Characteristic Classes

Abstract

The goal of this lecture notes is to introduce to Characteristic Classes. This is an important tool of the contemporary mathematics, indispensable to work in geometry and topology, and also useful in number theory. Classical roots of characteristic classes overlap: the Euler characteristic, indices of vector fields and the Poincaré–Hopf theorem, Plücker formulas for plane curves, the Euler characteristic of the Milnor fibre, Riemann–Roch and Riemann–Hurwitz theorems for curves and Schubert calculus. The present approach to the characteristic classes treats them as elements of the cohomology rings and their analogues. We shall discuss the Chern classes of complex vector bundles, characteristic classes of real vector bundles, various characteristic classes of singular analytic varieties. The fundamental theorems on characteristic classes will be proven, in particular, the Grothendieck– –Hirzebruch–Riemann–Roch theorem. Characteristic classes mark out the place where many domains of the contemporary mathematics meet: geometry, topology, singularities, representation theory, algebra and combinatorics. As for what concerns these last three domains, we shall discuss the basic properties of Schur functions. To the memory of my Father (1928–2012 )