The following lemma is due to Campana, [The class $\mathcal C$ is not stable by small deformations](https://link.springer.com/article/10.1007/BF01459236)



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> Let $\mathcal X\rightarrow \Delta$ be a smooth family, if $K\_{X\_0}$ is nef and big, then so is every $K\_{X\_t}$ for $t$ sufficiently small.
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Here is the step. Invoking Kawamata-Viehweg vanishing theorem, one has
$$H^i(X\_0,NK\_{X\_0})=0$$
for $N\geq 2$. Then the similarly higher cohomology vanishing holds for $K\_{X\_t}$ by Grauert direct image theorem.
Therefore,
$$h^0(X\_t,NK\_{X\_t})=h^0(X\_0,NK\_{X\_0})\,\,\,\,\,\,(\*)$$ for $N\geq 2$ thanks to deformation invariance of Euler characteristic.
Also, the base point free theorem tells us that
$|NK\_{X\_0}|$ is globally generated for sufficiently big $N$.


Then the author claims that this fact combines (\*) can infer that
$|NK\_{X\_s}|$ is also globally generated for sufficiently big $N$ (this will imply the nefness of $X\_t$ then).
Why this holds? How the size of global sections determine the base locus with respect to $X\_t$ in our setting?


Any suggestions will be appreciated. Thanks in advance.



semiample of canonical bundle in a smooth family (Campana's proof)
Let $f:X\rightarrow \Delta $ be your family. $\ (\*)$ implies that $f\_\*K\_{X/\Delta }^{N}$ is a vector bundle on $\Delta $, with fiber $H^0(X\_t, K\_{X\_t}^N)$ at $t\in\Delta$. The canonical homomorphism $\ f^\*f\_\*K\_{X/\Delta }^{N}\rightarrow K\_{X/\Delta }^{N}$ is surjective on $X\_0$ by hypothesis, hence also on $X\_t$ for small $t$.


