On the behavior of stringy motives under Galois quasi-\'etale covers

Abstract

We investigate the behavior of stringy motives under Galois quasi-\'etale covers. We prove that they descend under such covers in a sense defined via their Poincar\'e realizations. Further, we show that such descent is strict in the presence of ramification. As a corollary, we reduce the problem regarding the finiteness of the \'etale fundamental group of KLT singularities to a DCC property for their stringy motives. We verify such DCC property for surfaces in arbitrary characteristic. As an application, we give a characteristic-free proof for the finiteness of the \'etale fundamental group of log terminal surface singularities, which was unknown in equal characteristics 2 and 3 and in mixed characteristics.