The Bochner-Lichnerowicz formula and the Atiyah-Singer Index Formula for the Dirac operator have been used to find an obstruction (the $\widehatA$-genus) to producing metrics of positive scalar curvature on spin manifolds. Here the technique is applied to twisted Dirac operators in order to obtain upper bounds on the minimum of the scalar curvature for Riemannian manifolds which admit certain contractive spin mappings into a fixed Riemannian manifold. The principal application is to obtain such upper bounds for algebraic varieties equipped with arbitrary metrics, which admit contractive maps into P^n(C) homotopic to inclusions.