TY - JOUR

T1 - Quantum (non-commutative) toric geometry

T2 - Foundations

AU - Katzarkov, Ludmil

AU - Lupercio, Ernesto

AU - Meersseman, Laurent

AU - Verjovsky, Alberto

N1 - Funding Information:
E.L. would like to thank FORDECYT (CONACYT), IMATE-UNAM, NRU HSE, RF government grant, ag. 14.641.31.0001, the Institute for Mathematical Sciences of the Americas, the Simon's Foundation and the Moshinsky Foundation, the University of Geneva, the QUANTUM project from the University of Angers, the Laboratory of Mirror Symmetry HSE Moscow and the proyecto de ciencia básica de Conacyt CB-2017-2018-A1-S-30345-F-3125.
Funding Information:
L.M. would like to thank the kind support of the QUANTUM project from the University of Angers , UMI CNRS 2001 LaSol and the Institute for Mathematical Sciences of the Americas (Simons Foundation and University of Miami).
Funding Information:
A.V. would like to thank the support of the University of Angers , QUANTUM project UMI CNRS 2001 LaSo, IMSA in Miami and DGAPA PAPIIT project IN108120 UNAM , Mexico.
Funding Information:
Acknowledgments. L.K. was supported by a Simons Investigator Award, the Simons collaborative Grant - HMS , the National Science Foundation , as well as he has been partially supported by the NRU HSE , RF government grant, ag. 14.641.31.0001 , Simons Principal Investigator Grant, CKGA VIHREN grant Project no. KP-06-DV-7. Much of the research was conducted while the authors enjoyed the hospitality of IMSA Miami and Laboratory of Mirror Symmetry HSE Moscow.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/11/19

Y1 - 2021/11/19

N2 - In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we show that their category is equivalent to a certain category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.

AB - In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we show that their category is equivalent to a certain category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.

KW - Irrational fans and polytopes

KW - Mirror symmetry

KW - Moduli spaces

KW - Non-commutative geometry

KW - Quantum integrable systems

KW - Toric geometry

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U2 - 10.1016/j.aim.2021.107945

DO - 10.1016/j.aim.2021.107945

M3 - Article

AN - SCOPUS:85112737593

VL - 391

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107945

ER -