Local volumes on normal algebraic varieties

In this paper we study a notion of volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study a volume for normal isolated singularities, generalizing work on surfaces done by Wahl. We also compare this volume of isolated singularities to a different generalization due to Boucksom, de Fernex and Favre.