Non-Rationality of Projective Approximations for Classifying Spaces

Let BG be the classifying space of an algebraic group G over a subfield k of the complex numbers ℂ.We compute a new stable birational invariant defined by Benoist and Ottem [Duke Math.J. 170 (2021), 2719-2753] as the difference of two coniveau filtrations of a smooth projective (Ekedahl) approximation X of BG × ℙ ∞ .Then we show (without and with the unramified cohomology) that in many cases X are not stable rational.