Global 2-forms on regular 3-folds of general type
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.
scientific article; zbMATH DE number 558989
scientific article