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

الأصل · EN

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 Y₁∪D Y₂ that arises from a degeneration W/ A¹ 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 Wₜ of W/ A¹ to the Gromov-Witten invariants of the singular fiber W₀=Y₁∪D Y₂ via gluing the relative pairs (Y₁,D) and (Y₂,D). The finite sum mentioned above depends on a relative ample line bundle H on W/ A¹. 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 β in A(Wₜ) or H₂(Wₜ; 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/ A¹ that arises from blowing up a trivial family X× A¹, we explain how the details of Jun Li's work can be employed to reach such a desired degeneration formula. The related set Ω₍g,ₖ;ᵦ₎ of admissible triples adapted to (g,k;β) 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;β) and the normal bundle NZ/ₓ of the smooth subscheme Z in X to be blown up.