In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form Y_1\cup_D Y_2 that arises from a degeneration Y_1\cup_D Y_2 and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair Y_1\cup_D Y_2 . As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber Y_1\cup_D Y_2 of Y_1\cup_D Y_2 to the Gromov-Witten invariants of the singular fiber Y_1\cup_D Y_2 via gluing the relative pairs Y_1\cup_D Y_2 and Y_1\cup_D Y_2 . The finite sum mentioned above depends on a relative ample line bundle Y_1\cup_D Y_2 on Y_1\cup_D Y_2 . His theory has already applications to string theory and mathematics alike. For other new applications of Jun Li's theory, one needs a refined degeneration formula that depends on a curve class Y_1\cup_D Y_2 in Y_1\cup_D Y_2 or Y_1\cup_D Y_2 , rather than on the line bundle Y_1\cup_D Y_2 . Some monodromy effect has to be taken care of to deal with this. For the simple but useful case of a degeneration Y_1\cup_D Y_2 that arises from blowing up a trivial family Y_1\cup_D Y_2 , we explain how the details of Jun Li's work can be employed to reach such a desired degeneration formula. The related set Y_1\cup_D Y_2 of admissible triples adapted to Y_1\cup_D Y_2 that appears in the formula can be obtained via an analysis on the intersection numbers of relevant cycles and a study of Mori cones that appear in the problem. This set is intrinsically determined by Y_1\cup_D Y_2 and the normal bundle Y_1\cup_D Y_2 of the smooth subscheme Y_1\cup_D Y_2 in Y_1\cup_D Y_2 to be blown up.

For projective conifold transitions between Calabi-Yau threefolds X and Y, with X close to Y in the moduli, we show that the combined information provided by the A model (Gromov--Witten theory in all genera) and B model (variation of Hodge structures) on X, linked along the vanishing cycles, determines the corresponding combined information on Y. Similar result holds in the reverse direction when linked with the exceptional curves.

The paper is Part III of our ongoing project to study a case of Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth base. An essential ingredient in the proof is a quantum splitting principle, which reduces a statement in Gromov--Witten theory on non-split bundles to the case of split bundles.

We present a series of results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the <i>n</i>-point functions for Witten9s classes, an effective recursion formula to compute higher WeilPetersson volumes, several new recursion formulae of intersection numbers and our proof of a conjecture of Itzykson and Zuber concerning denominators of intersection numbers. We also present Virasoro and KdV properties of generating functions of general mixed and intersections.

We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda–Pandharipande–Pixton–Zvonkine, we construct non-trivial tautological relations parallel to Pixton’s double ramification cycle relations for the moduli of curves. Examples and applications are discussed.