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We prove that compact K\"ahler manifolds whose sectional curvatures are closeto 1/4-pinched have ratios of Chern numbers close to the corresponding ratiosof a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (andtheir three-dimensional analogues constructed by the first author) do not admitK\"ahler metrics with pinching close to 1/4.

Almost quarter-pinched K\"ahler metrics and Chern numbers