On the Virasoro Constraints for the Product Space <i>X × C</i>

Let [Formula: see text] be a smooth projective variety with [Formula: see text] = 0, and let [Formula: see text] be a smooth projective curve of genus [Formula: see text]. We study the Virasoro constraints for the fiber-degreezero sector of the Gromov–Witten theory of [Formula: see text]. We prove that, if the full-genus Virasoro constraints hold for [Formula: see text], then the full-genus fiber-degree-zero Virasoro constraints hold for [Formula: see text]. As an application, the full Gromov–Witten theory of [Formula: see text] satisfies the Virasoro constraints through genus [Formula: see text].