CMSA Algebraic Geometry in String Theory: Modularity of mirror families of log Calabi–Yau surfaces

 In 'Mirror symmetry for log Calabi--Yau surfaces I', given a smooth log Calabi--Yau surface pair (Y,D), Gross--Hacking--Keel constructed its mirror family as the spectrum of an explicit algebra whose structure coefficients are determined by the enumerative geometry of (Y,D). As a follow-up of the work of Gross--Hacking--Keel, when (Y,D) is positive, we prove the modularity of the mirror family as the universal family of log Calabi-Yau surface pairs deformation equivalent to (Y,D) with at worst du Val singularities. As a corollary, we show that the ring of regular functions of a smooth affine log Calabi--Yau surface has a canonical basis of theta functions. The key step towards the proof of the main theorem is the application of the tropical construction of singular cycles and explicit formulas of period integrals given in the work of Helge--Siebert. This is joint work with Jonathan Lai.

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