Deformations of canonical double covers

In this paper, we show that if $X$ is a smooth variety of general type of dimension $m \geq 2$, for which its canonical map induces a double cover onto $Y$, where $Y$ is a projective bundle over $\mathbf P^1$, or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series, then the general deformation of the canonical morphism of $X$ again is canonical and again induces a double cover. The second part of the article deals with the existence or non existence of canonical double structures on rational varieties. The negative result in this article has consequences for the moduli of varieties of general type of arbitrary dimension. The results here show that there is an entire component, that is hyperelliptic in infinitely many moduli spaces of higher dimensional varieties of general type. This is in sharp contrast with the case of curves or surfaces of lower Kodaira dimensions.