, can be equiped with a 2-form giving it a symplectic structure ruled by curves that are  holomorphic with respect to its almost-complex structure, and such that the symplectic area of the base is smaller that  the capacity of the variety. In the proof, we blow up a ball in the 4-variety, and obtain a fibration containing two distinct spheres with  a self-intersection equal to -1: the pre-image of the point where the usual complex blow-up is done, and  the proper transform of the fiber. These two are exceptional, so it is possible to do the inverse operation - the blow down -  on each of them. By blowing down the latter, we get a minimal variety, and by combining information about the  symplectic area of its homology classes and of those of the original variety, we obtain the result.
Classical algebraic geometry Real blow up Complex blow up Symplectic blow up Blow up along a submanifold