Let $(\mathcal{X},\mathcal L)$ be a deformation over a (smooth) base $B$ of the pair $(X,\mathcal O_X)$ where $X$ is a smooth projective variety (over $\mathbb C$).
Is the class $c_1(\mathcal L_b)\in \mathrm{NS}(X_b)$ always trivial?
(we can assume the $X_b$ to be projective).
$\begingroup$
$\endgroup$
4
-
5$\begingroup$ Assuming $B$ is connected, then yes, because the $C^\infty$ structure of the pair $(X_b, L_b)$ won't change. $\endgroup$– Donu ArapuraNov 15, 2021 at 18:53
-
2$\begingroup$ By the way, this is false over a regular base scheme of mixed characteristic, cf. Enriques surfaces that are "singular" and "supersingular". $\endgroup$– Jason StarrNov 16, 2021 at 12:20
-
$\begingroup$ @Donu Arapura Thank you very much for your answer. $\endgroup$– pi_1Nov 16, 2021 at 17:13
-
$\begingroup$ @Jason Starr Thanks also for the information. $\endgroup$– pi_1Nov 16, 2021 at 19:47
Add a comment
|