Virtual Clemens conjecture for Calabi–Yau threefolds
Virtual Clemens conjecture for Calabi–Yau threefolds
For a Calabi–Yau threefold , an isolated rational curve is a rational curve that is isolated in the relevant moduli space, and a smooth curve is a smooth rational curve with normal bundle isomorphic to . The curves are denoted by .
Virtual Clemens conjecture. For the purpose of computing genus zero enumerative invariants on Calabi–Yau threefolds , one may assume that there are only finitely many isolated rational curves . Furthermore, are all smooth curves.
This folklore conjecture is used as a heuristic starting point for relating genus-zero Gopakumar–Vafa invariants to quantum -invariants through a multiple-cover formula. The source does not state a resolution of the conjecture.
Sources & referencesView supporting material
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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