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Three-dimensional flops and noncommutative rings. (English) Zbl 1074.14013
This paper gives a new proof, based on noncommutative rings, of T. Bridgeland’s theorem [Invent. Math. 147, No. 3, 613–632 (2002; Zbl 1085.14017)] that says that two three dimensional smooth varieties \(Y\), \(Y^+\) related by a flopping transformation \(Y\to X\leftarrow Y^+\) have equivalent bounded derived categories of coherent sheaves. There are extensions of this result for normal varieties with isolated smooth singularities, due to J.-C. Chen [J. Differ. Geom. 61, No. 2, 227–261 (2002; Zbl 1090.14003)] and for some non-Gorenstein singularities, due to Y. Kawamata [in: Algebraic geometry, de Gruyter, Berlin, 197–215 (2002; Zbl 1092.14023)]. The proof in this paper is really nice, and is based in the construction of vector bundles \(\mathcal P\) and \(Y\) and \(\mathcal Q^+\) on \(Y^+\) such that the direct image on \(X\) of the endomorphism sheaves algebras of \(\mathcal P\) and \(\mathcal Q^+\) are isomorphic. If \(\mathcal A\) is this noncommutative algebra, then the author proves that the bounded derived categories of coherent sheaves on \(Y\) and on \(Y^+\) are both equivalent to the bounded derived category of right \(\mathcal A\)-modules, being thus equivalent. The bundles \(\mathcal P\) and \(\mathcal Q^+\) are related to certain categories of perverse sheaves associated with the flop which appear also in the original Bridgeland proof. The author also describes the projective generators of these categories of perverse sheaves. Actually, some more general results are proven, among them, some higher dimensional generalizations (also considered by Chen).

MSC:
14E05 Rational and birational maps
14E30 Minimal model program (Mori theory, extremal rays)
18E30 Derived categories, triangulated categories (MSC2010)
14A22 Noncommutative algebraic geometry
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