semiample of canonical bundle in a smooth family (Campana's proof)

Question

The following lemma is due to Campana, The class $\mathcal C$ is not stable by small deformations

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.

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.

Answer 1

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 $t$.