On the existence of polarized varieties without weakly Ulrich bundles

Source abstract

We give an obstruction to the existence of bundles $\mathcal E$ on a smooth $n$-dimensional polarized variety such that $\mathcal E$ is $a$-regular and ${\mathcal E}^*(K_X+(n+1)H)$ is $b$-regular. In particular, this gives an obstruction to the existence of weakly Ulrich bundles. Using the latter and some recent examples of [A1] in dimension $2$, we prove that for any $n \ge 2$ there exist many smooth $n$-dimensional varieties $X$ with a very ample line bundle $H$, such that there is no weakly Ulrich bundle for $(X,H)$.