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Log Calabi-Yau mirror symmetry and non-archimedean disks

Sean Keel, Tony Yue YU

Abstract

We construct the mirror algebra to a smooth affine log Calabi-Yau variety with maximal boundary, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are given as naive counts of non-archimedean analytic disks. More generally, we studied the enumeration of non-archimedean analytic curves with boundaries, associated to a given transverse spine in the essential skeleton of the log Calabi-Yau variety. The moduli spaces of such curves are infinite dimensional. In order to obtain finite counts, we impose a boundary regularity condition so that the curves can be analytically continued into tori, that are unrelated to the given log Calabi-Yau variety. We prove the properness of the resulting moduli spaces, and show that the mirror algebra is a finitely generated commutative associative algebra, giving rise to a mirror family of log Calabi-Yau varieties.

Log Calabi-Yau mirror symmetry and non-archimedean disks

Abstract

We construct the mirror algebra to a smooth affine log Calabi-Yau variety with maximal boundary, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are given as naive counts of non-archimedean analytic disks. More generally, we studied the enumeration of non-archimedean analytic curves with boundaries, associated to a given transverse spine in the essential skeleton of the log Calabi-Yau variety. The moduli spaces of such curves are infinite dimensional. In order to obtain finite counts, we impose a boundary regularity condition so that the curves can be analytically continued into tori, that are unrelated to the given log Calabi-Yau variety. We prove the properness of the resulting moduli spaces, and show that the mirror algebra is a finitely generated commutative associative algebra, giving rise to a mirror family of log Calabi-Yau varieties.

Paper Structure

This paper contains 46 sections, 101 theorems, 72 equations, 4 figures.

Table of Contents

  1. Introduction and statements of main results
  2. Application to Moduli of Log Calabi-Yau Pairs
  3. Application to the Cox ring of a positive Looijenga pair
  4. Application to counts of tropical curves
  5. Geometric motivation for the definition of the multiplication rule
  6. Related works
  7. Acknowledgments
  8. Log Calabi-Yau pairs
  9. Non-archimedean SYZ fibration
  10. Integer linear structure
  11. Models
  12. Bounding realizable tropical curves
  13. Ios tropicalisation
  14. Spines
  15. Rigidity and transversality of spines
  16. ...and 31 more sections

Key Result

Theorem 1.3

Let $\mathcal{N}$ be the moduli space of all such curves in $Z$. The map taking domain and evaluation at $q$ is finite and étale over a dense Zariski open subset. Its degree is independent of all the auxiliary choices in the construction, and gives the structure constant $\chi(P_1,\dots,P_n,Q,\beta)$.

Figures (4)

  • Figure 1: The red curve is a heuristic depiction of the non-archimedean analytic disk (different from the Berkovich space). The green segments depict the associated spine.
  • Figure 2: The glued target space $Z = Y^\mathrm{an} \cup_{\mathcal{G}} T^\mathrm{an}$ and a rational curve in it.
  • Figure 3:
  • Figure 4:

Theorems & Definitions (254)

  • Theorem 1.3: \ref{['thm:scthm']}, \ref{['prop:scwd']}
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Conjecture 1.8
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • ...and 244 more