Mirrors to Toric Degenerations via Intrinsic Mirror Symmetry

We explore the connection between two mirror constructions in Gross-Siebert mirror symmetry: toric degeneration mirror symmetry and intrinsic mirror symmetry. 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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2023

Author dc:creator

• Goncharov, Evgeny

Advisor dc:contributor.advisor

• Gross, Mark

• Logarithmic geometry
• Mirror symmetry
• Resolution of singularities
• Toric degenerations

Licence

• https://www.repository.cam.ac.uk/bitstreams/e0bb9e61-8407-4cb4-a4b1-d81419e7ab99/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.102243

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/358793