Two questions about line bundles over Kuranishi families

Question

i'm studying the article "Variétés Kahleriennes dont la première classe de chern est nulle" by Arnaud Beauville and i have a couple of questions i would like to ask you, hoping they are not too trivial.

They are about Corollaire 1 and 2 at pages 776 and 777:

1) Let's say we have $X$ an irreducible symplectic manifold and $L$ an holomorphic not trivial line bundle on $X$. We have $f:\mathcal{X}\rightarrow B$ a Kuranishi family of $X$, with $f^{-1}(0)=X$. I think (i may be wrong) that this is sufficient for Beauville to say that if we have two line bundles $\mathcal{L}$ and $\mathcal{L}'$ on $\mathcal{X}$ such that $\mathcal{L}|_X=\mathcal{L}'|_X$, then we can say that $\mathcal{L}$ and $\mathcal{L}'$ differ by an element in $Pic(B)$.

Again i'm afraid this is trivial, but i can't understand this point.. if i restrict $\mathcal{L}$ and $\mathcal{L}'$ to another fiber $\mathcal{X}_b$ for me a priori the two bundles can differ by an element in $Pic(\mathcal{X}_b)$..

2) With the same hypothesis on $X$, in corollaire 2 Beauville says that if $l$ is the class of an ample line bundle $L$ on $X$ (which gives the embedding $X\rightarrow \mathbb{P}^N$) and $\mathcal{K}$ is the Hilbert scheme of $X$ in $\mathbb{P}^N$, with natural map $h:\mathcal{K}\rightarrow B$ ($B$ again is the base of the Kuranishi family), then the image of $h$ is contained in the set $\widetilde{B}:=\{b\in B| 0\neq{u_b^{-1}}^*(l)\in H^{1,1}(\mathcal{X}_b,\mathbb{Z})\}$.

Of course $u_b:X\rightarrow \mathcal{X}_b$ is the diffeomorphism between the fibers of the Kuranishi family and of course i think that the image of $h$ should be given by the points $b\in B$ for which $\mathcal{X}_b$ is projective.

On $\widetilde{B}$ i know by corollaire 1 that i can extend the line bundle $L$ to a line bundle $\mathcal{L}$ on $\mathcal{X}|_{\widetilde{B}}$, so now by Beauville it should follow that the restriction $\mathcal{L}|_{\mathcal{X}_b}$, $b\in\widetilde{B}$, should be ample. Why is that?

Thank you

Answer 1

1): The manifolds considered have $h^{1,0}=0$, so two line bundles $L$ and $L'$ on $\mathcal{X}_b$ are isomorphic if and only if $c_1(L)=c_1(L')$ in $H^2(\mathcal{X}_b,\mathbb{Z})$. If this holds at one point, it holds everywhere.

2): Being ample is an open property, so reducing $B$ if necessary you may indeed assume that $\mathcal{L}|_{\mathcal{X}_b}$ is ample for all $b$ in $\tilde{B} $.