We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties $(X,\mathcal{ O}_X(1))$, 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 $\mathcal{ F}$ is GV. Here we answer the question in the affirmative for many pairs $(X,\mathcal{ O}_X(1))$ which includes the case of any polarized abelian variety. Moreover, for these pairs, we show that if $\mathcal{ F}$ is continuously $k$-regular for some integer $1\leq k\leq \dim X$, then $\mathcal{ F}$ is a GV$_{ -(k-1)}$ sheaf. Further, we extend the notion of continuous CM-regularity to a real valued function on the $\mathbb{ Q}$-twisted bundles on polarized abelian varieties $(X,\mathcal{ O}_X(1))$, and we show that this function can be extended to a continuous function on $N^1(X)_{ \mathbb{ R}}$. We also provide syzygetic consequences of our results for $\mathcal{ O}_{ \mathbb{ P}(\mathcal{ E})}(1)$ on $\mathbb{ P}(\mathcal{ E})$ associated to a $0$-regular bundle $\mathcal{ E}$ on polarized abelian varieties. In particular, we show that $\mathcal{ O}_{ \mathbb{ P}(\mathcal{ E})}(1)$ satisfies $N_p$ property if the base-point freeness threshold of the class of $\mathcal{ O}_X(1)$ in $N^1(X)$ is less than $\frac{ 1}{ p+2}$. This result is obtained using a theorem in the Appendix written by Atsushi Ito.
Categories: math.AG