Non-commutative complex differential geometry. (English) Zbl 1286.32008
The non-commutative algebraic geometry has its roots in the seminal paper by J.-P. Serre [Ann. Math. (2) 61, 197–278 (1955; Zbl 0067.16201)]; the main result says that the category of coherent sheaves over an algebraic variety is equivalent to such of the projective modules over its coordinate ring. The bottom line is that projective modules “don’t care” whether the ring is commutative or not, thus in principle all algebraic geometry can be rephrased in terms of the non-commutative algebras. This observation would remain a mere curiosity if not an astounding example of the non-commutative coordinate rings of elliptic curve (the Sklyanin algebras) was constructed by E. K. Sklyanin [Funct. Anal. Appl. 16, 263–270 (1983); translation from Funkts. Anal. Prilozh. 16, No. 4, 27–34 (1982; Zbl 0513.58028)]. For a survey of this already formidable area of mathematics we refer the reader to J. T. Stafford and M. Van den Bergh [Bull. Am. Math. Soc., New Ser. 38, No. 2, 171–216 (2001; Zbl 1042.16016)] and A. V. Odesskij [Russ. Math. Surv. 57, No. 6, 1127–1162 (2002); translation from Usp. Mat. Nauk 57, No. 6, 87–122 (2002; Zbl 1062.16035)].
The paper under review uses the Chow Embedding Theorem linking the algebraic geometry with compact complex manifolds to recast the latter in terms of non-commutative algebras; the study is focused on the non-commutative analogues of almost complex structures, holomorphic curvature, cohomology and holomorphic sheaves. The paper is clearly written and well explained; it is highly recommended to the graduate students as well as experts in the field of noncommutative geometry.
The paper under review uses the Chow Embedding Theorem linking the algebraic geometry with compact complex manifolds to recast the latter in terms of non-commutative algebras; the study is focused on the non-commutative analogues of almost complex structures, holomorphic curvature, cohomology and holomorphic sheaves. The paper is clearly written and well explained; it is highly recommended to the graduate students as well as experts in the field of noncommutative geometry.
Reviewer: Igor V. Nikolaev (Southbridge)
MSC:
| 32J99 | Compact analytic spaces |
| 14A22 | Noncommutative algebraic geometry |
| 32L10 | Sheaves and cohomology of sections of holomorphic vector bundles, general results |
| 32Q60 | Almost complex manifolds |
| 58B34 | Noncommutative geometry (à la Connes) |
| 46L87 | Noncommutative differential geometry |
Keywords:
non-commutative algebraic geometry; non-commutative analogues of almost complex structures; non-commutative analogues of holomorphic sheavesReferences:
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