The B-B decomposition via Sumihiro's theorem (Q1919998)

This paper consists of an elementary proof of a result of Białynicki-Birula (now called the B-B-decomposition). It describes the decomposition of a nonsingular complete variety \(X\) defined over the algebraically field \(k\) relative to an algebraic action of the multiplicative group \(T\) of \(k\). The main tools are: (A) The theorem of Sumihiro: If \(X\) is a normal algebraic variety with an action of \(T\), then for each point \(x\in X\) there exists an open affine \(T\)-invariant neighbourhood \(U\) of \(x\) in \(X\). \(U\) can be embedded in an equivariant way as a closed subvariety in some vector space with a linear action of \(T\). (B) Every algebraic representation of \(T\) is completely reducible, i.e., may be diagonalized with respect to some basis.