Almost quarter-pinched Kähler metrics and Chern numbers

Given $n \in {\mathbb{Z}}^+$ and $\varepsilon >0$, we prove that there exists $\delta = \delta (\varepsilon,n) >0$ such that the following holds: If $(M^n,g)$ is a compact Kähler $n$-manifold whose sectional curvatures $K$ satisfy

$$-1 - \delta \le K \le - \frac {1}{4}$$

and $c_I(M)$, $c_J(M)$ are any two Chern numbers of $M$, then

$$\Bigl \vert \frac {c_I(M)}{c_J(M)} - \frac {c_I^0}{c_J^0} \Bigr \vert < \varepsilon,$$

where $c_I^0$, $c_J^0$ are the corresponding characteristic numbers of a complex hyperbolic space form.

It follows that the Mostow-Siu surfaces and the threefolds of Deraux do not admit Kähler metrics with pinching close to $\frac {1}{4}$.