## 50.9 First Chern class in de Rham cohomology

Let $X \to S$ be a morphism of schemes. There is a map of complexes

$\text{d}\log : \mathcal{O}_ X^*[-1] \longrightarrow \Omega ^\bullet _{X/S}$

which sends the section $g \in \mathcal{O}_ X^*(U)$ to the section $\text{d}\log (g) = g^{-1}\text{d}g$ of $\Omega ^1_{X/S}(U)$. Thus we can consider the map

$\mathop{\mathrm{Pic}}\nolimits (X) = H^1(X, \mathcal{O}_ X^*) = H^2(X, \mathcal{O}_ X^*[-1]) \longrightarrow H^2_{dR}(X/S)$

where the first equality is Cohomology, Lemma 20.6.1. The image of the isomorphism class of the invertible module $\mathcal{L}$ is denoted $c^{dR}_1(\mathcal{L}) \in H^2_{dR}(X/S)$.

We can also use the map $\text{d}\log : \mathcal{O}_ X^* \to \Omega ^1_{X/S}$ to define a Chern class in Hodge cohomology

$c_1^{Hodge} : \mathop{\mathrm{Pic}}\nolimits (X) \longrightarrow H^1(X, \Omega ^1_{X/S}) \subset H^2_{Hodge}(X/S)$

These constructions are compatible with pullbacks.

Lemma 50.9.1. Given a commutative diagram

$\xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S }$

of schemes the diagrams

$\xymatrix{ \mathop{\mathrm{Pic}}\nolimits (X') \ar[d]_{c_1^{dR}} & \mathop{\mathrm{Pic}}\nolimits (X) \ar[d]^{c_1^{dR}} \ar[l]^{f^*} \\ H^2_{dR}(X'/S') & H^2_{dR}(X/S) \ar[l]_{f^*} } \quad \xymatrix{ \mathop{\mathrm{Pic}}\nolimits (X') \ar[d]_{c_1^{Hodge}} & \mathop{\mathrm{Pic}}\nolimits (X) \ar[d]^{c_1^{Hodge}} \ar[l]^{f^*} \\ H^1(X', \Omega ^1_{X'/S'}) & H^1(X, \Omega ^1_{X/S}) \ar[l]_{f^*} }$

commute.

Proof. Omitted. $\square$

Let us “compute” the element $c^{dR}_1(\mathcal{L})$ in Čech cohomology (with sign rules for Čech differentials as in Cohomology, Section 20.25). Namely, choose an open covering $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ such that we have a trivializing section $s_ i$ of $\mathcal{L}|_{U_ i}$ for all $i$. On the overlaps $U_{i_0i_1} = U_{i_0} \cap U_{i_1}$ we have an invertible function $f_{i_0i_1}$ such that $f_{i_0i_1} = s_{i_1}|_{U_{i_0i_1}} s_{i_0}|_{U_{i_0i_1}}^{-1}$1. Of course we have

$f_{i_1i_2}|_{U_{i_0i_1i_2}} f_{i_0i_2}^{-1}|_{U_{i_0i_1i_2}} f_{i_0i_1}|_{U_{i_0i_1i_2}} = 1$

The cohomology class of $\mathcal{L}$ in $H^1(X, \mathcal{O}_ X^*)$ is the image of the Čech cohomology class of the cocycle $\{ f_{i_0i_1}\}$ in $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{O}_ X^*)$. Therefore we see that $c_1^{dR}(\mathcal{L})$ is the image of the cohomology class associated to the Čech cocycle $\{ \alpha _{i_0 \ldots i_ p}\}$ in $\text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \Omega _{X/S}^\bullet ))$ of degree $2$ given by

1. $\alpha _{i_0} = 0$ in $\Omega ^2_{X/S}(U_{i_0})$,

2. $\alpha _{i_0i_1} = f_{i_0i_1}^{-1}\text{d}f_{i_0i_1}$ in $\Omega ^1_{X/S}(U_{i_0i_1})$, and

3. $\alpha _{i_0i_1i_2} = 0$ in $\mathcal{O}_{X/S}(U_{i_0i_1i_2})$.

Suppose we have invertible modules $\mathcal{L}_ k$, $k = 1, \ldots , a$ each trivialized over $U_ i$ for all $i \in I$ giving rise to cocycles $f_{k, i_0i_1}$ and $\alpha _ k = \{ \alpha _{k, i_0 \ldots i_ p}\}$ as above. Using the rule in Cohomology, Section 20.25 we can compute

$\beta = \alpha _1 \cup \alpha _2 \cup \ldots \cup \alpha _ a$

to be given by the cocycle $\beta = \{ \beta _{i_0 \ldots i_ p}\}$ described as follows

1. $\beta _{i_0 \ldots i_ p} = 0$ in $\Omega ^{2a - p}_{X/S}(U_{i_0 \ldots i_ p})$ unless $p = a$, and

2. $\beta _{i_0 \ldots i_ a} = (-1)^{a(a - 1)/2} \alpha _{1, i_0i_1} \wedge \alpha _{2, i_1 i_2} \wedge \ldots \wedge \alpha _{a, i_{a - 1}i_ a}$ in $\Omega ^ a_{X/S}(U_{i_0 \ldots i_ a})$.

Thus this is a cocycle representing $c_1^{dR}(\mathcal{L}_1) \cup \ldots \cup c_1^{dR}(\mathcal{L}_ a)$ Of course, the same computation shows that the cocycle $\{ \beta _{i_0 \ldots i_ a}\}$ in $\check{\mathcal{C}}^ a(\mathcal{U}, \Omega _{X/S}^ a))$ represents the cohomology class $c_1^{Hodge}(\mathcal{L}_1) \cup \ldots \cup c_1^{Hodge}(\mathcal{L}_ a)$

Remark 50.9.2. Here is a reformulation of the calculations above in more abstract terms. Let $p : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If we view $\text{d}\log$ as a map

$\mathcal{O}_ X^*[-1] \to \sigma _{\geq 1}\Omega ^\bullet _{X/S}$

then using $\mathop{\mathrm{Pic}}\nolimits (X) = H^1(X, \mathcal{O}_ X^*)$ as above we find a cohomology class

$\gamma _1(\mathcal{L}) \in H^2(X, \sigma _{\geq 1}\Omega ^\bullet _{X/S})$

The image of $\gamma _1(\mathcal{L})$ under the map $\sigma _{\geq 1}\Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}$ recovers $c_1^{dR}(\mathcal{L})$. In particular we see that $c_1^{dR}(\mathcal{L}) \in F^1H^2_{dR}(X/S)$, see Section 50.7. The image of $\gamma _1(\mathcal{L})$ under the map $\sigma _{\geq 1}\Omega ^\bullet _{X/S} \to \Omega ^1_{X/S}[-1]$ recovers $c_1^{Hodge}(\mathcal{L})$. Taking the cup product (see Section 50.7) we obtain

$\xi = \gamma _1(\mathcal{L}_1) \cup \ldots \cup \gamma _1(\mathcal{L}_ a) \in H^{2a}(X, \sigma _{\geq a}\Omega ^\bullet _{X/S})$

The commutative diagrams in Section 50.7 show that $\xi$ is mapped to $c_1^{dR}(\mathcal{L}_1) \cup \ldots \cup c_1^{dR}(\mathcal{L}_ a)$ in $H^{2a}_{dR}(X/S)$ by the map $\sigma _{\geq a}\Omega ^\bullet _{X/S} \to \Omega ^\bullet _{X/S}$. Also, it follows $c_1^{dR}(\mathcal{L}_1) \cup \ldots \cup c_1^{dR}(\mathcal{L}_ a)$ is contained in $F^ a H^{2a}_{dR}(X/S)$. Similarly, the map $\sigma _{\geq a}\Omega ^\bullet _{X/S} \to \Omega ^ a_{X/S}[-a]$ sends $\xi$ to $c_1^{Hodge}(\mathcal{L}_1) \cup \ldots \cup c_1^{Hodge}(\mathcal{L}_ a)$ in $H^ a(X, \Omega ^ a_{X/S})$.

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 (50.4.0.1) 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,

[1] The Čech differential of a $0$-cycle $\{ a_{i_0}\}$ has $a_{i_1} - a_{i_0}$ over $U_{i_0i_1}$.

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