Mirrors to toric degenerations via intrinsic mirror symmetry

Abstract

We explore the connection between two mirror constructions in Gross-Siebert mirror symmetry: toric degeneration mirror symmetry (arXiv:1212.4220, arXiv:math/0309070, arXiv:0709.2290, arXiv:math/0703822) and intrinsic mirror symmetry (arXiv:1909.07649, arXiv:2105.02502). After briefly exploring the case of degenerations of elliptic curves, we show that the Gross-Siebert mirror construction for minimal relative log Calabi-Yau degenerations generalizes that for divisorial toric degenerations $\bar{\mathfrak{X}}\to \mathcal{S}$ of K3-s that have a smooth generic fibre. We achieve this by constructing a resolution of $\bar{\mathfrak{X}}\to \mathcal{S}$ to a relative minimal log Calabi-Yau degeneration $\mathfrak{X}\to \mathcal{S}$ and comparing the algorithmic scattering diagram $\bar{\mathfrak{D}}$ giving rise to the toric degeneration mirror $\check{\stackrel{ˉ}{\mathfrak{X}}}$ and the canonical scattering diagram $\mathfrak{D}$ giving rise to the intrinsic mirror $\check{\mathfrak{X}}$. Moreover, we vastly expand the construction and obtain a correspondence between the restriction of the intrinsic mirror to the (numerical) minimal relative Gross-Siebert locus and the universal toric degeneration mirror. We also discuss generalizing the results to higher dimensions. In particular, we construct log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds.

arXiv comment260 pages, 36 figures