I have the following question. Let $U\subset X$ be an open subset of $X$ such that the complement $X\setminus U$ has codimension $\ge2$ in $X$. Suppose $L$ is a line bundle on $U$ such that $c_1(L)^2=0$. Now let $j:U\to X$ be the inclusion map and let $L'=j_*L$ be the extension of $L$ as a reflexive sheaf over $X$. Is it true that $c_1(L')^2=0$?
Any suggestions/comments are appreciated, thanks!
The answer is no. For instance, let $X \subset \mathbb{P}^3$ be a smooth quintic surface, let $Z \subset X$ be the intersection of $X$ with a general line (so, this is a finite scheme of length 5), let $U = X \setminus Z$, and let $L$ be the restriction of $\mathcal{O}_{\mathbb{P}^3}(1)$. Then $c_1(L)^2 = 0$ but $c_1(L')^2 = 5$.