Global holomorphic 2-forms and pluricanonical systems on threefolds
The change of zero locus of a global holomorphic 2-form on a threefold under birational transformations is investigated. Let \(X\) be a complex projective threefold and let \(\omega\in H^0(X,\Omega_X)\) be a nontrivial \((\omega\neq 0)\) global holomorphic 2-form. Let \(Z(\omega)\) be the zero locus of \(\omega\). In the smooth case the following result is obtained: Theorem 2.2: Let \(X\) be a smooth threefold of nonnegative Kodaira dimension with \(h^0(X, \Omega^2_X) =h^2(X,{\mathcal O}_X)=l\geq 4\). Assume that there is a subspace \(V\subset H^0(X, \Omega^2_X)\) of dimension 4 and the dimension of \(Z(\omega)\) is bounded from above by 1 for \(0\neq\omega\in V\). Then \(X\) has a smooth minimal model. Further, it is proved that existence of 2-forms with certain conditions on their zero loci on a threefold of non-negative Kodaira dimension limits types of terminal singularities appearing on its minimal models. As a consequence of the restriction on the types of terminal singularities and Reid's Riemann-Roch formula, the following result is obtained: Let \(X\) be a smooth complex projective threefold of general type. Let \({\mathcal E}\) be the subsheaf generated by global 2-forms in \(\Omega^2_X\). Assume \(h^2(X,{\mathcal O}_X)\geq 2\) and \(\dim D({\mathcal E})\leq 1\), where \(D({\mathcal E})\) is the degenerating locus of \({\mathcal E}\). Then there is a universal \(N\) such that \(NK_X\) defines a birational map from \(X\).
scientific article; zbMATH DE number 1552929