Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs

Graefnitz, Tim

doi:10.48550/arxiv.2005.14018

Cited by 3 publications

(25 citation statements)

References 16 publications

(41 reference statements)

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“…Broken lines are certain piecewise straight lines with possible bends when crossing a wall due to interaction with the tropical disks carried by the wall. The present paper also gives a systematic treatment of the geometric information carried by the wall structure, which sometimes contains interesting enumerative information in its own right [9,20,32,40].…”

Section: The Canonical Wall Structure and The Syz Interpretation Of I...mentioning

confidence: 99%

The canonical wall structure and intrinsic mirror symmetry

Gross¹,

Siebert

2022

Invent. math.

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As announced in Gross and Siebert (in Algebraic geometry: Salt Lake City 2015, Proceedings of Symposia in Pure Mathematics, vol 97, no 2. AMS, Providence, pp 199–230, 2018) in 2016, we construct and prove consistency of the canonical wall structure. This construction starts with a log Calabi–Yau pair (X, D) and produces a wall structure, as defined in Gross et al. (Mem. Amer. Math. Soc. 278(1376), 1376, 1–103, 2022). Roughly put, the canonical wall structure is a data structure which encodes an algebro-geometric analogue of counts of Maslov index zero disks. These enumerative invariants are defined in terms of the punctured invariants of Abramovich et al. (Punctured Gromov–Witten invariants, 2020. arXiv:2009.07720v2 [math.AG]). There are then two main theorems of the paper. First, we prove consistency of the canonical wall structure, so that, using the setup of Gross et al. (Mem. Amer. Math. Soc. 278(1376), 1376, 1–103, 2022), the canonical wall structure gives rise to a mirror family. Second, we prove that this mirror family coincides with the intrinsic mirror constructed in Gross and Siebert (Intrinsic mirror symmetry, 2019. arXiv:1909.07649v2 [math.AG]). While the setup of this paper is narrower than that of Gross and Siebert (Intrinsic mirror symmetry, 2019. arXiv:1909.07649v2 [math.AG]), it gives a more detailed description of the mirror.

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Section: The Canonical Wall Structure and The Syz Interpretation Of I...mentioning

confidence: 99%

The canonical wall structure and intrinsic mirror symmetry

Gross¹,

Siebert

2022

Invent. math.

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“…We do not give proofs as the results are already proven in the preceding sections. We introduce the concepts and construction of the Gross-Siebert programme that we need and refer the interested reader to [7,13,14] for more details.…”

Section: Proofmentioning

confidence: 99%

“…8. Indeed, the (−1)-curves correspond to the lowest degree tropical curves of [7,13]. These are formed by choosing two of the compact edges and on each edge a focus-focus singularity.…”

Section: Proofmentioning

confidence: 99%

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Prelog Chow rings and degenerations

Böhning

Bothmer

Garrel

2022

Rend. Circ. Mat. Palermo, II. Ser

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For a simple normal crossing variety X, we introduce the concepts of prelog Chow ring, saturated prelog Chow group, as well as their counterparts for numerical equivalence. Thinking of X as the central fibre in a (strictly) semistable degeneration, these objects can intuitively be thought of as consisting of cycle classes on X for which some initial obstruction to arise as specializations of cycle classes on the generic fibre is absent. Cycle classes in the generic fibre specialize to their prelog counterparts in the central fibre, thus extending to Chow rings the method of studying smooth varieties via strictly semistable degenerations. After proving basic properties for prelog Chow rings and groups, we explain how they can be used in an envisaged further development of the degeneration method by Voisin et al. to prove stable irrationality of very general fibres of certain families of varieties; this extension would allow for much more singular degenerations, such as toric degenerations as occur in the Gross–Siebert programme, to be usable. We illustrate that by looking at the example of degenerations of elliptic curves, which, although simple, shows that our notion of prelog decomposition of the diagonal can also be used as an obstruction in cases where all components in a degeneration and their mutual intersections are rational. We also compute the saturated prelog Chow group of degenerations of cubic surfaces.

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“…One might expect to generalize our proof of Theorem 0.2.1 to treat that case. This would require to generalize [Gra20] and [Bou19a] from (P 2 , E) to (Y, D). We leave the question open for the present paper.…”

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confidence: 99%

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

Bousseau

2023

Duke Math. J.

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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

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Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs

Cited by 3 publications

References 16 publications

The canonical wall structure and intrinsic mirror symmetry

The canonical wall structure and intrinsic mirror symmetry

Prelog Chow rings and degenerations

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

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