A study of irregular singular algebraic connections on curves : Monodromies, canonical decompositions and an example of the determinant of period integrals
Abstract : In this thesis, we study three aspects of irregular connections on curves: monodromies, canonical decompositions and the period determi-nant. In chapter 1, we recall a theorem due to Deligne that any rank 1 represen-tation of the fundamental group of an affine curve can appear as a connection on the trivial bundle. Then we make its generalization into two directions: firstly, we show the obstruction of the problem for rank 1 representations is the torsion of H 2 (U, Z ) for an affine variety U of any dimension. Secondly, over an affine curve the theorem holds for a representation of any rank. In chapter 2, from the view of Beilinson, Bloch and Esnault, we show the classical decomposition theorems of formal connections in one variable known as the slope decomposition and the Turritin-Levelt decomposition are absolute if the connection is integrable. Finally, in chapter 3, we calculate the determinant of the period integrals for an irregular connection r = d + dy over a Legendre elliptic curve E : y2 = x(x - 1)(x - lambda).