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On threefolds with trivial canonical bundle. (English) Zbl 0753.14035
Complex geometry and Lie theory, Proc. Symp., Sundance/UT (USA) 1989, Proc. Symp. Pure Math. 53, 103-134 (1991).
[For the entire collection see Zbl 0741.00047.]
The paper is a survey on the state of knowledge of the subject announced in the title. It begins with a discussion on K3 surfaces [in fact an updated report of some conjectures appearing in A. Weil’s “Final report on contract AF 18(603)–57”, see Oeuvres scientifiques, Collected papers. Vol. II, 1951-1964 (1979; Zbl 0424.01028); 390-395] and using this as a guide moves to threefolds with trivial canonical bundle. – The following aspects are discussed: (a) Kähler manifolds with trivial canonical class; (b) examples and first questions; (c) smooth classification of certain 6 manifolds; (d) the period map; (e) rational curves: existence, elementary modifications and smoothings.
The report is also of great interest for theoretical physicists since it touches the connections between the subject and the string theory.

14J30 \(3\)-folds
32J17 Compact complex \(3\)-folds
32J27 Compact Kähler manifolds: generalizations, classification