TITLE: Do we know any examples of complex surfaces where we have explicit knowledge of the Chern–Weil functions?
QUESTION [8 upvotes]: Let $X$ be a compact complex surface (smooth). Let $\gamma_1, \gamma_2$ denote the Chern–Weil functions. That is, if $\omega$ is a Kähler form on $X$ with volume form $\omega^2$, then $\gamma_1, \gamma_2$ are functions such that $$c_1(X) = \int_X \gamma_1 \omega^2, \quad c_2(X)=\int_X\gamma_2\omega^2.$$ In other words, the Chern–Weil functions are the functions for which the Chern classes are given by their $L^1$-norms. We have a good list of examples where we know constraints of the form $$a c_1^2(X) \leq b c_2(X) \leq k c_1^2(X)$$ for $a,b,k\in \mathbb{R}$. I'm asking for something stronger; namely:

Do we know concrete examples of surfaces for which $\gamma_1^2$ and $\gamma_2$ are known?

If you give me, for instance, a Barlow surface, or Beauville surface of some type, or some other surface, can we get information on the relations between $\gamma_1^2$ and $\gamma_2$? I do not mean "well, you can't have this relation, because integrating it would violate the Chern class inequality" — I'm not asking for non-relations, I'm asking for relations on the Chern–Weil functions.
I'd be very surprised and pleased if such knowledge exists, but I'm doubtful.

REPLY [6 votes]: If you know the volume form, then you are asking for explicit formulas for the Chern-Weil representatives of $c_1$ and $c_2$. These would come from explicit formulas for the curvature. The calculation of the curvature for a hypersurface of ${\mathbb C}P^3$, with respect to the (restriction of) the Fubini-Study metric was carried out by Al Vitter in On the curvature of complex hypersurfaces, Indiana Univ. Math. J. 23 (1973/74), 813–826."