Indices of vector fields and Chern classes for singular varieties

Speaker:José Antonio Seade Kuri (UNAM)
Schedule:Wed., 9:50-11:25 am; Fri., 1:30-3:05 pm; Apr. 19-May 12, 2023
Venue:Room A-404, Science Building, Department of Mathematical Sciences, Tsinghua Univerisity

Note: There are no lectures on May 3 & May 5. The lecture on May 3 is moved to 9:50-11:25 am, May 7. 

Course Description:

The Poincare- Hopf local index of a vector field at an isolated singularity on a smooth manifold is a classical invariant that has given rise to a vast literature. The Poincare-Hopf theorem stating that the total index of a vector field on a closed smooth manifold equals the Euler characteristic is a cornerstone in several areas of mathematics. These are much related with the theory of Chern classes on smooth manifolds, which play a fundamental role in geometry and topology. In this course we shall discuss generalizations to the case of complex analytic singular varieties. In this setting there is not a unique concept of index of a vector field, nor a unique notion of Chern classes, but several different ones, each with its own properties and interest. On a singular variety one does not have a tangent bundle over the singular set, and the various notions of index and Chern classes somehow correspond to different ways of extending the tangent bundle over the singular part.

In this course we will start by reviewing the classical theory. Then we shall move toward singular varieties, starting with examples, the basic definitions and properties, stratifications, the local conical structure of analytic sets and Milnor’s fibration theorem.


José Seade obtained his BSc in mathematics from the National University of Mexico (UNAM) in 1976 and his Masters and PhD from the University of Oxford in 1977 and 1980, respectively. Since then, he has worked at the Institute of Mathematics, UNAM, where he has been director since April of 2014.

His present research interests are on singularities theory and complex geometry and he has authored several publications in algebraic topology, algebraic and differential geometry and geometric analysis. He has been awarded the Ferran Sunyer i Balaguer Prize twice (2005 and 2012), was President of the Mexican Mathematical Society (1986-87) and founded the Mexican Mathematics Olympiades. He also founded the Solomon Lefschetz International Laboratory of Mathematics, in Cuernavaca, Mexico, which is associated with the CNRS of France, and is the current Scientific Coordinator of that laboratory. He was a member of the Scientific Council of UMALCA, the Latin American and Caribbean Union of Mathematicians (2001-2009) and since then has been a member of the Executive Committee of UMALCA.