On irregular threefolds and fourfolds with numerically trivial canonical bundle

Abstract

We prove that for a smooth projective irregular $3$-fold $X$ with $K_X\equiv 0$ and a nef and big divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\ge 3$ and all $P\in {\text{Pic}}^0(X)$. We also use the same method to deal with $4$-folds, and prove that for a smooth projective irregular $4$-fold $X$ with $K_X\equiv 0$ and an ample divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\ge 5$ and all $P\in {\text{Pic}}^0(X)$. These results are also optimal.

21 pages. Final version, to appear in Math. Zeit

References in corpus (2)

• On birational geometry of minimal threefolds with numerically trivial canonical divisors
• Pluricanonical maps of varieties of Albanese fiber dimension two