Minimal compactifications of the affine plane with nef canonical divisors

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. In this article, we show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.