Gross-Siebert Mirror Ring for Smooth log Calabi-Yau Pairs
In this paper, we exhibit a formula relating punctured Gromov-Witten invariants used by Gross and Siebert in [GS2] to 2-point relative/logarithmic Gromov-Witten invariants with one point-constraint for any smooth log Calabi-Yau pair (W, D). Denote by Na,b the number of rational curves in W meeting D in two points, one with contact order a and one with contact order b with a point constraint. (Such numbers are defined within relative or logarithmic Gromov-Witten theory). We then apply a modified version of deformation to the normal cone technique and the degeneration formula developed in [KLR] and [ACGS1] to give a full understanding of Ne−1,1 with D nef where e is the intersection number of D and a chosen curve class. Later, by means of punctured invariants as auxiliary invariants, we prove, for the projective plane with an elliptic curve (P2, D), that all standard 2-pointed, degree d, relative invariants with a point condition, for each d, can be determined by exactly one of these degree d invariants, namely N3d−1,1, plus those lower degree invariants. In the last section, we give full calculations of 2-pointed, degree 2, one-point-constrained relative Gromov-Witten invariants for (P2, D).