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 the integral on a function involving the holomorphic sectional curvature alone by 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, we characterise the n-dimensional complex torus by certain numerical 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 construct a new example of projective manifold of dimension n \((n\ge 2)\) which is with ample canonical bundle, but does not carry any Hermitian metric with negative holomorphic sectional curvature.
