Repository logo
 

Mirrors to Toric Degenerations via Intrinsic Mirror Symmetry


Type

Thesis

Change log

Authors

Goncharov, Evgeny 

Abstract

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 X¯S of K3-s that have a smooth generic fibre. We achieve this by constructing a resolution of X¯S to a relative minimal log Calabi-Yau degeneration XS and comparing the algorithmic scattering diagram D¯ giving rise to the toric degeneration mirror X¯ˇ and the canonical scattering diagram D giving rise to the intrinsic mirror Xˇ. 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.

Description

Date

2023-06-01

Advisors

Gross, Mark

Keywords

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

Qualification

Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge