Algebraic Geometry Seminar

Shrikant Bhatwadekar (Tata Institute of Fundamental Research)

Zero cycles and the Euler class groups of smooth affine varieties

Let $X = Spec (A)$ be a smooth affine variety of dimension $d >1$ over a field $k$ and let $P$ be a projective $A$-module. Motivated by a result on topological vector bundles, Serre proved that if rank $P >d$, then $P$ has a unimodular element (i.e. $P$ splits of a free summand of rank one ). This the best possible result in general. Therefore it is natural to ask: If rank $P =d$, under what conditions does $P$ have a unimodular element ?

In my talk, I would show how to associate an abelian group $E(A)$ (the Euler class group) to $A$ and how to associate an invariant $e(P)$ (the Euler class of $P$ which takes value in $E(A)$ ) to $P$ and discuss the connection between existence of unimodular elements in $P$ and vanishing of the $d$th Chern class $C_d(P)$ and the Euler class $e(P)$ of $P$.