2014/07/31 by Travis Mandel · 6 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Character (mathematics) #Dual (grammatical number) #Fourier analysis #Fourier series #Minkowski addition #Minkowski space #Monomial #Pairing #Polynomial and algebraic computation #Theta function #math.AG #msc:13F60 #msc:14J33 #msc:14M25
paper · pdf · doi:10.1007/s00029-015-0221-y
published in Selecta Mathematica 22(3), 1289-1335 (Birkhäuser) · 40 pages, 2 figures. The final publication is available at Springer via http://dx.doi.org/10.1007/s00029-015-0221-y, Selecta Math. (2016)
openalex publication_date 2016/01/13 · arxiv created 2016/01/15 · arxiv updated 2016/01/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We generalize the standard combinatorial techniques of toric geometry to the study of log Calabi-Yau surfaces. The character and cocharacter lattices are replaced by certain integral linear manifolds described by Gross, Hacking, and Keel, and monomials on toric varieties are replaced with the canonical theta functions which GHK defined using ideas from mirror symmetry. We describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher-rank cluster varieties.