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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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##### References:
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