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Moishezon-Fano threefolds of index three. (English) Zbl 0816.14014

The author studies the Moishezon threefolds \(X\) with \(h^ 1 (X, {\mathcal O}_ X) = 0\), whose canonical bundle is \(K_ X = - nL\) \((n \geq 3)\) for some line bundle \(L\) with \(h^ 0 (X,L) \geq 2\). In particular he proves that:
(i) if \(n = 4\), \(X\) is isomorphic to \(\mathbb{P}^ 3\);
(ii) if \(n = 3\) and \(L\) has no fixed components, \(X\) is isomorphic either to a quadric or to a \(\mathbb{P}^ 2\)-bundle over \(\mathbb{P}^ 1\).
The situation differs from the analogous algebraic case: it is known that the unique projective Fano threefold of index 3 is, up to isomorphism, the quadratic hypersurface of \(\mathbb{P}^ 4\).

MSC:

14J30 \(3\)-folds
14J45 Fano varieties
32J17 Compact complex \(3\)-folds
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