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 $K$.
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.
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, 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?
[1] Keiji Oguiso, Picard number of the generic fiber of an abelian fibered hyperkähler manifold, 2009, arXiv:0803.1205
