Семинар НУЛ АГП "Modality and complexity of reductive group actions"

An important numerical invariant of an action of a connected group H on an irreducible variety X is its modality mod(X, H). It is the maximal number of parameters in a continuous family of H-orbits. Let X be equipped with an action of a connected reductive group G. Remarkably, the modality of the induced action of a Borel subgroup B (a regular invariant) is equal to the complexity c(X) of the G-action (a birational invariant) [E. Vinberg’86]: mod(X, B) = c(X).

If instead of families of B-orbits we consider families of G-orbits, the number of parameters may drop, i.e. it may happen that mod(X, G) < c(X). However, it turns out that there always exists a G-variety X’ birationally G-isomorphic to X such that mod(X’, G) = c(X), and therefore the complexity c(X) is equal to the maximal modality of the G-action among all G-birational models of X. This result is due to D. Akhiezer. In the talk we will discuss the notions of complexity and modality and give a proof of Akhiezer’s theorem following the paper «On modality and complexity of actions of reductive groups» by D. Akhiezer.