Continuous CM-regularity and generic vanishing

We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties (X, O X (1)), and its relation with the theory of generic vanishing. This continuous variant of the Castelnuovo–Mumford regularity was introduced by Mustopa, and he raised the question whether a continuously 1-regular such sheaf F is GV. Here we answer the question in the affirmative for many pairs (X, O X (1)) which includes the case of any polarized abelian variety. Moreover, for these pairs, we show that if F is continuously k-regular for some positive integer k ≤ dim X, then F is a GV−(k−1) sheaf. Further, we extend the notion of continuous CM-regularity to a real valued function on the ℚ-twisted bundles on polarized abelian varieties (X, O X (1)), and we show that this function can be extended to a continuous function on N 1(X)ℝ. We also provide syzygetic consequences of our results for Oℙ(E)(1) on ℙ(ɛ) associated to a 0-regular bundle ɛ on polarized abelian varieties. In particular, we show that Oℙ(E)(1) satisfies the Np property if the base-point freeness threshold of the class of O X (1) in N 1(X) is less than 1/(p + 2). This result is obtained using a theorem in the Appendix A written by Atsushi Ito.