An Explicit Five-Variable Counterexample to the Generalized Vanishing Conjecture

Abstract

We give an explicit counterexample in five variables to the Generalized Vanishing Conjecture. The construction is motivated by the recent counterexample to the Mathieu conjecture for SU(2). In the polynomial ring C[a,b,c,d,t], set P = (t+c)(ad+bt), Q = c, and let Λ = d/dt (d/da d/dd - d/db d/dc). We prove that Λ^m(P^m) = 0 for every m >= 1, whereas for every m >= 2, Λ^m(QP^m) = (-1)^m (m!)^2 (m+1)! t, which is nonzero. Thus the Generalized Vanishing Conjecture fails in dimension 5.