Some constructions around stability
of vector bundles on projective
Defended on December, 6th 2006, at Jussieu (Paris)
This is a work on stability of vector bundles.
It is made of four chapters.
The first one is introductive, there we define all the necessary notions
to deal with stability, such as intersection theory, Chern classes,
the motivations and definition of stability,
and some notions about metrics of a stable bundle.
There are no new results and all references to the existing constructions
in the literature are included.
The other three parts are independent but
In the second chapter we show that stable bundles can be taken as
generators of the Chow ring, the K-theory and the derived category
of a smooth projective variety.
We use a polystable resolution of ideal sheaves.
This construction leads to some natural questions,
which can be found in various other contexts in the literature,
which are partially answered in the other chapters.
The third chapter is dedicated to the investigation about stability
of vector bundles that we call transforms.
These are kernels of evaluation maps
on subspaces of the space of global sections of a sheaf.
Stability is shown for tranforms of line bundles
with repsect to high
The fourth chapter consists in a similar question,
concerning stability of tautological sheaves
and their transforms on a symmetric product of curves.