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There are some Calabi-Yau threefolds that do not contain any rational curves, e.g. Calabi-Yau threefold of type A in the paper "Calabi-Yau threefolds of quotient type" by Oguiso and Sakurai.

My question is "Is Gromov-Wittne theory for such Calabi-Yau threefolds trivial?" I cannot really imagine the situation where the genus zero GW invariants are zero while higher ones are not.

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