# Sign of $\int_X\operatorname{Tr}(F_h^2)$

Let $$(E,h)\to X$$ be a holomorphic Hermitian vector bundle over a compact Kähler manifold. Denote by $$F_h$$ the curvature of its Chern connection. Can we know a priori the sign of the quantity $$\int_X\operatorname{Tr}(F_h^2)\wedge \omega^{n-2}$$
or if $$X$$ is a surface the sign of $$\int_X\operatorname{Tr}(F_h^2)?$$

The longer answer: The $$(2,2)$$-form $$\operatorname{Tr}(F_h^2)$$ represents the second Chern class of $$E$$, or $$c_2(E)$$. In general, the integral of that class over the manifold doesn't have a sign independent of the bundle. What we do have in general is the pointwise identity $$\left| \frac{i}{2\pi} F_h \right|^2 \omega^{n}/n! = (2 c_2(E,h) - c_1(E,h)^2) \wedge \omega^{n-2} / (n-2)! + \left| \operatorname{Tr}_{\omega} \frac{i}{2\pi} F_h \right|^2 \omega^n / n!,$$ where $$c_k(E,h)$$ is the representative of the $$k$$-th Chern class of $$E$$ defined by the curvature form of the metric. We don't need $$\omega$$ to be Kahler for this to hold, nor $$X$$ to be compact; this holds on general Hermitian manifolds. (This is known, and usually implicit somewhere in any differential-geometric proof of the Kobayashi-Lubke inequality. See here for one proof.)
To be able to say anything more I think we have to assume that $$\omega$$ is Kahler and $$E = T_X$$, in which case $$\operatorname{Tr}_\omega \frac{i}{2\pi} F_h = c_1(T_X,\omega)$$. The above then reduces to $$\left| \frac{i}{2\pi} F_h \right|^2 \omega^{n}/n! = 2 c_2(T_X,\omega) \wedge \omega^{n-2} / (n-2)! + s(\omega)^2 \omega^n / n!,$$ where $$s(\omega)$$ is the scalar curvature of $$\omega$$. This still doesn't force $$c_2(T_X,\omega) \wedge \omega^{n-2}$$ to have a definite sign unless we assume the scalar curvature of $$\omega$$ is constantly zero, in which case $$\omega$$ will have to be a Ricci-flat Kahler-Einstein metric. In that case, yes, $$c_2(T_X,\omega) \wedge \omega^{n-2} \geq 0$$ with strict inequality unless the metric is flat.
• Shouldn't it be $Tr(F^2_h)=c_1^2(E)-c_2(E)$? Commented Dec 29, 2021 at 5:14