Pinched theorem and the reverse Yau's inequalities for compact Kähler-Einstein manifolds

TL;DR

The paper derives a sharp Chern-number identity for compact Kähler–Einstein manifolds by expressing $-c_1^n+\frac{2(n+1)}{n}c_2c_1^{n-2}$ as a global integral of the variance of the holomorphic sectional curvature, computed via invariant theory and sphere integration. This yields a concrete reverse Yau inequality, alongside dimension-dependent pinching constants, and leads to curvature-based characterizations (e.g., $\mathbb{P}^n$, ball quotients, and complex tori) plus partial confirmations of Yau’s and Siu–Yang’s conjectures in regimes of small curvature. The work further shows that an ample canonical bundle does not guarantee the existence of a metric with negative holomorphic sectional curvature by constructing ramified-cover examples where the reverse inequality fails. These results advance the understanding of the interplay between Chern-number inequalities, curvature pinching, and the global geometry of Kähler manifield.

Abstract

For a compact Kähler-Einstein manifold $M$ of dimension $n\ge 2$, we explicitly write the expression $-c_1^n(M)+\frac{2(n+1)}{n}c_2(M)c_1^{n-2}(M)$ in the form of certain integral on the holomorphic sectional curvature and its average at a fixed point in $M$ using the invariant theory. As applications, we get a reverse Yau's inequality and improve the classical $\frac{1}{4}$-pinched theorem and negative $\frac{1}{4}$-pinched theorem for compact Kähler-Einstein manifolds to smaller pinching constant depending only on the dimension and the first Chern class of $M$. If $M$ is not with positive or negative holomorphic sectional curvature, then there exists a point $x\in M$ such that the average of the holomorphic sectional curvature at $x$ vanishes. In particular, we characterise the $2$-dimensional complex torus by certain curvature condition. Moreover, we confirm Yau's conjecture for positive holomorphic sectional curvature and Siu-Yang's conjecture for negative holomorphic sectional curvature even for higher dimensions if the absolute value of the holomorphic sectional curvature is small enough. Finally, using the reverse Yau's inequality, we can judge if a projective manifold doesn't carry any hermitian metric with negative holomorphic sectional curvature.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Definition 2.1
• Lemma 2.2
• Theorem 2.3