Smoothing, scattering, and a conjecture of Fukaya

Abstract

In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the multi-valued Morse theory on the base $\check{B}$ of an SYZ fibration $\check{p}:\check{X}\to \check{B}$, and the other between deformation theory of the mirror $X$ and the same multi-valued Morse theory on $\check{B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base $\check{B}$ is replaced by tropical geometry on the Legendre dual $B$. In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part $X_{\text{sf}}\subseteq X$ allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.

arXiv comment65 pages, 9 figures; v4: final version

References in corpus (12)

• Mirror Symmetry is T-Duality
• Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields
• Covariantly functorial wrapped Floer theory on Liouville sectors
• Mirror symmetry for very affine hypersurfaces
• Smoothing toroidal crossing spaces
• Smoothness Theorem for Differential BV Algebras
• Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties
• Scattering diagrams from asymptotic analysis on Maurer-Cartan equations
• Homological mirror symmetry at large volume
• Tropical counting from asymptotic analysis on Maurer-Cartan equations
• Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces
• Global logarithmic deformation theory