On the Zero Set of Holomorphic Sectional Curvature

An important example due to Heier, Lu, Wong, and Zheng shows that there exist complex Kaehler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi-de nite and vanishes along high-dimensional linear subspaces in every tangent space. The main result of this dissertation is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real valued bihomogeneous polynomial of degree 2 on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into di erences of squares. Our bound involves an invariant that we call the holomorphic decomposition length, and our arguments work as long as the the holomorphic sectional curvature is semi-de nite, be it negative or positive.