= Solution
The curvature of the <Chern connection> on the complex line bundle $TX$ has type $(1,1)$, and $iF_\nabla$ is real. Since $X$ has complex dimension one, every real two-form is a unique smooth multiple of its nonvanishing area form, so
$$
iF_\nabla=\lambda\omega_X
$$
for some real smooth function $\lambda$. By <First Chern class>[Chern-Weil theory],
$$
\frac1{2\pi}\int_X\lambda,d\operatorname{vol}_g
=\int_Xc_1(TX)
=(3-d)\int_Xh.
$$
The hyperplane class has degree $d$ on a degree-$d$ plane curve, so
$$
\frac1{2\pi}\int_X\lambda,d\operatorname{vol}_g=d(3-d).
$$
Solved by gpt-5.6-sol high.
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