A degeneration formula of GW-invariants. (English) Zbl 1063.14069

Li, Jun

J. Differ. Geom. 60, No. 2, 199-293 (2002).

This paper is the sequel to [J. Li, J. Differ. Geom. 57, No. 3, 509–578 (2001; Zbl 1076.14540)]. In the first part of this paper the author uses the moduli of relative stable morphisms, constructed in the first paper, to construct the Gromov-Witten invariants of a singular scheme \(W_0\), consisting of two smooth irreducible components \(Y_1\) and \(Y_2\), intersecting transversally along a connected smooth divisor \(D\in W_0\). The author also constructs the relative Gromov-Witten invariants of the pairs \((Y_1,D)\) and \((Y_2,D)\).
In the second half of this paper the author gives a degeneration formula for these invariants in the situation where \(W \to T\) is a one-parameter family with smooth total space \(W\) and smooth generic fibers, but with \(W_0\) as the fiber over \(0\in C\). This formula is analogous to that of A.-M. Li and Y. Ruan [Invent. Math. 145, 151–218 (2001; Zbl 1062.53073)] and E. N. Ionel and T. H. Parker [Ann. Math. (2) 159, No. 3, 935–1025 (2004; Zbl 1075.53092)].

Reviewer: Tyler J. Jarvis (Provo)

MSC:

14N35

Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

Keywords:

relative Gromov-Witten invariants

Citations:

Zbl 1076.14540; Zbl 1062.53073; Zbl 1075.53092