A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups

Abstract

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 $W/{\Bbb A}^1$ and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair $(Y,D)$. As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber $W_t$ of $W/{\Bbb A}^1$ to the Gromov-Witten invariants of the singular fiber $W_0=Y_1\cup_D Y_2$ via gluing the relative pairs $(Y_1,D)$ and $(Y_2,D)$. The finite sum mentioned above depends on a relative ample line bundle $H$ on $W/{\Bbb A}^1$. 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 $\beta$ in $A_{\ast}(W_t)$ or $H_2(W_t;{\Bbb Z})$, rather than on the line bundle $H$. Some monodromy effect has to be taken care of to deal with this. For the simple but useful case of a degeneration $W/{\Bbb A}^1$ that arises from blowing up a trivial family $X\times{\Bbb A}^1$, we explain how the details of Jun Li's work can be employed to reach such a desired degeneration formula. The related set $\Omega_{(g,k;\beta)}$ of admissible triples adapted to $(g,k;\beta)$ 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 $(g,k;\beta)$ and the normal bundle ${\cal N}_{Z/X}$ of the smooth subscheme $Z$ in $X$ to be blown up.