A proof of N. Takahashi’s conjecture for (P2,E) and a refined sheaves/Gromov–Witten correspondence

Abstract

We prove N. Takahashi's conjecture determining the contribution of each contact point in genus-0 maximal contact Gromov-Witten theory of P 2 relative to a smooth cubic E. This is a new example of a question in Gromov-Witten theory which can be fully solved despite the presence of contracted components and multiple covers. The proof relies on a tropical computation of the Gromov-Witten invariants and on the interpretation of the tropical picture as describing wall-crossing in the derived category of coherent sheaves on P 2 , giving a translation of the original Gromov-Witten question into a known statement about Euler characteristics of moduli spaces of one-dimensional Gieseker semistable sheaves on P 2 .The same techniques allow us to prove a new sheaves/Gromov-Witten correspondence, relating Betti numbers of moduli spaces of one-dimensional Gieseker semistable sheaves on P 2 , or equivalently refined genus-0 Gopakumar-Vafa invariants of local P 2 , with higher-genus maximal contact Gromov-Witten theory of (P 2 , E). The correspondence involves the non-trivial change of variables y = e i ̵ h , where y is the refined/cohomological variable on the sheaf side, and ̵ h is the genus variable on the Gromov-Witten side. We explain how this correspondence can be heuristically motivated by a combination of mirror symmetry and hyperkähler rotation. d,χ 2.2. Comparison with Donaldson-Thomas invariants of K P 2 . MATHEMATICS SUBJECT CLASSIFICATION: 14N35 1 2.3. Proof of the χ-independence of Ω P 2 d,χ 3. Comparison of the scattering diagrams of [Bou19a] and [CPS10] 3.1. The integral affine manifold of [CPS10] 3.2. Comparison with the integral affine manifold of [Bou19a] 3.3. Comparison of the scattering diagrams 3.4. Stringy Kähler moduli space 4. Unrefined sheaf counting and genus-0 Gromov-Witten theory of (P 2 , E) 4.1. Vertical rays of S(D in cl +) and one-dimensional sheaves on P 2 4.2. Vertical rays of S(D in cl + ) and genus-0 Gromov-Witten invariants of (P 2 , E) 4.3. Proof of Theorem 0.3.2 4.4. Proof of Theorem 0.3.3 5. Refined sheaf counting and higher-genus Gromov-Witten invariants of (P 2 , E) 5.1. Vertical rays of S(D in y + ) and one-dimensional sheaves on P 2 5.2. Vertical rays of S(D in y + ) and higher-genus Gromov-Witten invariants 5.3. Proof of Theorem 0.4.4 5.4. Proof of Theorem 0.4.5 5.5. Proof of Theorem 0.4.6 6. Further results 6.1. Compatibility with the local-relative correspondence 6.2. A conjectural real sheaves/Gromov-Witten correspondence 7. Some heuristic explanation 7.1. Hyperkähler rotation for V 7.2. Suspension from dimension 2 to dimension 3 7.3. Mirror symmetry for K P 2 7.4. Conclusion References

Rubber tori in the boundary of expanded stable maps

“…The development of logarithmic Gromov-Witten theory over the past decade has seen it grow to occupy a key position within contemporary enumerative geometry [1,12,17,20,34]. This growth has been aided crucially by a corpus of techniques unique to the logarithmic theory, chief among them correspondence theorems involving tropical curve counts and scattering diagrams [5,8,10,15,16,24,27,32].…”