Noncommutative two-tori with real multiplication as noncommutative projective varieties

By Alexander Polishchuk. 2002

48 citations recorded by the source corpus.

Abstract We define analogues of homogeneous coordinate algebras for noncommutative two-tori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative two-torus with real multiplication. These results are based on the techniques of [Categories of holomorphic bundles on noncommutative two-tori. math.AG/0211262 ] allowing to interpret all the data in terms of autoequivalences of the derived categories of coherent sheaves on elliptic curves.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7b2faf46462301f41f2ee19730120f51da684d34

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Classification of holomorphic vector bundles on noncommutative two-tori (2003, methodology): “Then we combine these results with the techniques of [4] where the category C′ was described in terms of coherent modules over a certain algebra.”
Noncommutative Geometry, Quantum Fields and Motives (2007, background): “At the algebraic level, noncommutative tori can be described through the algebraic noncommutative space obtained by Polishchuk [270] from the holomorphic bundles of [105].”
Derived categories of irreducible projective curves of arithmetic genus one (2005, background): “follows that none of the components of f : ⊕ Xi → B or g : B → ⊕ Yj is zero, because none of the Xi could be a direct summand of Y [−1] and none of the Yj could be a summand of X[1].”