Full Record

Author | Mandel, Travis Glenn |

Title | Tropical theta functions and log Calabi-Yau surfaces |

URL | http://hdl.handle.net/2152/24935 |

Publication Date | 2014 |

Date Accessioned | 2014-07-01 17:57:08 |

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 | 2019-09-11 |

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…