A local cohomology obstruction to small Cohen--Macaulay modules

We propose a local-cohomology obstruction to the existence of small Cohen–Macaulay modules over completed section rings of surfaces. Let $Y$ be a smooth connected complex projective surface, and let $H$ be an ample globally generated divisor with $K_Y\equiv 4H$ numerically. Put $c(Y,H)=15H^2\mathrm{/}8-\chi ({\text{OO}}_Y)$. The main claim is that, when $c(Y,H)>0$, every nonzero finite reflexive module $M$ over the completed vertex local ring of ${⨁}_{n\ge 0}{H}^0(Y,{\text{OO}}_Y(nH))$ satisfies ${\dim }_{\text{C}}{H}_{\text{mm}}^2(M)\ge c(Y,H)\text{rk}M$. The argument combines elementary transformations of reflexive extensions, Harder–Narasimhan slopes, Bogomolov’s inequality, and a two-section Koszul estimate. A fixed-source kernel estimate transfers the resulting cohomology to the punctured spectrum without requiring a grading on $M$. For the degree-six member of the six-line Hirzebruch–Kummer family, $H^2=1080$ and $\chi ({\text{OO}}_Y)=1926$, so the bound is $99\text{rk}M$. We give an explicit complete-intersection model for this application and spell out the argument leading to the claimed nonexistence of nonzero finite maximal Cohen–Macaulay modules.