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Flops. (English) Zbl 0645.14004
A flop is an operation that starts with a three dimensional algebraic (or complex analytic) variety X with at worst canonical singularities, removes a compact curve C which has zero intersection with the canonical class and replaces it with a differently embedded curve C’ to obtain X’. The present article generalizes earlier results of M. Reid [in Algebraic Varieties and Analytic Varieties, Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1, 131-180 (1983; Zbl 0558.14028)] and Y. Kawamata [Ann. Math., II. Ser. 127, No.1, 93-163 (1988)].
The article has three main results. First it is proved that if X has terminal singularities then X’ has the same analytic singularities as X. Second, some of the proofs of the above articles are simplified and generalized to analytic threefolds. Third, it is shown that if X and X’ are compact complex threefolds such that their canonical classes have nonnegative intersection with any curve then any bimeromorphic map between X and X’ is a composite of flops.
Reviewer: J.Kollár

MSC:
14E99 Birational geometry
14J30 \(3\)-folds
32S05 Local complex singularities
14B05 Singularities in algebraic geometry
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