A Vanishing Theorem for the Cohomology of a Vector Bundle Whose Curvature Can Change Its Sign

This chapter deals with a vanishing theorem for the cohomology of a vector bundle whose curvature can change its sign. The famous Kodaira-Nakano theorem states the vanishing of the cohomology groups H^q for q > 0 under a strict positivity assumption for the curvature of E. Later Grauert and Riemenschneider showed the same result under the weaker assumption that the curvature form is nonnegative everywhere and strictly positive at least at one point. The chapter proves the above vanishing when the curvature form can change its sign, provided the set of points where it is positive prevails over that of points where it is negative.