리만 곡면 위의 계량 텐서

개요

<math>ds^2=\lambda^2(z,\overline{z})\, dz\,d\overline{z}</math>

여기서 λ는 양의 값을 갖는 <math>z</math>와 <math>\overline{z}</math>의 함수.

<math>4\frac{\partial}{\partial z}

\frac{\partial}{\partial \overline{z}} \Phi(z,\overline{z})=\lambda^2(z,\overline{z})</math>

<math>

\Delta=4\frac{\partial}{\partial z} \frac{\partial}{\partial \overline{z}}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} </math>

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

X : 리만 곡면
holomorphic line bundle <math>H\to X</math> 에 대한 에르미트 metric <math>e^{-\phi}</math>