$\DeclareMathOperator\BMY{BMY}$Let $(X,H)$ be a smooth, complex, projective, polarized variety of dimension $n\geq3$, whose canonical bundle $K_X$ is big and nef.

- Is it know whether the inequality $\displaystyle \BMY(X,H)=\int_X\left[c_2(X)-\frac{n}{2(n+1)}c_1(X)^2\right]\cdot H^{n-2}\geq0$ holds?
- If $\BMY(X,H)=0$ then $X$ is uniformized by the unit ball $\mathbb{B}^n$, and in particular $X$ is hyperbolic. Is it kwnow wether the inverse implication holds?, I mean if $X$ is uniformized by $\mathbb{B}^n$ then does $\BMY(X,H)$ vanish?

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[2] S. Kobayashi (2005) *Hyperbolic Manifolds And Holomorphic Mappings. An Introduction. (Second Edition)* World Scientific.

[3] C. T. Simpson - *Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization*, J. Amer. Math. Soc. **1** (1988) 867-918.