Yau-Uhlenbeck inequality works for higher Chern class? If the holomorphic vector bundle of $E$ on a Kähler manifold $M$ admit Hermitian-Yang-Mills metric then we have the following known universal inequality of Yau-Uhlenbeck $$\int_Mc_2(End E)\wedge \omega^{n-2}\geq 0$$
Is it known for other chern classes $c_{2n}$?
$$\int_Mc_{2k}(End E)\wedge \omega^{n-2k}\geq 0$$
We know
$ch(E) = r + c_1(E) + 1/2(c_1(E)^2 − 2c_2(E)) + · · · ,$
and 
$$ch(End(E)) = ch(E ⊗ E^∗
) = ch(E)ch(E^∗
)
= (r + ch_1(E) + ch_2(E) + · · ·)(r − ch_1(E) + ch_2(E) − · · ·)
= (r^2 + 2r · ch_2(E) − (ch_1(E))^2 + · · ·).$$
We obtain,
$$c_1(End(E)) = 0, c_2(End(E)) = 2rc_2(E) − (r − 1)c_1(E)^
2,...$$
 A: This is not true, as we can check by calculating the intersection number $\int_X c_4(\operatorname{End}(T_X)) \cup \omega^{n-4}$ for some easy examples of Kahler-Einstein spaces:
This intersection number seems to be always positive for $\operatorname{End}(T_{\mathbb{P}^n})$ by numerical calculations. There it is given by a degree-$5$ polynomial in $n$ that has a positive leading coefficient and is postive for the first 10 million $n$.
For a Calabi-Yau hypersurface $X \subset \mathbb{P}^{n+1}$, that is, a hypersurface of degree $n+2$, and the Kahler class of the restriction of the Fubini-Study metric, the intersection number is negative for the first few thousand dimensions $n$ and seems to decrease with $n$.
For a hypersurface $X \subset \mathbb{P}^{n+1}$ with ample canonical bundle, that is, a hypersurface of degree $> n+2$, this intersection number can be either positive or negative. For example, for hypersurfaces in $5$-dimensional projective space, it is negatives in degrees $6, 7, 8, 9$ and positive in degrees $10$-$1000$.
This follows by noting that we have
$$
12 c_4(\operatorname{End} E)
= (r-1) c_1(E)^4
-(r-1) c_1(E)^2c_2(E)
+(r+1) c_2(E)^2
+(r-1) c_1(E)c_3(E)
-r c_4(E)
$$
for a holomorphic vector bundle $E \to X$ of rank $r$, and recalling that the Chern classes of projective space are
$$
c_k(\mathbb{P}^n) = \binom{n}{k} h^k,
$$
where $h$ is the hyperplane class, and the Chern classes of a hypersurface $X$ of degree $d$ in $(n+1)$-dimensional projective space satisfy
$$
c_{k+1}(X) = \binom{n+1}{k+1} h^{k+1} - d c_k(X) \cup h.
$$
We can then simply have a computer crunch through the intersection numbers
$$
p(n,d) := 12 \int_X c_4(\operatorname{End} T_X) \cup h^{n-4} \Bigm/ \int_X h^n
$$
and tell us when they are positive or negative. This is done here.
