AI 中文总结
该研究探究Gross-Siebert镜像对称中两种镜像构造的联系，将极小相对对数Calabi-Yau退化的Gross-Siebert镜像构造推广至K3曲面的除子型环面退化情形，还得到对应关系并讨论高维推广。
AI 中文摘要
我们探究Gross-Siebert镜像对称中两种镜像构造之间的联系：环面退化镜像对称（arXiv:1212.4220、arXiv:math/0309070、arXiv:0709.2290、arXiv:math/0703822）与内在镜像对称（arXiv:1909.07649、arXiv:2105.02502）。在简要探究椭圆曲线退化的情况后，我们证明，极小相对对数Calabi-Yau退化的Gross-Siebert镜像构造可推广至具有光滑一般纤维的K3曲面的除子型环面退化$\bar{\frak{X}} \to \frak{S}$的情形。我们通过将$\bar{\frak{X}} \to \frak{S}$构造为相对极小对数Calabi-Yau退化$\frak{X} \to \frak{S}$的消解，并对比生成环面退化镜像$\breve{\bar{\frak{X}}}$的算法散射图$\bar{\frak{D}}$与生成内在镜像$\breve{\frak{X}}$的典范散射图$\frak{D}$来实现这一点。此外，我们大幅扩展了该构造，得到内在镜像在（数值）极小相对Gross-Siebert轨迹上的限制与通用环面退化镜像之间的对应关系。我们还讨论了将结果推广至高维情形，特别地，我们为Calabi-Yau三维簇的一类自然环面退化构造了对数光滑消解。
英文摘要
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 $\check{\bar{\mathfrak{X}}}$ 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.
Comments260 pages, 36 figures
