Elliptic GW invariants of blowups along curves and surfaces

Over the last few years, many mathematicians contributed their efforts to establish the mathematical foundation of the theory of quantum cohomology or Gromov-Witten (GW) invariants. In 1995, Ruan and Tian [13, 15] first established for the semipositive symplectic manifolds. Recently, the semipositivity condition has been removed by many authors. Now, the focus turned to the calculations and applications. Many Fano manifolds were computed. We think that it is important to study the change of GW invariants under surgery. Some recent research indicated that there is a deep amazing relation between quantum cohomology and birational geometry. The quantum minimal model conjecture [14] leads to attempt to find quantum cohomology of a minimal model without knowing minimal model. This problem requires a thorough understanding of blowup type formula of GW invariants and quantum cohomology. According to McDuff [11], the blowup operation in symplectic geometry amounts to a removal of an open symplectic ball followed by a collapse of some boundary directions. Lerman [8] gave a generalization of blowup construction, “the symplectic cut.” Let M be a compact symplectic manifold of dimension 2n, M̃ the blowup of M along a smooth submanifold. Denote by p : M̃ →M the natural projection. Denote by Ψ(A,g)(α1, . . . ,αm) the genus g GW invariant of M. The authors refer the interested reader to [15] for the definition of GW invariants. In [5, 6], we mainly concentrated on the changes of genus zero GW invariants under blowup along smooth curves and surfaces. In this paper, we mainly concentrate on the elliptic GW invariants. These invariants were first discussed in [1] and since then have been studied in various contexts, see [3, 12]. An elegant recursion was predicted in [2] using the method of Virasoro constraints. In this paper, we will generalize some results