As seen in Section 4.2, the structure of connected solvable linear algebraic groups is well-understood. We intend to exploit this by studying a particular family of connected solvable subgroups of an arbitrary linear algebraic group.

By the Lie–Kolchin Theorem any connected solvable subgroup G ≤ GLn can be embedded in Tn. In particular G stabilizes a flag ℱ : 0 = V0 ⊂ V1 ⊆ … ⊂ Vn = kn of subspaces. Moreover, for G an arbitrary closed subgroup of GLn, it is clear that the stabilizer Gℱ ≤ G of any such flag is a solvable group, and the quotient variety G/Gℱ is a quasi-projective variety, i.e., an open subset of a projective space. (See the remarks at the beginning of Chapter 5 and Proposition 5.4.) Now if we choose ℱ such that its G-orbit is of minimal dimension, then this orbit is in fact closed (loc. cit.) and so a projective space. We will obtain such minimal-dimensional orbits by choosing Gℱ of maximal possible dimension among flag stabilizers. This leads us to our definition of Borel subgroups (see Definition 6.3 below).

The Borel fixed point theorem

The principal ingredient for the study of Borel subgroups is the following fixed point theorem:

Theorem 6.1 (Borel fixed point theorem) Let G be a connected, solvable linear algebraic group acting on a non-empty projective G-space X. Then there exists x ∈ X such that g.x = x for all g ∈ G.