Comptes Rendus
Géométrie algébrique, Physique mathématique
Toric mirror symmetry revisited
Comptes Rendus. Mathématique, Volume 360 (2022), 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
Vivek Shende 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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2022__360_G7_751_0,
     author = {Vivek Shende},
     title = {Toric mirror symmetry revisited},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {751--759},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.304},
     language = {en},
}
TY  - JOUR
AU  - Vivek Shende
TI  - Toric mirror symmetry revisited
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 751
EP  - 759
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.304
LA  - en
ID  - CRMATH_2022__360_G7_751_0
ER  - 
%0 Journal Article
%A Vivek Shende
%T Toric mirror symmetry revisited
%J Comptes Rendus. Mathématique
%D 2022
%P 751-759
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.304
%G en
%F CRMATH_2022__360_G7_751_0
Vivek Shende. Toric mirror symmetry revisited. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 751-759. doi : 10.5802/crmath.304. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.304/

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

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

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

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

[6] Bohan Fang; Chiu-Chu Melissa Liu; David Treumann; Eric Zaslow 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] Dennis Gaitsgory 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] Dennis Gaitsgory; Nick Rozenblyum A study in derived algebraic geometry: Volume I: Correspondences and duality, Mathematical Surveys and Monographs, 221, American Mathematical Society, 2017 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :

Commentaires - Politique