Seminars

Second Lecture

The main topic of this lecture is the heuristic principle stated at the end of the first lecture:

the lines on the character varieties (affine cubic surfaces) play a central role in the theory of Painlevé equations.

This principle works very well for PVI. Surprisingly it seems that this remains unnoticed until very recently.

In a first part (mainly in XIX-th algebraic geometry style), we will describe the lines on the cubic surfaces, in particular the 27 lines in the case of nonsingular (complete) cubic surfaces.

In the second part of the lecture I will explain some relations between the 24 lines on the character variety of PVI and some partial reducibility properties of the monodromy representations of the linearized equations.

Afterwards I will sketch a generalization for the other Painlevé equations.