Continuous CM-regularity and generic vanishing with an appendix by Atsushi Ito (Okayama University)

Abstract

We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties (X, OX(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, OX(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, OX(1)), and we show that this function can be extended to a continuous function on N1(X)ℝ. We also provide syzygetic consequences of our results for Oℙ(E)(1) on ℙ (ϵ) associated to a 0-regular bundle ϵ 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 OX(1) in N1(X) is less than 1/(p + 2). This result is obtained using a theorem in the Appendix A written by Atsushi Ito. © 2024 Walter de Gruyter GmbH, Berlin/Boston.