Positivity and the Kodaira embedding theorem

摘要

The Kodaira embedding theorem provides an effective characterization of projectivity of a K"ahler manifold in terms the second cohomology. X Yang (2018) proved that any compact K"ahler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms of its curvature. We prove that any compact K"ahler manifold with positive $2^{text{nd}}$ scalar curvature (which is the average of holomorphic sectional curvature over $2$--dimensional subspaces of the tangent space) must be projective. In view of generic $2$--tori being nonabelian, this new curvature characterization is sharp in certain sense.