Quantum (non-commutative) toric geometry: Foundations

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-...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2021-11, Vol.391, p.107945, Article 107945
Main Authors: Katzarkov, Ludmil, Lupercio, Ernesto, Meersseman, Laurent, Verjovsky, Alberto
Format: Article
Language:English
Subjects:
Algebraic Geometry
Complex Variables
Irrational fans and polytopes
Mathematics
Mirror symmetry
Moduli spaces
Non-commutative geometry
Quantum integrable systems
Symplectic Geometry
Toric geometry
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Summary: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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2021.107945