The formulation of Virasoro constraints for compact Kahler manifolds yields an identity involving genus 1 Gromov-Witten invariants
and Hodge numbers. Together with exact evaluations of these Gromov-Witten invariants this yields an identity involving Chern numbers and Hodge numbers, which was proven by Libgober and Wood (in an unrelated work). We'll explain how this works, and how to deal with this when one tries to formulate Virasoro constraints in orbifold Gromov-Witten theory.