Semistability of Syzygy Bundles Associated to Ulrich Bundles on Projective Varieties of Arbitrary Dimension

Let $X$ be a smooth irreducible projective variety of dimension $n\ge 3$ over an algebraically closed field of characteristic zero, polarized by a very ample line bundle ${\text{OO}}_X(1)$. Let $\text{E}$ be an Ulrich bundle on $X$. We prove that there exists an explicitly computable integer $M\gg 0$ such that for every $m\ge M$ the global syzygy bundle $S_{\text{E}(m)}$ is slope semistable with respect to ${\text{OO}}_X(1)$. This confirms Conjecture~3.11 of Miró-Roig.