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The Kähler cone on Calabi-Yau threefolds. (English) Zbl 0766.14035
Let \(X\) be a Calabi-Yau threefold (i.e. a smooth projective threefold with \(K_ X=0\) and \(h^ 1({\mathcal O}_ X)=0=h^ 2({\mathcal O}_ X))\) and let \(K\) be the Kähler cone in \(H^ 2(X,\mathbb{R})\). The author studies the behaviour of \(K\) under deformations. In particular he proves that, if \(\pi:X\to B\) is the Kuranishi family of \(X\) over a polydisc \(B\) in \(H^ 1(X,T_ X)\), \(K\) is invariant over the dense subset of \(b\in B\) for which \(X_ b\) does not contain a smooth elliptic ruled surface. — The proof is based on the study of the primitive birational contractions which contract some irreducible surface to a curve of
genus \(g>0\).

MSC:
14J30 \(3\)-folds
14E05 Rational and birational maps
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