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Universally counting curves in Calabi--Yau threefolds
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Universally counting curves in Calabi--Yau threefolds
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We show that curve enumeration invariants of complex threefolds with nef anti-canonical bundle are determined by their values on local curves. This implies the MNOP conjecture of Maulik, Nekrasov, Okounkov, and Pandharipande relating Gromov--Witten and Donaldson--Pandharipande--Thomas invariants, for all complex threefolds with nef anti-canonical bundle (in particular, all Calabi--Yau threefolds) and primary insertions (no descendents), given its known validity for local curves due to Bryan, Okounkov, and Pandharipande. The main new technical ingredient in our work is a generic transversality result for holomorphic curves in complex manifolds. Due to the rigidity of complex structures, this result is necessarily weaker than the corresponding generic transversality property for holomorphic curves in almost complex manifolds. Despite this weaker nature, it is enough to obtain our main result by following the proof of the Gopakumar--Vafa integrality conjecture by Ionel and Parker.
Forward citations
Cited by 6 Pith papers
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The GW/PT conjectures for toric pairs
Proves the GW/PT correspondence for logarithmic toric pairs, including singular divisors, and shows the PT series is a Laurent polynomial under positivity.
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The refined K-theoretic DT and PT theories of local curves are solved explicitly via localization to skew nested Hilbert schemes, yielding three universal series from the equivariant vertex and confirming the DT/PT co...
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Generating series of stable pairs descendent invariants on Fano 3-folds are rational and q ↔ q^{-1} symmetric.
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A Pardon Algebra for Zero-cycles
An analogous Pardon homology algebra is defined for zero-cycles in d-folds, supplying a new perspective on point-counting enumerative problems including the degree-zero MNOP conjecture.
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The Pandharipande-Thomas rationality conjecture for superpositive curve classes on projective complex 3-manifolds
Proves that generating functions of Pandharipande-Thomas invariants with descendent insertions are rational with controlled poles for superpositive curve classes on projective complex 3-manifolds.
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The Pandharipande-Thomas rationality conjecture for superpositive curve classes on projective complex 3-manifolds
PT generating functions with descendents are rational, with controlled poles, for superpositive curve classes on projective complex 3-folds.
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