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

  1. 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?
  2. 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?

[1] R. Brody - Compact Manifolds and Hyperbolicity, Trans. Amer. Math. Soc. 235 (1978) 213-219.

[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.

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1 Answer 1

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Since $X$ admits a Kähler-Einstein metric [Theorem 2,3] and [Theorem 1,2], if $K_X$ is ample the inequality holds with $H=K_X$, by [Theorem IV.4.16,1].


[1] S. Kobayashi (1987) Differential Geometry of Complex Vector Bundles, Iwanami Shoten Publishers and Princeton University Press.

[2] S.-T. Yau - Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. 74 (1977) 1798-1799.

[3] S.-T. Yau - On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978) 339-411.

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