2022/08/27 by Debaditya Raychaudhury, Raychaudhury, Debaditya
Mathematics · Medicine · #14F06 #14F17 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Intracerebral and Subarachnoid Hemorrhage Research #Neurosurgical Procedures and Complications
paper · pdf · doi:10.48550/arxiv.2208.13096
openalex publication_date 2022/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 integer 1≤ 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(E) associated to a 0-regular bundle E on polarized abelian varieties. In particular, we show that Oℙ(E)(1) satisfies 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 written by Atsushi Ito.