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If X is a nonsingular curve in a Calabi--Yau threefold Y whose normal bundleN_{X/Y} is a generic semistable bundle, are the local Gromov-Witten invariantsof X well defined? For X of genus two or higher, the issues are subtle. We willformulate a precise line of inquiry and present some results, some positive andsome negative.Comment: This is the version published by Geometry & Topology Monographs on 22  April 200

On the rigidity of stable maps to Calabi–Yau threefolds