Pinched theorem and the reverse Yau's inequalities for compact K\"{a}hler-Einstein manifolds

Abstract

For a compact K\"{a}hler-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 K\"{a}hler-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.