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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 118 / 1 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-118/1/d/solution