Continuous CM-regularity and generic vanishing

TL;DR

This paper explores the relationship between continuous CM-regularity and generic vanishing on polarized irregular varieties, extending the concept to real-valued functions and deriving syzygetic consequences for certain projective bundles.

Contribution

It proves that continuously 1-regular sheaves are GV on many pairs, extends continuous CM-regularity to real functions, and links these to syzygetic properties of projective bundles.

Findings

Continuously 1-regular sheaves are GV on many pairs including polarized abelian varieties.

Continuous CM-regularity can be extended to a real-valued function on $N^1(X)_{\mathbb{R}}$.

$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ satisfies $N_p$ property under certain conditions.

Abstract

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.