Białynicki-Birula decomposition for reductive groups

We generalize the Białynicki-Birula decomposition from actions of G_m on smooth varieties to actions of linearly reductive group G on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks. We define the Białynicki-Birula decomposition functorially: for a fixed G-scheme X and a monoid Gbar which compactifies G, the BB decomposition parameterizes G-schemes over X for which the G-action extends to the Gbar-action. The freedom of choice of Gbar makes the theory richer than the G_m-case.