Torelli theorem for the product of moduli spaces of vector bundles over a curve

We prove a Torelli-type theorem for a product of the moduli spaces of semistable vector bundles over smooth projective curves of genus $g\ge 4$. A similar result is proved for the product of the moduli stacks of semistable vector bundles. This is proved using a decomposition theorem for the product of the normal projective varieties with Picard rank one and discrete Picard group. As an application, we compute the automorphism group of the product of the moduli spaces and moduli stacks.