# The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

@article{Keel2019TheFS, title={The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus}, author={S. Keel and Tony Yue Yu}, journal={arXiv: Algebraic Geometry}, year={2019} }

We show that the naive counts of rational curves in any affine log Calabi-Yau variety $U$, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a compatible multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary… Expand

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#### References

SHOWING 1-10 OF 41 REFERENCES

Canonical bases for cluster algebras

- Mathematics
- 2014

In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical… Expand

Skeletons of stable maps I: rational curves in toric varieties

- Mathematics, Computer Science
- J. Lond. Math. Soc.
- 2017

The Nishinou--Siebert correspondence theorem is shown to be a consequence of this geometric connection between the algebraic and tropical moduli spaces, giving a large new collection of examples of faithful tropicalizations for moduli. Expand

Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I

- Mathematics
- 2015

We define the counting of holomorphic cylinders in log Calabi–Yau surfaces. Although we start with a complex log Calabi–Yau surface, the counting is achieved by applying methods from non-archimedean… Expand

Homological mirror symmetry and torus fibrations

- Mathematics, Physics
- 2000

In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya… Expand

Gromov compactness in non-archimedean analytic geometry

- Mathematics
- 2014

Gromov's compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded… Expand

Essential skeletons of pairs and the geometric P=W conjecture.

- Mathematics
- 2018

We construct weight functions on the Berkovich analytification of a variety over a trivially-valued field of characteristic zero, and this leads to the definition of the Kontsevich-Soibelman… Expand

Skeletons and tropicalizations

- Mathematics
- 2014

Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model… Expand

Intrinsic mirror symmetry and punctured Gromov-Witten invariants

- Mathematics, Physics
- 2016

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… Expand

Functors and Computations in Floer Homology with Applications, I

- Mathematics
- 1999

Abstract. This paper is concerned with Floer cohomology of manifolds with contact type boundary. In this case, there is no conjecture on this ring, as opposed to the compact case, where it is… Expand

Mirror symmetry for log Calabi-Yau surfaces I

- Mathematics
- 2011

We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is… Expand