Equivariant multiplicities and torsion at attractive fixed points

Abstract

Let \(X\) be a rationally smooth complex affine variety with a torus action and an attractive fixed point \(x\). Suppose that \(X\setminus\{x\}\) is \(p\)-smooth and that its integral equivariant cohomology has no \(p\)-torsion. We prove that the order of its total \(p\)-primary cohomology torsion is the \(p\)-part of the reduced numerator of the equivariant multiplicity at \(x\). In particular, this resolves the torsion-order conjecture of Juteau--Williamson and its local version for normal slices in Schubert varieties, using the equivariant torsion-freeness result of Fiebig--Williamson.