A lower bound on the essential dimension of a connected linear group

Abstract

Let G be a connected linear algebraic group defined over an algebraically closed field k and H be a finite abelian subgroup of G whose order does not divide char(k). We show that the essential dimension of G is bounded from below by \mathrm{rank}(H) − \mathrm{rank} C_G(H)^0 , where \mathrm{rrank} C_G(H)^0 denotes the rank of the maximal torus in the centralizer C_G(H) . This inequality, conjectured by J.-P. Serre, generalizes previous results of Reichstein–Youssin (where \mathrm{char}(k) is assumed to be 0 and C_G(H) to be finite) and Chernousov–Serre (where H is assumed to be a 2-group).