From the differential geometric proof of Yau and the algebraic proof of Miyaoka for minimal varieties of general type $\kappa (X)=\dim X$, we know that $$(-1)^nc_1^n(X)\leq (-1)^n\frac{2(n+1)}{n} c_1^{n-2}(X)c_2(X)$$

What can we say about the Bogomolov–Miyaoka–Yau inequality for minimal varieties with an intermediate Kodaira dimension $0<\kappa(X)<\dim X$?

So I guess we need to prove the following inequality:

$$\left(\frac{2(n+1)}{n}c_2(\mathcal T_{X/X_{can}})-c_1^2(\mathcal T_{X/X_{can}})\right).[\omega]^{n-2}\geq 0$$

where $\omega$ is a Kahler form on the minimal projective variety $X=X_{min}$ and $X_{can}=\text{Proj}\bigoplus_{m\geq 0}H^0(X,K_X^{m})$ is the canonical model of $X$ (here $\mathcal T_{X/X_{can} }=Hom(Ω^1_{X/X_{can}}, \mathcal O_X)$ mean relative tangent sheaf) via Iitaka fibration $\pi: X\to X_{can}$.

Certainly we must require stability in order that this inequality holds true. The stability must be equivalent with the fact that the following flow converges in $C^\infty$$$\frac{\partial\omega(t)}{\partial t}=-Ric_{X/X_{can}}(\omega(t))-\omega(t)$$

Here $Ric_{X/X_{can}}=dd^c\log \Omega_{X/X_{can}}$(where $\Omega_{X/X_{can}}$ is the relative volume form) means relative Ricci form. Note that if such relative Kahler Ricci flow has solution then $K_{X/X_{can}}$ is psudo-effective I think that the analytical minimal model program can prove this.

In fact I guess if we have relative Kahler-Einstein metric $Ric_{X/X_{can}}\omega=-\omega$ , then Bogomolov–Miyaoka–Yau inequality for minimal varieties with intermediate Kodaira dimension $0<\kappa (X)<\dim X$ holds true.

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    $\begingroup$ See the following paper for logarithmic version ;) H. Tsuji, Stability of tangent bundles of minimal algebraic varieties, Topology, 27, no. 4 (1988), 429-442 . The question for positive Kodaira dimension is still open $\endgroup$ – user21574 Apr 22 '17 at 18:20
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    $\begingroup$ @Henri , if you apply the method of the following paper for conical Kahler Ricci flow you can get much simpler proof ams.org/journals/proc/2009-137-08/S0002-9939-09-09838-4/… $\endgroup$ – user21574 Apr 22 '17 at 18:24
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    $\begingroup$ I have assumed $\kappa (X)\neq \dim X$ and fibers are CY here. So for intermidiate Kodaira dimension varieties you think we need to use pull back of tangent sheaf of canonical model instead relative tangent sheaf $\mathcal T_{X/X_{can} }$? $\endgroup$ – user21574 Apr 23 '17 at 19:48
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    $\begingroup$ see short survey paper of Demailly www-fourier.ujf-grenoble.fr/~demailly/manuscripts/chern $\endgroup$ – user21574 Apr 23 '17 at 20:26
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    $\begingroup$ @Yury Ustinovskiy, I guess you are confusing about my notation, first chern class of $c_1(TX_s)$ where $X_s$ is Calabi-Yau is zero, but $c_1(T_{X/X_{can}})$ in general is not zero even fibers be Calabi-Yau, and depence to hermitian metric and by choosing fiberwise Calabi-Yau form, such first Chern class of relative tangent sheaf is Weil-Petersson metric for instance. So I think my notation is correct $\endgroup$ – user21574 Apr 23 '17 at 21:29

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