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Suppose that $X$ is a smooth projective variety of dimension at least $3$, and that $D$ is a smooth ample divisor. I am wondering to about the status of the Lefschetz hyperplane theorem for the map $\operatorname{NS}(X) \to \operatorname{NS}(D)$ on Neron-Severi groups.

It seems to me that if I work over $\mathbb C$, and ignore torsion, then $\operatorname{NS}_{\mathbb R}(X) \to \operatorname{NS}_{\mathbb R}(D)$ is injective; this should follow from the theorem for $H^{1,1}$ and the Lefschetz $(1,1)$ theorem. (Is this right?)

What happens if I do care about torsion, or I'm working over a different field? I'm especially interested in the borderline case $\dim X = 3$, when we won't get an isomorphism.

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  • $\begingroup$ The Lefschetz $(1,1)$-theorem is valid with $\mathbb{Z}$-coefficients, so it also works if you do care about torsion. I am not sure what happens in other characteristics, but I believe your conjecture is correct (i.e., the map is injective). $\endgroup$
    – jmc
    Mar 24, 2015 at 14:49
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    $\begingroup$ You don't need the Lefschetz (1,1) theorem, it suffices to know that $H^2(X,\mathbb{Z})\rightarrow H^2(D,\mathbb{Z})$ is injective (Lefschetz hyperplane theorem) and that $\mathrm{NS}(X)$ injects into $H^2(X,\mathbb{Z})$. $\endgroup$
    – abx
    Mar 24, 2015 at 14:57
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    $\begingroup$ Hard to argue with that -- thanks all. (If anybody wants to post it this an answer I'll accept it.) $\endgroup$
    – user47305
    Mar 24, 2015 at 16:29

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