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A note on complex projective threefolds admitting holomorphic contact structures. (English) Zbl 0801.14014
From the introduction: “Let \(X\) be a \((2n + 1)\)-dimensional complex manifold. A (holomorphic) contact structure on \(X\) is a \(2n\)-dimensional non-integrable holomorphic distribution on \(X\). Dually, we can think of a contact structure on \(X\) as a holomorphic line subbundle \(L\) of \(\Omega^ 1_ X\) (the holomorphic cotangent bundle of \(X)\) such that if \(\theta\) is a local section of \(L\), then \(\theta \wedge (d \theta)^ n\) is everywhere nonzero”. This in particular implies that the canonical bundle \(K_ X\) is isomorphic to \((n + 1)L\). In the paper a complete classification of complex projective contact threefolds is given. Precisely, it is proved that if \(X\) is a complex projective contact threefold, then \(X\) is isomorphic to either \(\mathbb{C} \mathbb{P}^ 3\), or \(\mathbb{P} (T_ M)\) for some smooth projective surface \(M\). Interesting examples are also given.

MSC:
14J30 \(3\)-folds
32J17 Compact complex \(3\)-folds
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