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Author
Title Tropical theta functions and log Calabi-Yau surfaces
URL
Publication Date
Date Accessioned
Degree PhD
Discipline/Department Mathematics
Degree Level doctoral
University/Publisher University of Texas – Austin
Abstract We describe combinatorial techniques for studying log Calabi-Yau surfaces. These can be viewed as generalizing the techniques for studying toric varieties in terms of their character and cocharacter lattices. These lattices are replaced by certain integral linear manifolds described in [GHK11], and monomials on toric varieties are replaced with the canonical theta functions defined in [GHK11] using ideas from mirror symmetry. We classify deformation classes of log Calabi-Yau surfaces in terms of the geometry of these integral linear manifolds. We then 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.
Subjects/Keywords Theta function; Log Calabi-Yau; Surfaces; Tropical; Toric; Cluster; Mirror symmetry; Polytope
Contributors Keel, Seán (advisor)
Language en
Country of Publication us
Record ID handle:2152/24935
Repository texas
Date Retrieved
Date Indexed 2019-09-12
Grantor The University of Texas at Austin
Note [] text; [department] Mathematics;

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…mentioned, a point q ∈ U trop (Z) corresponds to a boundary divisor Dq for some compactification of U , and also to a canonical theta function ϑq on the mirror V. Let V be a generic fiber of the mirror ([GHK] shows that V is…

…deformation equivalent 1 to U ).1 We similarly have that v ∈ V trop (Z) corresponds to a boundary divisor Dv of certain compactifications of V , and also to a theta function ϑv on the mirror U to V . We may view U as a fiber of U. For f a…

…Calabi-Yau surfaces, with the integer points of U trop serving as a generalization of the character lattice for V. That is, the global sections of the family V admit a canonical module-basis of “theta functions,” parametrized by the integer points U trop…

…language of cluster varieties). The tropical functions form a min-plus algebra, and we call a tropical function ϕ indecomposable if it cannot be written as a minimum of two other tropical functions, neither of which is ϕ. The tropical functions…

…x5B;FG09] conjectures that the theta functions (not their tropicalizations) satisfy a related indecomposability condition (now known to be false in general). For the log Calabi-Yau surface cases, we show: 1 We assume…

…regular functions, and the indecomposable tropical functions are exactly the tropicalizations of theta functions. We also generalize the notion of Newton polytopes by defining Newt( q∈S ϑq ) to be the “strong” convex hull of S in U trop . We…

…cones and s-tuples are needed, so Minkowski sums really are computable. We view this as a tropicalized version of [GHK11]’s formula for multiplying theta functions. 1.0.2 Setup Throughout this paper, Y will denote a smooth, projective…

…Dvi on a compactification (Y, D = Dvi ) of U , and the “bending parameter” of ϕ across ρvi is the intersection number Wϕ · Dvi . Let βv1 ,...,vs denote a function which has bending parameter |vi | along the ray ρvi generated by vi , for…

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