Lemma 50.15.6. Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \subset X$ over $S$. Let $b \in H^ m_{dR}(X/S)$ be a de Rham cohomology class whose restriction to $Y$ is zero. Then $c_1^{dR}(\mathcal{O}_ X(Y)) \cup b = 0$ in $H^{m + 2}_{dR}(X/S)$.
Proof. This follows immediately from Lemma 50.15.5. Namely, we have
\[ c_1^{dR}(\mathcal{O}_ X(Y)) \cup b = -c_1^{dR}(\mathcal{O}_ X(-Y)) \cup b = -\xi '(b) = -\delta (b|_ Y) = 0 \]
as desired. For the second equality, see Remark 50.4.3. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: