Stability conditions on product varieties

Given a stability condition on a smooth projective variety $X$, we construct a family of stability conditions on $X\times C$, where $C$ is a smooth projective curve. In particular, this gives the existence of stability conditions on arbitrary products of curves. The proof uses, by following an idea of Toda, the positivity lemma established by Bayer and Macr\`i and weak stability conditions on the Abramovich-Polishchuk heart of a bounded t-structure in $D(X\times C)$.