On the rigidity of stable maps to Calabi-Yau threefolds

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If X is a nonsingular curve in a Calabi--Yau threefold Y whose normal bundle N_{X/Y} is a generic semistable bundle, are the local Gromov-Witten invariants of X well defined? For X of genus two or higher, the issues are subtle. We will formulate a precise line of inquiry and present some results, some positive and some negative.
This is the version published by Geometry & Topology Monographs on 22 April 2006

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