Flops, motives and invariance of quantum rings

Details Flops, motives and invariance of quantum rings Flops, motives and invariance of quantum rings

2006-08-15

arxiv.org/abs/math/0608370v2

Annals of Math. 172 (2010), no.1, 243-290

Mathematics

Algebraic Geometry

51 pages, 2 figures, final version to appear in Annals of Mathematics

Scientific paper

For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincar\'e pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended K\"ahler moduli space. For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish.

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