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This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings.
S.-T. Yau: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., 31
1978
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S.-Y. Cheng and S.-T. Yau, The real Monge-Ampère equation and affine flat structures , in Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations , Vol. 1, 2, 3 (Beijing, 1980), 339–370, Science Press, Beijing, 1982
1982
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S. Amari: Differential-geometric methods in statistics , Lecture Notes in Statistics, 28
1985
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K. Kato: Logarithmic structures of Fontaine–Illusie , in: Algebraic analysis, geometry, and number theory
1989
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P. Candelas, M. Lynker, R. Schimmrigk, Calabi-Yau manifolds in weighted 𝐏 4 {\bf P}_{4} , Nuclear Phys. B 341
1990
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B. Greene and M. Plesser, Duality in Calabi-Yau moduli space, Nuclear Phys. B 338
1990
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P. Candelas, X. de la Ossa, P. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359
1991
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D. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, in Journées de Géométrie Algébrique d’Orsay (Orsay, 1992) , Astérisque 218
1993
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V. Batyrev: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3
1994
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L. Illusie: Logarithmic spaces (according to K. Kato) , in Barsotti Symposium in Algebraic Geometry
1994
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Y. Kawamata, Y. Namikawa: Logarithmic deformations of normal crossing varieties and smooothing of degenerate Calabi-Yau varieties , Invent. Math. 118
1994
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M. Kontsevich: Homological algebra of mirror symmetry , Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120–139, Birkhäuser, Basel, 1995
1995
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V. Batyrev, and L. Borisov: On Calabi-Yau complete intersections in toric varieties, in Higher-dimensional complex varieties (Trento, 1994)
1996
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A. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 13
1996
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F. Kato: Log smooth deformation theory , Tohoku Math. J. 48
1996
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A. Strominger, S.-T. Yau, and E. Zaslow, Mirror Symmetry is T T -duality, Nucl. Phys. B479
1996
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M. Gross, and P.M.H. Wilson: Mirror symmetry via 3 3 -tori for a class of Calabi-Yau threefolds, Math. Ann. 309
1997
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N. Hitchin: The Moduli Space of Special Lagrangian Submanifolds , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25
1997
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B. Lian, K. Liu, S-T. Yau, Mirror principle. I, Asian J. Math. 1
1997
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M. Gross: Special Lagrangian Fibrations I: Topology, in: Integrable Systems and Algebraic Geometry
1998
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N.C. Leung, C. Vafa: Branes and toric geometry , Adv. Theor. Math. Phys. 2
1998
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R. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6
1998
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P. Petersen: Riemannian geometry, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998
1998
Earlier work this paper cites.
M. Gross: Special Lagrangian Fibrations II: Geometry, in: Surveys in Differential Geometry
1999
Earlier work this paper cites.
M. Gross, and P.M.H. Wilson: Large complex structure limits of K 3 K3 surfaces, J. Differential Geom. 55
2000
Earlier work this paper cites.
W.-D. Ruan: Lagrangian torus fibration and mirror symmetry of Calabi-Yau hypersurface in toric variety , preprint 2000, math.DG/0007028
2000
Cited alongside, same era.
E. Goldstein: A construction of new families of minimal Lagrangian submanifolds via torus actions, J. Differential Geom. 58
2001
Cited alongside, same era.
M. Gross: Topological Mirror Symmetry , Invent. Math. 144
2001
Cited alongside, same era.
M. Gross: Examples of special Lagrangian fibrations, in Symplectic geometry and mirror symmetry (Seoul, 2000) , 81–109, World Sci. Publishing, River Edge, NJ, 2001
2001
Cited alongside, same era.
M. Kontsevich, and Y. Soibelman: Homological mirror symmetry and torus fibrations , in: Symplectic geometry and mirror symmetry
2001
Cited alongside, same era.
M. Kontsevich, and Y. Soibelman: Affine structures and non-archimedean analytic spaces , The unity of mathematics, 321–385, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006
2006
Later among the works it cites.
T. Nishinou, B. Siebert: Toric degenerations of toric varieties and tropical curves , Duke Math. J., 135
2006
Later among the works it cites.
S. Schröer, B. Siebert: Toroidal crossings and logarithmic structures , Adv. Math. 202
2006
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D. Auroux: Mirror symmetry and T T -duality in the complement of an anticanonical divisor , J. Gökova Geom. Topol. GGT 1
2007
Later among the works it cites.
P. Aspinwall, T. Bridgeland, A. Craw, M. Douglas, M. Gross, A. Kapustin, G. Moore, G. Segal, B. Szendroi, P. Wilson, Dirichlet branes and mirror symmetry. Clay Mathematics Monographs, 4. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2009. x+681 pp
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A.-M. Li, Y. Ruan: Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds , Invent. Math., 145
2001
Cited alongside, same era.
B. Siebert: Gromov-Witten invariants in relative and singular cases. Lecture given at the workshop “Algebraic aspects of mirror symmetry,” Univ. Kaiserslautern, Germany, June 2001
2001
Cited alongside, same era.
C. Haase, and I. Zharkov: Integral affine structures on spheres and torus fibrations of Calabi-Yau toric hypersurfaces I , preprint 2002, math.AG/0205321
2002
Cited alongside, same era.
J. Li: A degeneration formula of GW-invariants, J. Differential Geom. 60
2002
Cited alongside, same era.
W.-D. Ruan: Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces. II. Technical results on gradient flow construction, J. Symplectic Geom. 1
2002
Cited alongside, same era.
S. Schröer, B. Siebert: Irreducible degenerations of primary Kodaira surfaces, in Complex geometry (Göttingen, 2000), 193–222, Springer, Berlin, 2002
2002
Cited alongside, same era.
M. Gross, and B. Siebert: Affine manifolds, log structures, and mirror symmetry , Turkish J. Math. 27
2003
Cited alongside, same era.
2009
Later among the works it cites.
R. Castaño-Bernard and D. Matessi, Lagrangian 3-torus fibration , J. Differential Geom., 81
2009
Later among the works it cites.
M. Gross: The Strominger-Yau-Zaslow conjecture: from torus fibrations to degenerations, Algebraic geometry—Seattle 2005. Part 1, 149–192, Proc. Sympos. Pure Math., 80
2009
Later among the works it cites.
Y. Zhang: Collapsing of Calabi-Yau manifolds and special Lagrangian submanifolds, preprint, 2009
2009
Later among the works it cites.
Q. Chen: Stable logarithmic maps to Deligne–Faltings pairs I, preprint, 2010
2010
Later among the works it cites.
D. Gaiotto, G. Moore, A. Neitzke: Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299
2010
Later among the works it cites.
M. Gross: Mirror symmetry for ℙ 2 \mathbb{P}^{2} and tropical geometry , Adv. Math., 224
2010
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M. Gross, R. Pandharipande, B. Siebert: The tropical vertex , Duke Math. J. 153
2010
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M. Gross, and B. Siebert: Mirror symmetry via logarithmic degeneration data II , J. Algebraic Geom., 19
2010
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M. Reineke: Poisson automorphisms and quiver moduli , J. Inst. Math. Jussieu 9
2010
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V. Tosatti, Adiabatic limits of Ricci-flat Kähler metrics, J. Differential Geom., 84
2010
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D. Abromovich, Q. Chen: Stable logarithmic maps to Deligne–Faltings pairs II, preprint, 2011
2011
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M. Gross: Tropical geometry and mirror symmetry , CBMS Regional Conference Series in Mathematics, 114
2011
Later among the works it cites.
M. Gross, P. Hacking, S. Keel: Mirror symmetry for log Calabi-Yau surfaces I , preprint, 2011
2011
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M. Gross and B. Siebert: From affine geoemtry to complex geometry , Annals of Mathematics , 174
2011
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M. Gross and B. Siebert: Logarithmic Gromov-Witten invariants , preprint, 2011, to appear in J. of the AMS
2011
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M. Gross and B. Siebert: Theta functions and mirror symmetry , preprint, 2011
2011
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M. Gross, V. Tosatti, Y. Zhang: Collapsing of abelian fibred Calabi-Yau manifolds , preprint, 2011. To appear in Duke Math. J
2011
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B. Parker: Gromov-Witten invariants of exploded manifolds , preprint, 2011
2011
Later among the works it cites.
B. Parker: Exploded manifolds , Adv. in Math., 229
2012
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