The Stacks project

Remark 50.9.3. Let $p : X \to S$ be a morphism of schemes. For $i > 0$ denote $\Omega ^ i_{X/S, log} \subset \Omega ^ i_{X/S}$ the abelian subsheaf generated by local sections of the form

\[ \text{d}\log (u_1) \wedge \ldots \wedge \text{d}\log (u_ i) \]

where $u_1, \ldots , u_ n$ are invertible local sections of $\mathcal{O}_ X$. For $i = 0$ the subsheaf $\Omega ^0_{X/S, log} \subset \mathcal{O}_ X$ is the image of $\mathbf{Z} \to \mathcal{O}_ X$. For every $i \geq 0$ we have a map of complexes

\[ \Omega ^ i_{X/S, log}[-i] \longrightarrow \Omega ^\bullet _{X/S} \]

because the derivative of a logarithmic form is zero. Moreover, wedging logarithmic forms gives another, hence we find bilinear maps

\[ \wedge : \Omega ^ i_{X/S, log} \times \Omega ^ j_{X/S, log} \longrightarrow \Omega ^{i + j}_{X/S, log} \]

compatible with ( and the maps above. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Using the map of abelian sheaves $\text{d}\log : \mathcal{O}_ X^* \to \Omega ^1_{X/S, log}$ and the identification $\mathop{\mathrm{Pic}}\nolimits (X) = H^1(X, \mathcal{O}_ X^*)$ we find a canonical cohomology class

\[ \tilde\gamma _1(\mathcal{L}) \in H^1(X, \Omega ^1_{X/S, log}) \]

These classes have the following properties

  1. the image of $\tilde\gamma _1(\mathcal{L})$ under the canonical map $\Omega ^1_{X/S, log}[-1] \to \sigma _{\geq 1}\Omega ^\bullet _{X/S}$ sends $\tilde\gamma _1(\mathcal{L})$ to the class $\gamma _1(\mathcal{L}) \in H^2(X, \sigma _{\geq 1}\Omega ^\bullet _{X/S})$ of Remark 50.9.2,

  2. the image of $\tilde\gamma _1(\mathcal{L})$ under the canonical map $\Omega ^1_{X/S, log}[-1] \to \Omega ^\bullet _{X/S}$ sends $\tilde\gamma _1(\mathcal{L})$ to $c_1^{dR}(\mathcal{L})$ in $H^2_{dR}(X/S)$,

  3. the image of $\tilde\gamma _1(\mathcal{L})$ under the canonical map $\Omega ^1_{X/S, log} \to \Omega ^1_{X/S}$ sends $\tilde\gamma _1(\mathcal{L})$ to $c_1^{Hodge}(\mathcal{L})$ in $H^1(X, \Omega ^1_{X/S})$,

  4. the construction of these classes is compatible with pullbacks,

  5. add more here.

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