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Global 2-forms on regular 3-folds of general type. (English) Zbl 0838.14032
Let $$V$$ be a smooth projective variety of $$\mathbb{C}$$-dimension = 3. The author is motivated by the following problem: Assume that $$V$$ is of general type. Then find some universal integer $$N$$ such that $$|NK |$$ defines a birational map where $$K$$ is the canonical bundle of $$V$$. For irregular 3-folds $$V$$ of general type, the problem was settled by Kollár. Therefore, in this paper, the author concentrates on regular 3-folds $$V$$ of general type, i.e. $$h^1 (V,{\mathcal O}_V) = 0$$. Let $${\mathcal H}^2 : = H^0 (V, \Omega^2_V) \otimes_\mathbb{C} {\mathcal O}_V \subset \Omega^2_V$$ be the subsheaf of $$\Omega^2_V$$ spanned by $$H^0 (V, \Omega^2_V)$$. It is clear that $\text{rank } {\mathcal H}^2 \leq 3.$ The main result of this paper is the following theorem:
The above problem admits a positive answer for regular 3-folds $$V$$ of general type, provided rank $${\mathcal H}^2 = 1$$ or 3. Some discussion when rank $${\mathcal H}^2 = 2$$ is carried out.

MSC:
 14J30 $$3$$-folds
Full Text:
References:
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