Géométrie algébrique, Physique mathématique
Toric mirror symmetry revisited
Comptes Rendus. Mathématique, Tome 360 (2022) no. G7, pp. 751-759.

The Cox construction presents a toric variety as a quotient of affine space by a torus. The category of coherent sheaves on the corresponding stack thus has an evident description as invariants in a quotient of the category of modules over a polynomial ring. Here we give the mirror to this description, and in particular, a clean new proof of mirror symmetry for smooth toric stacks.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.304
Shende, Vivek 1, 2

1 Center for Quantum Mathematics, University of Southern Denmark, Campusvej 55, Odense 5230, Denmark
2 Department of Mathematics, UC Berkeley, Berkeley CA 94720, USA
@article{CRMATH_2022__360_G7_751_0,
     author = {Shende, Vivek},
     title = {Toric mirror symmetry revisited},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {751--759},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     number = {G7},
     year = {2022},
     doi = {10.5802/crmath.304},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.304/}
}
TY  - JOUR
AU  - Shende, Vivek
TI  - Toric mirror symmetry revisited
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 751
EP  - 759
VL  - 360
IS  - G7
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.304/
DO  - 10.5802/crmath.304
LA  - en
ID  - CRMATH_2022__360_G7_751_0
ER  - 
%0 Journal Article
%A Shende, Vivek
%T Toric mirror symmetry revisited
%J Comptes Rendus. Mathématique
%D 2022
%P 751-759
%V 360
%N G7
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.304/
%R 10.5802/crmath.304
%G en
%F CRMATH_2022__360_G7_751_0
Shende, Vivek. Toric mirror symmetry revisited. Comptes Rendus. Mathématique, Tome 360 (2022) no. G7, pp. 751-759. doi : 10.5802/crmath.304. http://www.numdam.org/articles/10.5802/crmath.304/

[1] Abouzaid, Mohammed Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol., Volume 10 (2006), pp. 1097-1156 | DOI | MR | Zbl

[2] Abouzaid, Mohammed Morse homology, tropical geometry, and homological mirror symmetry for toric varieties, Sel. Math., New Ser., Volume 15 (2009) no. 2, pp. 189-270 | DOI | MR | Zbl

[3] Bondal, Alexei Derived categories of toric varieties, Convex and Algebraic Geometry, Oberwolfach Conference Reports, Volume 3, European Mathematical Society, 2006, pp. 284-286

[4] Cox, David A. The homogeneous coordinate ring of a toric variety, J. Algebr. Geom., Volume 4 (1995) no. 1, pp. 17-50 | MR | Zbl

[5] Fang, Bohan; Liu, Chiu-Chu Melissa; Treumann, David; Zaslow, Eric A categorification of Morelli’s theorem, Invent. Math., Volume 186 (2011) no. 1, pp. 79-114 | DOI | MR | Zbl

[6] Fang, Bohan; Liu, Chiu-Chu Melissa; Treumann, David; Zaslow, Eric The coherent-constructible correspondence for toric Deligne-Mumford stacks, Int. Math. Res. Not., Volume 2014 (2014) no. 4, pp. 914-954 | DOI | MR | Zbl

[7] Gaitsgory, Dennis Sheaves of categories and the notion of 1-affineness, Stacks and categories in geometry, topology, and algebra (Contemporary Mathematics), Volume 643, American Mathematical Society, 2015, pp. 127-225 (CATS4 conference on higher categorical structures and their interactions with algebraic geometry, algebraic topology and algebra, CIRM, Luminy, France, July 2–7, 2012) | DOI | MR | Zbl

[8] Gaitsgory, Dennis; Rozenblyum, Nick A study in derived algebraic geometry: Volume I: Correspondences and duality, Mathematical Surveys and Monographs, 221, American Mathematical Society, 2017 | Zbl

[9] Gammage, Benjamin Mirror symmetry for Berglund–Hübsch Milnor fibers (2020) (https://arxiv.org/abs/2010.15570)

[10] Gammage, Benjamin Local mirror symmetry via SYZ (2021) (https://arxiv.org/abs/2105.12863)

[11] Gammage, Benjamin; Shende, Vivek Homological mirror symmetry at large volume (2021) (https://arxiv.org/abs/2104.11129)

[12] Gammage, Benjamin; Shende, Vivek Mirror symmetry for very affine hypersurfaces (2021) (https://arxiv.org/abs/1707.02959)

[13] Ganatra, Sheel; Pardon, John; Shende, Vivek Sectorial descent for wrapped Fukaya categories (2019) (https://arxiv.org/abs/1809.03427)

[14] Ganatra, Sheel; Pardon, John; Shende, Vivek Covariantly functorial wrapped Floer theory on Liouville sectors, Publ. Math., Inst. Hautes Étud. Sci., Volume 131 (2020) no. 1, pp. 73-200 | DOI | MR | Zbl

[15] Ganatra, Sheel; Pardon, John; Shende, Vivek Microlocal Morse theory of wrapped Fukaya categories (2020) (https://arxiv.org/abs/1809.08807)

[16] Hanlon, Andrew Monodromy of monomially admissible Fukaya–Seidel categories mirror to toric varieties, Adv. Math., Volume 350 (2019), pp. 662-746 | DOI | MR | Zbl

[17] Hanlon, Andrew; Hicks, Jeff Functoriality and homological mirror symmetry for toric varieties (2020) (https://arxiv.org/abs/2010.08817v1)

[18] Hori, Kentaro; Vafa, Cumrun Mirror symmetry (2000) (https://arxiv.org/abs/hep-th/0002222)

[19] Huang, Jesse; Zhou, Peng Variation of GIT and Variation of Lagrangian Skeletons II: Quasi-Symmetric Case (2020) (https://arxiv.org/abs/2011.06114)

[20] Kashiwara, Masaki; Schapira, Pierre Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer, 1990 | DOI | Zbl

[21] Kuwagaki, Tatsuki The nonequivariant coherent-constructible correspondence for toric stacks, Duke Math. J., Volume 169 (2020) no. 11, pp. 2125-2197 | DOI | MR | Zbl

[22] Lurie, Jacob Higher topos theory, Annals of Mathematics Studies, 170, Princeton University Press, 2009 | DOI | MR | Zbl

[23] Lurie, Jacob Higher algebra, 2017 (Available at https://www.math.ias.edu/~lurie/papers/HA.pdf)

[24] Nadler, David Mirror symmetry for the Landau–Ginzburg A-model M= n , W=z 1 z n , Duke Math. J., Volume 168 (2019) no. 1, pp. 1-84 | MR | Zbl

[25] Nadler, David; Shende, Vivek Sheaf quantization in Weinstein symplectic manifolds (2021) (https://arxiv.org/abs/2007.10154)

[26] Seidel, Paul Homological mirror symmetry for the quartic surface, Memoirs of the American Mathematical Society, American Mathematical Society, 2015 | Zbl

[27] Treumann, David Remarks on the nonequivariant coherent-constructible correspondence for toric varieties (2010) (https://arxiv.org/abs/1006.5756)

[28] Vaintrob, Dmitry Coherent-constructible correspondences and log-perfectoid mirror symmetry for the torus, 2017 (https://math.berkeley.edu/~vaintrob/toric.pdf)

[29] Zhou, Peng Sheaf quantization of Legendrian isotopy (2018) (https://arxiv.org/abs/1804.08928)

[30] Zhou, Peng Twisted polytope sheaves and coherent-constructible correspondence for toric varieties, Sel. Math. New Ser., Volume 25 (2019) no. 1, 1 | MR | Zbl

[31] Zhou, Peng Lagrangian skeleta of hypersurfaces in ( × ) n , Sel. Math. New Ser., Volume 26 (2020) no. 2, 26 | MR | Zbl

[32] Zhou, Peng Variation of GIT and variation of Lagrangian skeletons I: Flip and Flop (2020) (https://arxiv.org/abs/2011.03719)

Cité par Sources :