Pluricanonical maps of varieties of Albanese fiber dimension two (Q2509918)

Let \(X\) be a smooth complex projective irregular variety of general type, i.e., a variety with Kodaira dimension \(\kappa(X)=\dim X\) and \(q(X)= h^1(X, {\mathcal{O}}_X)>0\). Let the Albanese fiber dimension \(e=\dim X- \dim a(X)\), where \(a: X\rightarrow \mathrm{Alb}(X)\) is the Albanese map. The \(n\)-th pluricanonical map \(\varphi_{|nK_X|}\) of \(X\) is defined by the basis of the vector space \(H^0(X, {\mathcal{O}}(nK_X))\), where \(K_X\) is the canonical divisor of \(X\). In this paper, with the above assumption, the author proves the following two theorems. Theorem 1.1. If \(e=2\), then \(\varphi_{|6K_X|}\) is birational. Theorem 1.2. If \(e=2\) and the translates through \(0\) of all components of \[ V^0(\omega_X) = \{\alpha \in \mathrm{Pic}^0(X), h^0(\omega_X\otimes \alpha)>0\} \] generates \(\mathrm{Pic}^0(X)\), then \(\varphi_{|5K_X|}\) is birational. The following are known results. Chen and Hacon proved that \(\varphi_{|6K_X|}\) is birational if \(e=0\) and \(\varphi_{|(e+5)K_X|}\) is birational when \(e=1\) or \(2\). Later Jiang, Lahoz and Tirabassi showed that \(\varphi_{|3K_X|}\) is birational when \(e=0\). Jiang and the author proved that \(\varphi_{|4K_X|}\) is birational when \(e=1\).