Hodge integrals and ${\lambda }_g$ conjecture with target varieties

In this paper, we propose ${\lambda }_g$ conjecture for Hodge integrals with target varieties. Then we establish relations between Virasoro conjecture and ${\lambda }_g$ conjecture, in particular, we prove ${\lambda }_g$ conjecture in all genus for smooth projective varieties with semisimple quantum cohomology or smooth algebraic curves. Meanwhile, we also prove ${\lambda }_g$ conjecture in genus zero for any smooth projective varieties. In the end, together with DR formula for ${\lambda }_g$ class, we obtain a new type of universal constraints for descendant Gromov-Witten invariants. As an application, we prove ${\lambda }_g$ conjecture in genus one for any smooth projective varieties.