Abstract
In this note, we prove that the syzygy bundle M L M_L is cohomologically stable with respect to L L for any ample and globally generated line bundle L L on an Enriques (resp. bielliptic) surface over an algebraically closed field of characteristic ≠ 2 \neq 2 (resp. ≠ 2 , 3 \neq 2,3 ). In particular our result on complex Enriques surfaces improves the result of Torres-López and Zamora [Beitr. Algebra Geom. (2021)] (Corollary 3.5) by removing the condition on Clifford index. Together with the results of Camere [Math. Z 271 (2012), pp. 499–507] and Caucci–Lahoz [Bull. Lond. Math. Soc. 53 (2021), pp. 1030–1036], it implies that M L M_L is stable with respect to L L for an ample and globally generated line bundle L L on any smooth minimal complex projective surface X X of Kodaira dimension zero.
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