Extension of first order deformations of a line bundle

Question

Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line bundle on $V_{\varepsilon} = V \times Spec[\mathbb{C}[\varepsilon]/(\varepsilon ^2)$. If we denote by $j_{\varepsilon}: V_{\varepsilon} \hookrightarrow X_{\varepsilon} = X \times Spec(\mathbb{C}[\varepsilon]/(\varepsilon ^2))$ the natural inclusion, then is it true that $\tilde{L}_{\varepsilon} = (j_{\varepsilon})_* (L_{\varepsilon})$ defines a line bundle on $X_{\varepsilon}$ ? If not, is there any other approach to extend $L_{\varepsilon}$ to $X_{\varepsilon}$ ?

Answer 1

Under some conditions on $X,V$, your line bundle can be extended to $X_{\varepsilon}$. Indeed, let $\imath_X:X\hookrightarrow X_{\varepsilon}$ and $\imath_V:V\hookrightarrow V_{\varepsilon}$ be two closed immersions and $\mathcal{I}_X,\mathcal{I}_V$ be the ideal sheaves respectively. By the following exact sequence $$ 0\to \mathcal{I}_X \to \mathcal{O}^{\times}_{X_{\varepsilon}}\to \mathcal{O}^{\times}_X\to 0,$$ if we consider universal $\delta$-functor theory and a natural transformation $H^0(X,-)\to H^0(U,-)$, we have a diagram of two exact sequences $$ \begin{aligned}H^1(X,\mathcal{I}_X)&\to \mathrm{Pic}(X_{\varepsilon})\to \mathrm{Pic}(X)\to H^2(X,\mathcal{I}_X) \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\downarrow \alpha&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow\beta\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow \gamma\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\downarrow\delta \\ H^1(V,\mathcal{I}_V)&\to \mathrm{Pic}(V_{\varepsilon})\to \mathrm{Pic}(V)\to H^2(V,\mathcal{I}_V) \end{aligned}$$ (see Hartshorne Exercises 3.4.6). Note that in the case, $\mathcal{I}_X\cong \mathcal{O}_X$ as $\mathcal{O}_{X_{\varepsilon}}$-modules and $\mathcal{I}_V\cong \mathcal{O}_V$ respectively.

Since $X$ is smooth, $\gamma$ is an isomorphism. Hence, if $\alpha$ is a surjection and $\delta$ is an injection, then $\beta$ is a surjection by five lemma and any line bundle on $V_{\varepsilon}$ can be extended to $X_{\varepsilon}$. Note that considering the local cohomology exact sequence $$ H^1_Z(X,\mathcal{O}_X)\to H^1(X,\mathcal{O}_X)\to H^1(V,\mathcal{O}_V)\to H^2_Z(X,\mathcal{O}_X)\to H^2(X,\mathcal{O}_X)\to H^2(V,\mathcal{O}_V),$$ where $Z:=X\setminus U$, if $H^2_Z(X,\mathcal{O}_X)=0$, then $\beta$ is a surjection (it is worth saying that if $\mathrm{codim}_X Z\ge 3$, then $H^2_Z(X,\mathcal{O}_X)=0$ by SGA2, III, Proposition 3.3).