In Part I, we presented the CMS-criterion giving a sufficient condition for bigness of the cotangent bundle of a birational class \({\mathscr {X}}\) of surfaces of general type. The CMS-criterion has the term \(Lh^1_\Omega ({\mathscr {X}})\) coming from the singularities of the canonical model of \({\mathscr {X}}\) , where each singularity contributes with \({h}_\Omega ^1(y)\) , the first cohomological \(\Omega \) -asymptotics of the singularity y. We give a method to find the invariant \({h}_\Omega ^1(A_n)\) , where y is an \(A_n\) singularity, and obtain a closed formula in n that allows us to understand the range of the CMS-criterion. To find \({h}_\Omega ^1(A_n)\) , we give a complete answer to the (holomorphic) extension problem along E for symmetric m-differentials on the complement \({{\widetilde{U}}}_{A_n}{\setminus }\hspace{1.111pt}E\) , where \({{\widetilde{U}}}_{A_n}\) is the resolution of a germ of an \(A_n\) singularity and E its exceptional locus. We show that, for fixed n, the function \(\hslash ^0(A_n,m)=\dim \hspace{0.55542pt}[H^0(\widetilde{U}_{A_n}{\setminus }\hspace{1.111pt}E,S^m\Omega ^1_{{\tilde{U}}_{A_n}})/H^0(\widetilde{U}_{A_n},S^m\Omega ^1_{{\tilde{U}}_{A_n}})]\) is a quasi-polynomial in m of degree 3 with constant cubic and quadratic coefficients, for which we give a formula. We also determine the precise extent to which the poles along E of the symmetric differentials on \({{\widetilde{U}}}_{A_n}{\setminus }\hspace{1.111pt}E\) are milder than logarithmic poles.
