Intrinsic Mirrors for Minimal Adjoint Orbit (ICM G&T 2022)
Elizabeth Gasparim
TL;DR
This note surveys intrinsic mirror symmetry for minimal semisimple adjoint orbits, focusing on geometric and homological facets of Landau–Ginzburg models and their Fukaya categories. It outlines HMS1, constructs symplectic Lefschetz fibrations from adjoint orbits, and discusses the interaction with flag manifolds and the Katzarkov–Kontsevich–Pantev program. A key result is that $LG(2)$ serves as a counterexample to projective mirrors (HMS1) while still fitting into HMS2 via an explicit Gross–Siebert mirror, $LG^∨(2)$, with a concrete equation and category equivalence $Fuk(LG(2)) ≈ D_{Sg} LG^∨(2)$. The paper also reports partial KKP verifications for LG models and SLFs, and demonstrates how intrinsic mirror symmetry can yield computable duals in this LG setting, clarifying the scope of HMS in noncompact cases.
Abstract
This text is contribution 77 to the ZAG Handbook of Modern Algebraic Geometry, edited by I. Cheltsov and J. Martinez-Garcia, and summarises the Short Communication I presented at the Geometry and Topology Session of the International Congress of Mathematicians which took place at the University of Copenhagen in 2022.