Doran–Harder–Thompson conjecture via SYZ mirror symmetry: Elliptic curves

Atsushi Kanazawa

doi:10.3842/SIGMA.2017.024

Doran–Harder–Thompson conjecture via SYZ mirror symmetry: Elliptic curves

Atsushi Kanazawa

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We prove the Doran–Harder–Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi– Yau manifold X degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi–Yau manifold of X can be constructed by gluing the two mirror Landau– Ginzburg models of the quasi-Fano manifolds. The two crucial ideas in our proof are to obtain a complex structure by gluing the underlying affine manifolds and to construct the theta functions from the Landau–Ginzburg superpotentials.

Original language	English
Article number	024
Journal	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume	13
DOIs	https://doi.org/10.3842/SIGMA.2017.024
Publication status	Published - 2017 Apr 11
Externally published	Yes

Keywords

Affine geometry
Calabi–Yau manifolds
Fano manifolds
Landau–Ginzburg models
SYZ mirror symmetry
Tyurin degeneration

ASJC Scopus subject areas

Analysis
Mathematical Physics
Geometry and Topology

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10.3842/SIGMA.2017.024

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title = "Doran–Harder–Thompson conjecture via SYZ mirror symmetry: Elliptic curves",

abstract = "We prove the Doran–Harder–Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi– Yau manifold X degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi–Yau manifold of X can be constructed by gluing the two mirror Landau– Ginzburg models of the quasi-Fano manifolds. The two crucial ideas in our proof are to obtain a complex structure by gluing the underlying affine manifolds and to construct the theta functions from the Landau–Ginzburg superpotentials.",

keywords = "Affine geometry, Calabi–Yau manifolds, Fano manifolds, Landau–Ginzburg models, SYZ mirror symmetry, Tyurin degeneration",

author = "Atsushi Kanazawa",

note = "Funding Information: The author would like to thank Yu-Wei Fan, Andrew Harder, Hansol Hong and Siu-Cheong Lau for useful conversations on related topics. Special thanks go to the referees for their valuable comments and improvements to this article. This research was supported by the Kyoto University Hakubi Project. Part of this work was carried out during the author{\textquoteright}s stay at BIRS in the fall of 2016. Publisher Copyright: {\textcopyright} 2017, Institute of Mathematics. All rights reserved.",

year = "2017",

month = apr,

day = "11",

doi = "10.3842/SIGMA.2017.024",

language = "English",

volume = "13",

journal = "Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)",

issn = "1815-0659",

publisher = "Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine",

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T1 - Doran–Harder–Thompson conjecture via SYZ mirror symmetry

T2 - Elliptic curves

AU - Kanazawa, Atsushi

N1 - Funding Information: The author would like to thank Yu-Wei Fan, Andrew Harder, Hansol Hong and Siu-Cheong Lau for useful conversations on related topics. Special thanks go to the referees for their valuable comments and improvements to this article. This research was supported by the Kyoto University Hakubi Project. Part of this work was carried out during the author’s stay at BIRS in the fall of 2016. Publisher Copyright: © 2017, Institute of Mathematics. All rights reserved.

PY - 2017/4/11

Y1 - 2017/4/11

N2 - We prove the Doran–Harder–Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi– Yau manifold X degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi–Yau manifold of X can be constructed by gluing the two mirror Landau– Ginzburg models of the quasi-Fano manifolds. The two crucial ideas in our proof are to obtain a complex structure by gluing the underlying affine manifolds and to construct the theta functions from the Landau–Ginzburg superpotentials.

AB - We prove the Doran–Harder–Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi– Yau manifold X degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi–Yau manifold of X can be constructed by gluing the two mirror Landau– Ginzburg models of the quasi-Fano manifolds. The two crucial ideas in our proof are to obtain a complex structure by gluing the underlying affine manifolds and to construct the theta functions from the Landau–Ginzburg superpotentials.

KW - Affine geometry

KW - Calabi–Yau manifolds

KW - Fano manifolds

KW - Landau–Ginzburg models

KW - SYZ mirror symmetry

KW - Tyurin degeneration

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DO - 10.3842/SIGMA.2017.024

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SN - 1815-0659

VL - 13

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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ER -

Doran–Harder–Thompson conjecture via SYZ mirror symmetry: Elliptic curves

Abstract

Keywords

ASJC Scopus subject areas

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