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.

中文翻译：

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关于全纯截面曲率的零集

Heier、Lu、Wong 和 Cheng 提出的一个著名例子表明，存在具有充足规范线束的紧复凯勒流形，使得全纯截面曲率是负半定的，并且在每个切空间中沿着高维线性子空间消失。本笔记的主要结果是这些子空间维度的上限。由于全纯截面曲率是每个切空间上的二阶 (2,2) 实值双齐次多项式，证明基于与 D'Angelo 关于实代数簇的复子簇和分解的工作的联系将多项式化为平方差。我们的界限涉及一个不变量，我们称之为全纯截面曲率平方分解长度，只要全纯截面曲率是半定的，无论是负数还是正数，我们的论证就有效。