Let X0 be a semi-flat Calabi-Yau manifold equipped with a Lagrangian torus fibration p : X0 → B0. We investigate the asymptotic behavior of Maurer-Cartan solutions of the Kodaira-Spencer deformation theory on X0 by expanding them into Fourier series along fibres of p over a contractible open subset U ⊂ B0, following a program set forth by Fukaya [21] in 2005. We prove that semi-classical limits (i.e. leading order terms in asymptotic expansions) of the Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to consistent scattering diagrams, which are tropical combinatorial objects that have played a crucial role in works of Kontsevich-Soibelman [37] and Gross-Siebert [28] on the reconstruction problem in mirror symmetry.1 Except in the semi-flat case when B = B0 where there are no singular fibres; see [40].5 Indeed Assumptions I and II (or more precisely Assumptions 5.48 and 5.49) are extracted from properties of the MC solutions we constructed.

We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer–Cartan elements of a (dg) Lie algebra constructed from a (not necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich–Soibelman, Gross–Siebert, and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer–Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of distinguished flat sections of the Maurer–Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with nondegenerate skew-symmetrized Euler forms.

Let X = XΣ be a toric surface and (X, W ) be its Landau-Ginzburg (LG) mirror where W is the Hori-Vafa potential [37]. We apply asymptotic analysis to study the extended deformation theory of the LG model (X, W ), and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in X with Maslov index 0 or 2, the latter of which produces a universal unfolding of W . For X = P 2 , our construction reproduces Gross' perturbed potential Wn [29] which was proven to be the universal unfolding of W written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of Wn across walls of the scattering diagram formed by Maslov index 0 tropical disks originally observed by Gross [29] (in the case of X = P 2 ). 1As in [10], a solution Φ to the extended MC equation 1.3 of (G/I) * , * can be constructed using Kuranishi's method [42], namely, by summing over directed ribbon weighted d-pointed k-trees (see Definition 2.6) with input Π. The MC solution Φ can then be decomposed aswhere Ξ i,i ∈ (G/I) i,i . A major discovery of this paper is that, as ℏ → 0, the correction terms Ξ 1,1 and Ξ 0,0 give rise to tropical disks of Maslov index 0 and 2 respectively: 1 (=Theorem 4.12). Each of the terms Ξ 0,0 , Ξ 1,1 of the Maurer-Cartan solution Φ can be expressed as a sum over tropical disks Γ whose moduli space M Γ is non-empty of codimensionhere Mono(Γ) is a holomorphic function and Log(Θ Γ ) is a holomorphic vector field defined explicitly for a tropical disk Γ, and α Γ is a Dolbeault (0, 1 − M I(Γ) 2 )-form with asymptotic support along the 1 These are the only nontrivial bulk deformations.

We apply the techniques developed in [2] to study smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded perfect complex of locally free sheaves over a degenerate Calabi–Yau variety $X$. In particular, if $X$ is a projective Calabi–Yau variety admitting the structure of a toroidal crossing space and with the higher tangent sheaf $\mathcal{T}^1_X$ globally generated, and $\mathfrak{F}$ is a locally free sheaf over $X$, then we prove, using the results in [ 8], that the pair $(X,\mathfrak{F})$ is formally smoothable when $\textrm{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.

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