The acyclicity criterion for semiorthogonal indecomposability of minimal surfaces

Let $X$ be a minimal surface. The structure sheaf $\mathcal{O}_X$ is acyclic when

$$H^i(X,\mathcal{O}_X)=0$$

for $i=1,2$.

Acyclicity criterion. The surface $X$ is not semiorthogonally indecomposable if and only if $\mathcal{O}_X$ is acyclic.

This conjecture makes precise in dimension two the proposed relationship between semiorthogonal indecomposability and the acyclicity of the structure sheaf. The paper proves semiorthogonal indecomposability for minimal surfaces of positive irregularity; the criterion concerns the remaining cases and is presented as an open conjecture.