Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero

We exhibit smooth polarized surfaces $(Y,H)$ over $\mathbf{C}$ carrying no nonzero arithmetically Cohen-Macaulay bundle of any rank. Equivalently, their section rings are three-dimensional normal $\mathbb{N}$-graded $\mathbf{C}$-domains, with an isolated singularity at the vertex, admitting no nonze