For standard flips, the extremal quantum spectrum is shown to be compatible with the Gamma-class decomposition and with the semi-orthogonal decompositions of Orlov and Belmans-Fu-Raedschelders.
Gromov-Witten invariants in family and quantum cohomology
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abstract
A moduli space of stable maps to the fibers of a fiber bundle is constructed. The new moduli space is a family version of the classical moduli space of stable maps to a non-singular complex projective variety. The virtual cycle for this moduli space is also constructed, and an analogue of Gromov-Witten invariants is defined. As an application, we recover the formula for the number of rational degree d curves in P3, whose image lies in a plane in P3 (known as planar curves in P3), intersecting r general lines while passing through given s general points, where r + 2s = 3d + 2, firstly proved by R. Mukherjee, R. Kumar Singh and the fourth named author.
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Quantum spectrum and Gamma structure for standard flips
For standard flips, the extremal quantum spectrum is shown to be compatible with the Gamma-class decomposition and with the semi-orthogonal decompositions of Orlov and Belmans-Fu-Raedschelders.