Higher Contou-Carrère symbol

The talk is based on a common work with D.V. Osipov. The Contou-Carrere symbol in dimension n is a way to construct an invertible element of an arbitrary commutative ring A using n+1 Laurent series of n variables over A. This symbol arises when considering n-dimensional varieties and complete flags on them, i.e. complete chains of irreducible subvarieties. The higher Contou-Carrere symbol satisfies a fundamental property — a so-called reciprocity law holds for it. All this will be discussed in detail in the talk. We will start with simple classical examples.