Degenerations of Calabi-Yau manifolds and integral affine geometry

Abstract: Let $X\rightarrow D^*$ be a maximal degeneration of $n$-dimensional Calabi-Yau varieties over the punctured disk. The SYZ conjecture, motivated by mirror symmetry, predicts that the general fiber $X_t$ admits a Lagrangian torus fibration $f_t : X_t \rightarrow B$ onto a base $B$ of real dimension $n$, and that as $t\rightarrow 0$ the variety $X_t$ endowed with its Ricci-flat Kähler metric collapses to the space $B$, endowed with a $Z$-affine structure. The goal of this talk is to explain how to construct the space $B$ with its extra structures using non-archimedean geometry. In particular, in the case of Fermat threefolds in $\mathbb{P}^4$, using the toric geometry of the ambient space, we are able to construct a non-archimedean SYZ fibration inducing on $B$ the affine structure naturally induced by the Gromov-Hausdorff convergence recently proved by Yang Li. This is based on work joint with Enrica Mazzon.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html