Let $X\_0$ be a smooth projective variety over the complex numbers and let $X$ be an infinitesimal deformation of $X\_0$ over the ring of dual numbers.


If the canonical bundle of $X\_0$ is ample (resp. anti-ample), does it follow that the canonical bundle of $X$ over $\mathbb C[\epsilon]$ 
is ample (resp. anti-ample)?



The canonical bundle of an infinitesimal deformation
This is true in a much more general setting:



> 
> **Fact** Let $X$ be a proper variety over a noetherian ring $A$ and let $\mathscr L$ be a line bundle on $X$. Then $\mathscr L$ is ample if and only if $\mathscr L\_{\mathrm{red}}\simeq \mathscr L\otimes \mathscr O\_{X\_{\mathrm{red}}}\ $ is ample on $X\_{\mathrm{red}}$. 
> 
> 
> 


See Exc.III.5.7(b) in [Hartshorne] for the proper case and EGA II, 4.5.13 without the proper assumption (thanks to @Laurent Moret-Bailly for the EGA reference).


In your case $X\_0\simeq X\_{\mathrm{red}}$ and $\omega\_{X\_0}\simeq \left(\omega\_{X} \right)\_{\mathrm{red}}\simeq \left(\omega\_{X/\mathbb C[\varepsilon]}\ \ \right)\_{\mathrm{red}}$ .


