Oguiso writes[1]



> 
> **Theorem 1.1** *Let $f: X \to \mathbf P^n$ be an abelian fibered HK [hyperkähler] manifold. Let $K = \mathbf C(\mathbf P^n)$ and let $A\_k$ be the generic fiber of $f$. Then, $\rho(A\_K)= 1$. Here $\rho(A\_K)$ is the Picard number of $A\_K$ over $K$.*
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> It can happen that $\rho(X\_t) \geq 2$ for all smooth closed fiber of $f$ ([...]). The statement is of arithmetical nature. Geometrically, it means that two horizontal divisors on $X$ are proportional in $NS(X)$ up to vertical divisors.
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A divisor is called horizontal if it dominates $\mathbf P^n$, otherwise it is called vertical.


I'm struggling with the last sentence of Oguiso. Is that correct? Given two line bundles $L\_1, L\_2$ on $X$, the theorem tells us that they are proportional in $NS(A\_K)$, so there exist $n,m \in \mathbb Z$ such that
$$ n L\_1|\_{A\_K} + m L\_2|\_{A\_K} = 0 \in NS(A\_K).$$
I guess that Oguiso then alludes to an exact sequence of the kind
$$Z^0(X \setminus U) \to NS(X) \to NS(U) \to 0,$$
where $U = f^{-1}(V)$ is the preimage of an open set $V \subset \mathbf P^n$, and $Z^0(X\setminus U)$ is the free group generated by the irreducible components of $X \setminus U$.


But I don't see a reason why $nL\_1 + mL\_2$ should vanish in $NS(U)$, for appropriate $U$.
[As far as I know](https://math.stackexchange.com/questions/4745392/line-bundles-which-are-fiber-wise-algebraically-trivial), a line bundle being trivial in the Neron-Severi group of all fibers does not mean that it is trivial in the total Neron-Severi group.


Did I miss anything here?




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[1] Keiji Oguiso, *Picard number of the generic fiber of an abelian fibered hyperkähler manifold*, 2009, [arXiv:0803.1205](https://arxiv.org/abs/0803.1205)



Are horizontal divisors on abelian fibered hyperkähler manifolds proportional in $NS(X)$ up to vertical divisors?
This is simply false, already for K3 surfaces with an elliptic fibration. The quotient of $NS(X)$ by the subgroup generated by the vertical components + the zero section is the Mordell-Weil group of the generic fiber, it can very well be nontrivial. See for instance *Elliptic surfaces* by Schütt and Shioda, Adv.Stud. Pure Math. 60.


