Smoothing, scattering, and a conjecture of Fukaya

Abstract

In 2002, Fukaya [16] proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture [41] by introducing two correspondences: one between the theory of pseudoholomorphic curves on a Calabi-Yau manifold X and the multi-valued Morse theory on the base B of an SYZ fibration p : X → B, and the other between deformation theory of the mirror X and the same multi-valued Morse theory on 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 B is replaced by tropical geometry on the Legendre dual B. In the proof, we apply techniques of asymptotic analysis developed in [6,7] to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety 0 X † introduced in [5]. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part X sf ⊆ X allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.