Virtual Clemens conjecture for Calabi–Yau threefolds

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For a Calabi–Yau threefold XX, an isolated rational curve is a rational curve that is isolated in the relevant moduli space, and a smooth (−1,−1)(-1,-1) curve is a smooth rational curve with normal bundle isomorphic to OP1(−1)⊕OP1(−1)\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1). The curves are denoted by Ci⊂XC_i\subset X.

Virtual Clemens conjecture. For the purpose of computing genus zero enumerative invariants on Calabi–Yau threefolds XX, one may assume that there are only finitely many isolated rational curves {Ci⊂X}\{C_i\subset X\}. Furthermore, CiC_i are all smooth (−1,−1)(-1,-1) curves.

This folklore conjecture is used as a heuristic starting point for relating genus-zero Gopakumar–Vafa invariants to quantum KK-invariants through a multiple-cover formula. The source does not state a resolution of the conjecture.

References

Primary source

You-Cheng Chou and Y. -P. Lee, “Quantum K-invariants and Gopakumar-Vafa invariants II. Calabi-Yau threefolds at genus zero”, arXiv:2305.08480 (2023).

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