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

RESUMO

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 nonzero finitely generated graded maximal Cohen-Macaulay module. To the author's knowledge no such ring was previously known. Their existence contrasts with the theorem of Hartshorne, Hochster and Peskine-Szpiro that a three-dimensional $\mathbb{N}$-graded domain over a perfect field of characteristic $p>0$ always has one. As a consequence, in characteristic zero Hochster's small Cohen-Macaulay conjecture admits no graded refinement. The nonexistence criterion is numerical: for $|H|$ base-point-free with finite morphism, $H^2<K_Y^2-8\chi(\mathcal{O}_Y)=\tau(Y)$, the signature of $Y$ over $\mathbf{C}$, leaves no nonzero $\mathcal{E}$ with $H^1(Y,\mathcal{E}(tH))=0$ for all $t$. For Ulrich bundles, being semistable, this is Bogomolov's inequality; what is new is that no stability hypothesis and no condition on the Hilbert polynomial are needed, the Harder-Narasimhan filtration of $\mathcal{E}$ being compared instead with the Horrocks splitting of its direct image on $\mathbb{P}^2$. Hirzebruch's Hesse surfaces with $H_n=4A_n-E_n$, $n\ge3$, qualify. In dimension $m$ the obstruction reads $(m+1)H^m\ge(K_X^2-2c_2(X))\cdot H^{m-2}$, the inequality Lopez proved for Ulrich bundles; products and complete intersections inside them give examples in every dimension $\ge3$, and the threefold examples show that the characteristic-$p$ hypothesis in the criterion of Shimomoto-Tavanfar is essential.