Abstract
We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees.
This gives complete control of the $B$-model side of mirror symmetry in terms of tropical geometry. For example, we expect that our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods.
-
[Alex]
V. Alexeev, "Complete moduli in the presence of semiabelian group action," Ann. of Math., vol. 155, iss. 3, pp. 611-708, 2002.@article {Alex, MRKEY = {1923963},
AUTHOR = {Alexeev, Valery},
TITLE = {Complete moduli in the presence of semiabelian group action},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {155},
YEAR = {2002},
NUMBER = {3},
PAGES = {611--708},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {14K10 (14D20 14M25)},
MRNUMBER = {1923963},
MRREVIEWER = {J{á}nos Koll{á}r},
DOI = {10.2307/3062130},
ZBLNUMBER = {1052.14017},
} -
[AlexNak] V. Alexeev and I. Nakamura, "On Mumford’s construction of degenerating abelian varieties," Tohoku Math. J., vol. 51, iss. 3, pp. 399-420, 1999.
@article {AlexNak, MRKEY = {1707764},
AUTHOR = {Alexeev, Valery and Nakamura, Iku},
TITLE = {On {M}umford's construction of degenerating abelian varieties},
JOURNAL = {Tohoku Math. J.},
FJOURNAL = {The Tohoku Mathematical Journal. Second Series},
VOLUME = {51},
YEAR = {1999},
NUMBER = {3},
PAGES = {399--420},
ISSN = {0040-8735},
CODEN = {TOMJAM},
MRCLASS = {14D06 (14D22 14K25)},
MRNUMBER = {1707764},
MRREVIEWER = {Antoine Chambert-Loir},
DOI = {10.2748/tmj/1178224770},
ZBLNUMBER = {0989.14003},
} -
[benzecri] J. -P. Benzécri, Variétés localement affines.
@misc{benzecri,
author={Benzécri, J.-P.},
TITLE={Variétés localement affines},
NOTE={Séminaire Ehresmann 1959},
} -
[douady] A. Douady, "Le problème des modules locaux pour les espaces ${\bf C}$-analytiques compacts," Ann. Sci. École Norm. Sup., vol. 7, pp. 569-602 (1975), 1974.
@article {douady, MRKEY = {0382729},
AUTHOR = {Douady, A.},
TITLE = {Le problème des modules locaux pour les espaces {${\bf C}$}-analytiques compacts},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {7},
YEAR = {1974},
PAGES = {569--602 (1975)},
ISSN = {0012-9593},
MRCLASS = {32G13},
MRNUMBER = {0382729},
MRREVIEWER = {H. W. Schuster},
URL = {http://www.numdam.org/item?id=ASENS_1974_4_7_4_569_0},
ZBLNUMBER = {0313.32036},
} -
[FriedmanB] R. Friedman, "Global smoothings of varieties with normal crossings," Ann. of Math., vol. 118, iss. 1, pp. 75-114, 1983.
@article {FriedmanB, MRKEY = {0707162},
AUTHOR = {Friedman, Robert},
TITLE = {Global smoothings of varieties with normal crossings},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {118},
YEAR = {1983},
NUMBER = {1},
PAGES = {75--114},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {32G11 (14D20 14J28 32G13)},
MRNUMBER = {0707162},
MRREVIEWER = {David R. Morrison},
DOI = {10.2307/2006955},
ZBLNUMBER = {0569.14002},
} -
[Fuk] K. Fukaya, "Multivalued Morse theory, asymptotic analysis and mirror symmetry," in Graphs and Patterns in Mathematics and Theoretical Physics, Providence, RI: Amer. Math. Soc., 2005, vol. 73, pp. 205-278.
@incollection {Fuk, MRKEY = {2131017},
AUTHOR = {Fukaya, Kenji},
TITLE = {Multivalued {M}orse theory, asymptotic analysis and mirror symmetry},
BOOKTITLE = {Graphs and Patterns in Mathematics and Theoretical Physics},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME = {73},
PAGES = {205--278},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2005},
MRCLASS = {53D40 (14J32 32Q25 53D12 58E05)},
MRNUMBER = {2131017},
MRREVIEWER = {Richard P. Thomas},
ZBLNUMBER = {1085.53080},
} -
[gola] J. E. Goodman and A. Landman, "Varieties proper over affine schemes," Invent. Math., vol. 20, pp. 267-312, 1973.
@article {gola, MRKEY = {0327772},
AUTHOR = {Goodman, Jacob Eli and Landman, Alan},
TITLE = {Varieties proper over affine schemes},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {20},
YEAR = {1973},
PAGES = {267--312},
ISSN = {0020-9910},
MRCLASS = {14E99},
MRNUMBER = {0327772},
MRREVIEWER = {G. Horrocks},
DOI = {10.1007/BF01391326},
ZBLNUMBER = {0304.14002},
} -
[GbZa] T. Graber and E. Zaslow, "Open-string Gromov-Witten invariants: Calculations and a mirror “theorem”," in Orbifolds in Mathematics and Physics, Providence, RI: Amer. Math. Soc., 2002, vol. 310, pp. 107-121.
@incollection {GbZa, MRKEY = {1950943},
AUTHOR = {Graber, Tom and Zaslow, Eric},
TITLE = {Open-string {G}romov-{W}itten invariants: Calculations and a mirror ``theorem''},
BOOKTITLE = {Orbifolds in Mathematics and Physics},
VENUE={{M}adison, {WI},
2001},
SERIES = {Contemp. Math.},
VOLUME = {310},
PAGES = {107--121},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2002},
MRCLASS = {53D45 (14N35)},
MRNUMBER = {1950943},
MRREVIEWER = {Jim A. Bryan},
ZBLNUMBER={1084.14518},
} -
[grauert] H. Grauert, "Der Satz von Kuranishi für kompakte komplexe Räume," Invent. Math., vol. 25, pp. 107-142, 1974.
@article {grauert, MRKEY = {0346194},
AUTHOR = {Grauert, Hans},
TITLE = {Der {S}atz von {K}uranishi für kompakte komplexe {R}äume},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {25},
YEAR = {1974},
PAGES = {107--142},
ISSN = {0020-9910},
MRCLASS = {32G05 (32G13)},
MRNUMBER = {0346194},
MRREVIEWER = {H. Kerner},
DOI = {10.1007/BF01390171},
ZBLNUMBER={0286.32015},
} -
[Gr1] M. Gross, "Topological mirror symmetry," Invent. Math., vol. 144, iss. 1, pp. 75-137, 2001.
@article {Gr1, MRKEY = {1821145},
AUTHOR = {Gross, Mark},
TITLE = {Topological mirror symmetry},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {144},
YEAR = {2001},
NUMBER = {1},
PAGES = {75--137},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14J32 (32Q25)},
MRNUMBER = {1821145},
MRREVIEWER = {Richard P. Thomas},
DOI = {10.1007/s002220000119},
ZBLNUMBER = {1077.14052},
} -
[Gr2] M. Gross, "Examples of special Lagrangian fibrations," in Symplectic Geometry and Mirror Symmetry, World Sci. Publ., River Edge, NJ, 2001, pp. 81-109.
@incollection {Gr2, MRKEY = {1882328},
AUTHOR = {Gross, Mark},
TITLE = {Examples of special {L}agrangian fibrations},
BOOKTITLE = {Symplectic Geometry and Mirror Symmetry},
VENUE={{S}eoul, 2000},
PAGES = {81--109},
PUBLISHER = {World Sci. Publ., River Edge, NJ},
YEAR = {2001},
MRCLASS = {53C38 (14J32 32Q25 53D12)},
MRNUMBER = {1882328},
MRREVIEWER = {Richard P. Thomas},
DOI = {10.1142/9789812799821_0004},
ZBLNUMBER = {1034.53054},
} -
[GBB] M. Gross, "Toric degenerations and Batyrev-Borisov duality," Math. Ann., vol. 333, iss. 3, pp. 645-688, 2005.
@article {GBB, MRKEY = {2198802},
AUTHOR = {Gross, Mark},
TITLE = {Toric degenerations and {B}atyrev-{B}orisov duality},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {333},
YEAR = {2005},
NUMBER = {3},
PAGES = {645--688},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {14J32 (14M25)},
MRNUMBER = {2198802},
MRREVIEWER = {Diego Matessi},
DOI = {10.1007/s00208-005-0686-7},
ZBLNUMBER = {1086.14035},
} -
[Gr4] M. Gross, "The Strominger-Yau-Zaslow conjecture: from torus fibrations to degenerations," in Algebraic Geometry—Seattle 2005. Part 1, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 149-192.
@incollection {Gr4, MRKEY = {2483935},
AUTHOR = {Gross, Mark},
TITLE = {The {S}trominger-{Y}au-{Z}aslow conjecture: from torus fibrations to degenerations},
BOOKTITLE = {Algebraic Geometry---{S}eattle 2005. {P}art 1},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME = {80},
PAGES = {149--192},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2009},
MRCLASS = {14J33 (14J32 14T05 53C38 53D37)},
MRNUMBER = {2483935},
MRREVIEWER = {Diego Matessi},
ZBLNUMBER = {1173.14031},
} -
[Announce] M. Gross and B. Siebert, "Affine manifolds, log structures, and mirror symmetry," Turkish J. Math., vol. 27, iss. 1, pp. 33-60, 2003.
@article {Announce, MRKEY = {1975331},
AUTHOR = {Gross, Mark and Siebert, Bernd},
TITLE = {Affine manifolds, log structures, and mirror symmetry},
JOURNAL = {Turkish J. Math.},
FJOURNAL = {Turkish Journal of Mathematics},
VOLUME = {27},
YEAR = {2003},
NUMBER = {1},
PAGES = {33--60},
ISSN = {1300-0098},
MRCLASS = {14J32 (32Q25)},
MRNUMBER = {1975331},
MRREVIEWER = {Diego Matessi},
ZBLNUMBER = {1063.14048},
} -
[logmirror] M. Gross and B. Siebert, "Mirror symmetry via logarithmic degeneration data. I," J. Differential Geom., vol. 72, iss. 2, pp. 169-338, 2006.
@article {logmirror, MRKEY = {2213573},
AUTHOR = {Gross, Mark and Siebert, Bernd},
TITLE = {Mirror symmetry via logarithmic degeneration data. {I}},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {72},
YEAR = {2006},
NUMBER = {2},
PAGES = {169--338},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {14J32 (32Q25)},
MRNUMBER = {2213573},
MRREVIEWER = {Diego Matessi},
URL = {http://projecteuclid.org/euclid.jdg/1143593211},
ZBLNUMBER = {1107.14029},
} -
[partII] M. Gross and B. Siebert, "Mirror symmetry via logarithmic degeneration data, II," J. Algebraic Geom., vol. 19, iss. 4, pp. 679-780, 2010.
@article {partII, MRKEY = {2669728},
AUTHOR = {Gross, Mark and Siebert, Bernd},
TITLE = {Mirror symmetry via logarithmic degeneration data, {II}},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {19},
YEAR = {2010},
NUMBER = {4},
PAGES = {679--780},
ISSN = {1056-3911},
MRCLASS = {14J33 (14C30 14F40)},
MRNUMBER = {2669728},
ZBLNUMBER = {1209.14033},
} -
[topology] M. Gross and B. Siebert, Torus fibrations and toric degenerations.
@misc{topology, SORTYEAR={2013},
AUTHOR = {Gross, Mark and Siebert, Bernd},
TITLE = {Torus fibrations and toric degenerations},
NOTE={in preparation},
} -
[EGAIII] A. Grothendieck, "Éléments de géométrie algébrique. III: Étude cohomologique des faiscaux cohérents," Publ. Math. Inst. Hautes Étud. Sci., vol. 17, 1963.
@article{EGAIII,
author={Grothendieck, A.},
TITLE={\'{E}léments de géométrie algébrique. {III}: {\'{E}}tude cohomologique des faiscaux cohérents},
JOURNAL={Publ. Math. Inst. Hautes Étud. Sci.},
VOLUME={17},
YEAR={1963},
MRNUMBER={0163911},
ZBLNUMBER={0122.16102},
URL={http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1963__17__5_0 },
} -
[haasezharkov] C. Haase and I. Zharkov, "Integral affine structures on spheres: complete intersections," Int. Math. Res. Not., vol. 2005, iss. 51, pp. 3153-3167, 2005.
@article {haasezharkov, MRKEY = {2187503},
AUTHOR = {Haase, Christian and Zharkov, Ilia},
TITLE = {Integral affine structures on spheres: complete intersections},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2005},
NUMBER = {51},
PAGES = {3153--3167},
ISSN = {1073-7928},
MRCLASS = {14J32 (14J81 52B70)},
MRNUMBER = {2187503},
MRREVIEWER = {Mark Gross},
DOI = {10.1155/IMRN.2005.3153},
VOLUME = {2005},
ZBLNUMBER = {1100.14042},
} -
[kato] K. Kato, "Logarithmic structures of Fontaine-Illusie," in Algebraic Analysis, Geometry, and Number Theory, Baltimore, MD: Johns Hopkins Univ. Press, 1989, pp. 191-224.
@incollection {kato, MRKEY = {1463703},
AUTHOR = {Kato, Kazuya},
TITLE = {Logarithmic structures of {F}ontaine-{I}llusie},
BOOKTITLE = {Algebraic Analysis, Geometry, and Number Theory},
VENUE={{B}altimore, {MD},
1988},
PAGES = {191--224},
PUBLISHER = {Johns Hopkins Univ. Press},
ADDRESS = {Baltimore, MD},
YEAR = {1989},
MRCLASS = {14F30 (14G20)},
MRNUMBER = {1463703},
MRREVIEWER = {Adolfo Quir{ó}s},
ZBLNUMBER = {0776.14004},
} -
[KawamataN] Y. Kawamata and Y. Namikawa, "Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties," Invent. Math., vol. 118, iss. 3, pp. 395-409, 1994.
@article {KawamataN, MRKEY = {1296351},
AUTHOR = {Kawamata, Yujiro and Namikawa, Yoshinori},
TITLE = {Logarithmic deformations of normal crossing varieties and smoothing of degenerate {C}alabi-{Y}au varieties},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {118},
YEAR = {1994},
NUMBER = {3},
PAGES = {395--409},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {32G05 (14D15 14J32 14J40 32S30)},
MRNUMBER = {1296351},
MRREVIEWER = {Claire Voisin},
DOI = {10.1007/BF01231538},
ZBLNUMBER = {0848.14004},
} -
[KoSo1] M. Kontsevich and Y. Soibelman, "Homological mirror symmetry and torus fibrations," in Symplectic Geometry and Mirror Symmetry, World Sci. Publ., River Edge, NJ, 2001, pp. 203-263.
@incollection {KoSo1, MRKEY = {1882331},
AUTHOR = {Kontsevich, Maxim and Soibelman, Yan},
TITLE = {Homological mirror symmetry and torus fibrations},
BOOKTITLE = {Symplectic Geometry and Mirror Symmetry},
VENUE={{S}eoul, 2000},
PAGES = {203--263},
PUBLISHER = {World Sci. Publ., River Edge, NJ},
YEAR = {2001},
MRCLASS = {32Q25 (14J32 14K05 18E30 32G05 53C38)},
MRNUMBER = {1882331},
MRREVIEWER = {Bal{á}zs Szendr{ö}i},
DOI = {10.1142/9789812799821_0007},
ZBLNUMBER = {1072.14046},
} -
[ks] M. Kontsevich and Y. Soibelman, "Affine structures and non-Archimedean analytic spaces," in The Unity of Mathematics, Boston, MA: Birkhäuser, 2006, vol. 244, pp. 321-385.
@incollection {ks, MRKEY = {2181810},
AUTHOR = {Kontsevich, Maxim and Soibelman, Yan},
TITLE = {Affine structures and non-{A}rchimedean analytic spaces},
BOOKTITLE = {The Unity of Mathematics},
SERIES = {Progr. Math.},
VOLUME = {244},
PAGES = {321--385},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {2006},
MRCLASS = {14J32 (14G22 32Q25)},
MRNUMBER = {2181810},
MRREVIEWER = {Mark Gross},
ZBLNUMBER = {1114.14027},
} -
[mikhalkin] G. Mikhalkin, "Enumerative tropical algebraic geometry in $\Bbb R^2$," J. Amer. Math. Soc., vol. 18, iss. 2, pp. 313-377, 2005.
@article {mikhalkin, MRKEY = {2137980},
AUTHOR = {Mikhalkin, Grigory},
TITLE = {Enumerative tropical algebraic geometry in {$\Bbb R\sp 2$}},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {18},
YEAR = {2005},
NUMBER = {2},
PAGES = {313--377},
ISSN = {0894-0347},
MRCLASS = {14N10 (05A99 14N35 52B70)},
MRNUMBER = {2137980},
MRREVIEWER = {Charles D. Cadman},
DOI = {10.1090/S0894-0347-05-00477-7},
ZBLNUMBER = {1092.14068},
} -
[milnor] J. Milnor, "On the existence of a connection with curvature zero," Comment. Math. Helv., vol. 32, pp. 215-223, 1958.
@article {milnor, MRKEY = {0095518},
AUTHOR = {Milnor, John},
TITLE = {On the existence of a connection with curvature zero},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {32},
YEAR = {1958},
PAGES = {215--223},
ISSN = {0010-2571},
MRCLASS = {53.00},
MRNUMBER = {0095518},
MRREVIEWER = {L. Auslander},
DOI = {10.1007/BF02564579},
ZBLNUMBER = {0196.25101},
} -
[mumford] D. Mumford, "An analytic construction of degenerating abelian varieties over complete rings," Compositio Math., vol. 24, pp. 239-272, 1972.
@article {mumford, MRKEY = {0352106},
AUTHOR = {Mumford, David},
TITLE = {An analytic construction of degenerating abelian varieties over complete rings},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {24},
YEAR = {1972},
PAGES = {239--272},
ISSN = {0010-437X},
MRCLASS = {14K20},
MRNUMBER = {0352106},
MRREVIEWER = {I. Dolgacev},
ZBLNUMBER = {0241.14020},
} -
[nisi] T. Nishinou and B. Siebert, "Toric degenerations of toric varieties and tropical curves," Duke Math. J., vol. 135, iss. 1, pp. 1-51, 2006.
@article {nisi, MRKEY = {2259922},
AUTHOR = {Nishinou, Takeo and Siebert, Bernd},
TITLE = {Toric degenerations of toric varieties and tropical curves},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {135},
YEAR = {2006},
NUMBER = {1},
PAGES = {1--51},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {14N10 (14M25 14N35)},
MRNUMBER = {2259922},
MRREVIEWER = {Diego Matessi},
DOI = {10.1215/S0012-7094-06-13511-1},
ZBLNUMBER = {1105.14073},
} -
[Oda] T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, New York: Springer-Verlag, 1988, vol. 15.
@book {Oda, MRKEY = {0922894},
AUTHOR = {Oda, Tadao},
TITLE = {Convex bodies and algebraic geometry. An introduction to the theory of toric varieties},
SERIES = {Ergeb. Math. Grenzgeb.},
VOLUME = {15},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1988},
PAGES = {viii+212},
ISBN = {3-540-17600-4},
MRCLASS = {14L32 (14-02 52A25 52A43)},
MRNUMBER = {0922894},
MRREVIEWER = {I. Dolgachev},
ZBLNUMBER = {0628.52002},
} -
[rockafellar] T. R. Rockafellar, Convex Analysis, Princeton, N.J.: Princeton Univ. Press, 1970, vol. 28.
@book {rockafellar, MRKEY = {0274683},
AUTHOR = {Rockafellar, R. Tyrrell},
TITLE = {Convex Analysis},
SERIES = {Princeton Math. Series},
VOLUME={28},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1970},
PAGES = {xviii+451},
MRCLASS = {26.52 (46.00)},
MRNUMBER = {0274683},
MRREVIEWER = {Ky Fan},
ZBLNUMBER = {0193.18401},
} -
[SYZ] A. Strominger, S. Yau, and E. Zaslow, "Mirror symmetry is $T$-duality," Nuclear Phys. B, vol. 479, iss. 1-2, pp. 243-259, 1996.
@article {SYZ, MRKEY = {1429831},
AUTHOR = {Strominger, Andrew and Yau, Shing-Tung and Zaslow, Eric},
TITLE = {Mirror symmetry is {$T$}-duality},
JOURNAL = {Nuclear Phys. B},
FJOURNAL = {Nuclear Physics. B},
VOLUME = {479},
YEAR = {1996},
NUMBER = {1-2},
PAGES = {243--259},
ISSN = {0550-3213},
CODEN = {NUPBBO},
MRCLASS = {32J17 (14J32 32J81 81T30)},
MRNUMBER = {1429831},
MRREVIEWER = {Mark Gross},
DOI = {10.1016/0550-3213(96)00434-8},
ZBLNUMBER = {0896.14024},
}
Print version