On Proper Descent of Smooth Affine Surfaces with Finite Homotopy Rank-Sum

Abstract: We study the descent behaviour of homotopy-theoretic properties of smooth complex affine surfaces under finite surjective morphisms. We first examine the Eilenberg-MacLane property and show, by means of an explicit counterexample, that it does not descend under proper morphisms in general. This negative result motivates the introduction of a weaker notion, the finite homotopy rank-sum property. Our main theorem establishes that this property does descend under proper morphisms between smooth affine surfaces of logarithmic Kodaira dimension at most zero. The proof relies essentially on the recent classification of smooth complex affine surfaces of log non-general type characterized by these two properties. As a further application, we classify smooth affine surfaces properly dominated by the complex algebraic 2-torus, thereby clarifying an earlier remark of M. Furushima.