# Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs

@article{Graefnitz2022TropicalCF,
title={Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs},
author={Tim Graefnitz},
journal={Journal of Algebraic Geometry},
year={2022}
}
• Tim Graefnitz
• Published 28 May 2020
• Computer Science
• Journal of Algebraic Geometry
<p>Consider a log Calabi-Yau pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,D)</mml:annotation> </mml…

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## References

SHOWING 1-10 OF 75 REFERENCES

### Log BPS numbers of log Calabi-Yau surfaces

• Mathematics
Transactions of the American Mathematical Society
• 2020
<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S comma upper E right-parenthesis"> <mml:semantics>

### Smoothing toric Fano surfaces using the Gross–Siebert algorithm

• T. Prince
• Mathematics
Proceedings of the London Mathematical Society
• 2018
A toric del Pezzo surface XP with cyclic quotient singularities determines and is determined by a Fano polygon P . We construct an affine manifold with singularities that partially smooths the

### Decomposition of degenerate Gromov–Witten invariants

• Mathematics
Compositio Mathematica
• 2020
We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields

### Relative Gromov-Witten invariants and the mirror formula

Abstract. Let X be a smooth complex projective variety, and let $Y \subset X$ be a smooth very ample hypersurface such that $-K_Y$ is nef. Using the technique of relative Gromov-Witten

### The quantum tropical vertex

Gross-Pandharipande-Siebert have shown that the 2-dimensional Kontsevich-Soibelman scattering diagrams compute certain genus zero log Gromov-Witten invariants of log Calabi-Yau surfaces. We show that

### Intrinsic mirror symmetry and punctured Gromov-Witten invariants

• Mathematics
Algebraic Geometry: Salt Lake City 2015
• 2018
This contribution to the 2015 AMS Summer Institute in Algebraic Geometry (Salt Lake City) announces a general mirror construction. This construction applies to log Calabi-Yau pairs (X,D) with maximal

### Sheaves of maximal intersection and multiplicities of stable log maps

• Mathematics
Selecta Mathematica
• 2021
A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J

### Integrality of relative BPS state counts of toric Del Pezzo surfaces

• Mathematics
• 2013
Relative BPS state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert and conjectured by the authors to be integers. For toric Del Pezzo surfaces, we provide an

### Tropical refined curve counting from higher genera and lambda classes

It is shown that the result is a generating series of higher genus log Gromov–Witten invariants with insertion of a lambda class, which gives a geometric interpretation of the Block-Göttsche invariants and makes their deformation invariance manifest.

### Intrinsic Mirror Symmetry

• Mathematics
• 2019
We associate a ring R to a log Calabi-Yau pair (X,D) or a degeneration of Calabi-Yau manifolds X->B. The vector space underlying R is determined by the tropicalization of (X,D) or X->B, while the