Local rigidity of covering constructions and Weil–Petersson subvarieties of the moduli space of curves

Abstract We show that totally geodesic subvarieties of the moduli space M₆, ₍ of genus g curves with n marked points, endowed with the Weil–Petersson metric, are locally rigid. This implies that covering constructions—examples of totally geodesic subvarieties of M₆, ₍ endowed with the Teichmüller metric—are locally rigid. We deduce the local rigidity statement from a more general rigidity result for a class of orbifold maps to M₆, ₍.