리만 곡면 위의 계량 텐서

개요

\[ds^2=\lambda^2(z,\overline{z})\, dz\,d\overline{z}\] 여기서 λ는 양의 값을 갖는 $z$와 $\overline{z}$의 함수.

\[4\frac{\partial}{\partial z} \frac{\partial}{\partial \overline{z}} \Phi(z,\overline{z})=\lambda^2(z,\overline{z})\]

$$ \Delta=4\frac{\partial}{\partial z} \frac{\partial}{\partial \overline{z}}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} $$

$\Omega$ : curvature form
Chern class\[\det \left(\frac {it\Omega}{2\pi} +I\right) = \sum_k c_k(V) t^k\]
$c_k(V)=P^{i}\left(\frac{\sqrt{-1}}{2\pi}\Omega\right)\in H^{2i}(M,\mathbb{Z})$
Chern class of line bundles on the complex projective line
V = $T\mathbb{C}P^1$ : the complex tangent bundle of the complex projective line
Kahler metric\[g=\frac{dz\otimes d\bar{z}}{(1+|z|^2)^2}\]
포텐셜 $\log (1+|z|^2)=\log (1+z\bar{z})$\[\partial \bar\partial \log (1+|z|^2)=\frac{dz\wedge d\bar{z}}{(1+|z|^2)^2}\]
Curvature form\[\Omega=\frac{2dz\wedge d\bar{z}}{(1+|z|^2)^2}\]
$c_1=\frac{\sqrt{-1}}{2\pi}\Omega$
$c_1(V) \not= 0$ 의 증명 (V 가 trivial vector bundle이 아님을 알 수 있다)\[\int c_1=\frac{\sqrt{-1}}{2\pi}\int \frac{2 dz\wedge d\bar{z}}{(1+|z|^2)^2}=\frac{\sqrt{-1}}{2\pi}\left(\int_{0}^{2\pi}\int_0^{\infty} \frac{-4\sqrt{-1}r}{\left(1+r^2\right)^2} \, dr d\theta \right)=4\left(\int_0^{\infty } \frac{r}{\left(1+r^2\right)^2} \, dr\right)=2\]
평사 투영(stereographic projection)