Infinitesimal Torelli theorem for double coverings of surfaces of general type

Let $X$ be a smooth complex projective surface subject to the following conditions:

(i)

the canonical divisor $K_X$ of $X$ is ample,

(ii)

the irregularity $q(X) = h^1(\mathcal{O}_X) =0$ and $p_g (X) =h^0 (\mathcal{O}_X (K_X)) \geq 2$,

(iii)

the canonical linear system $\mid K_X\mid$ contains a reduced irreducible divisor.

It is shown that if $K^2_X \geq 5$, then the Infinitesimal Torelli theorem holds for a double covering of $X$ branched along a smooth divisor in the linear system $\mid 2K_X\mid$.