欧拉正合列

正式表述

${\displaystyle 0\to \Omega _{\mathbb {P} _{A}^{n}/A}^{1}\to {\mathcal {O}}_{\mathbb {P} _{A}^{n}}(-1)^{\oplus n+1}\to {\mathcal {O}}_{\mathbb {P} _{A}^{n}}\to 0}$

几何表述

${\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} ^{n}}\to {\mathcal {O}}(1)^{\oplus (n+1)}\to {\mathcal {T}}_{\mathbb {P} ^{n}}\to 0}$

${\displaystyle V}$${\displaystyle k}$上的n维向量空间，于是有

${\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} (V)}\xrightarrow {f} {\mathcal {O}}_{\mathbb {P} (V)}(1)\otimes V\xrightarrow {g} {\mathcal {T}}_{\mathbb {P} (V)}\to 0}$

射影空间的典范线丛

${\displaystyle \omega _{\mathbb {P} _{A}^{n}/A}={\mathcal {O}}_{\mathbb {P} _{A}^{n}}(-(n+1))}$

${\displaystyle {\text{det}}({\mathcal {E}})={\text{det}}({\mathcal {E}}')\otimes {\text{det}}({\mathcal {E}}'')}$

用于陈类的计算

${\displaystyle 0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0}$,

{\displaystyle {\begin{aligned}c(\Omega _{\mathbb {P} ^{2}}^{1})&={\frac {c({\mathcal {O}}(-1)^{\oplus (2+1)})}{c({\mathcal {O}})}}\\&=(1-[H])^{3}\\&=1-3[H]+3[H]^{2}-[H]^{3}\\&=1-3[H]+3[H]^{2}\end{aligned}}}

${\displaystyle 0\to \Omega ^{2}\to {\mathcal {O}}(-2)^{\oplus 3}\to \Omega ^{1}\to 0}$

{\displaystyle {\begin{aligned}c(\Omega ^{2})&={\frac {c({\mathcal {O}}(-2)^{\oplus 3})}{c(\Omega ^{1})}}\\&={\frac {(1-2[H])^{3}}{1-3[H]+3[H]^{2}}}\\\end{aligned}}}

注释

1. ^ Vakil, Ravi. Rising Sea (PDF). 386. （原始内容 (PDF)存档于2019-11-30）.
2. ^ 3264 and all that (PDF): 169.
3. ^ Note that ${\displaystyle [H]^{3}=0}$ in the chow ring for dimension reasons.
4. ^ Arapura, Donu. Computation of Some Hodge Numbers (PDF). （原始内容 (PDF)存档于1 February 2020）.