Green function asymptotics for the ${\sigma }_2$-Yamabe problem

We establish, at every pole, the asymptotic expansion of the normalized ${\mathrm{\Gamma }}_2$-Green function with a finite pole set in every dimension $n\ge 5$. In dimensions $5\le n\le 7$ the first correction is a unique nonnegative constant, whereas dimension $8$ exhibits a Weyl-driven $\sqrt{\log }$ term. For $n>8$ we construct the finite local curvature parametrix through the first positive indicial resonance and identify the first genuinely global coefficient by a relative Newton flux.