Vanishing cycles and Picard Lefschetz formula

Vortrag von Thomas Jacob

Datum: 27.11.25  Zeit: 16.05 - 17.05  Raum: ETH HG G 26.3

In singularity theory, Picard-Lefschetz theory aims to study the
topology of a complex manifold \(X\) with the help of the critical points of
holomorphic functions defined on \(X\), and can be seen as a complex analog
of Morse theory. Let \(f\) be an holomorphic function on a complex manifold
\(X\), with values in the unit disk, having one isolated critical point \(x\)
such that \(f(x)=0\). For \(t\) different than \(0\) small enough, the fiber \(X_t\)
will contain a subvariety called the vanishing cycle subvariety, whose
homotopy type is one of a bouquet of sphere. The singular fiber \(X_0\) is
then obtained by "pinching" those spheres to a point. The
Picard-Lefschetz formula describe the monodromy around the critical
point in terms of these vanishing cycles. We will give a geometric
overview of the theory, and mention some applications.