On the zero set of the holomorphic sectional curvature

A notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi-definite and vanishes along high-dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real-valued bihomogeneous polynomial of bidegree (2,2) 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 differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi-definite, be it negative or positive.