Cohomology classes of complex approximable algebras

Huayi Chen introduces the notion of an approximable graded algebra, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asked if any such algebra is the graded ring of a big line bundle on a projective variety. This was proved to be false in a previous paper of the author's, who subsequently proved that any such algebra is associated to an infinite Weil divisor. In this paper, we show that over the complex numbers, this infinite Weil divisor necessarily has finite cohomology class.Comment: 8 pages. arXiv admin note: text overlap with arXiv:1709.0694

Volume functions of linear series

The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole N\'eron--Severi space, thus giving rise to a basic invariant of the underlying projective variety. Analogously, one can also define the volume function of a possibly non-complete multigraded linear series. In this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties. In the multigraded setting, relying on the work of Lazarsfeld and Musta\c{t}\u{a} (2009) on Okounkov bodies, we show that any continuous, homogeneous, and log-concave function appears as the volume function of a multigraded linear series. By contrast we show that there exists countably many functions which arise as the volume functions of projective varieties. We end the paper with an example, where the volume function of a projective variety is given by a transcendental formula, emphasizing the complicated nature of the volume in the classical case.Comment: 16 pages, minor revisio

The effect of low volume sprint interval training in patients with non-alcoholic fatty liver disease

Objectives: Exercise is an important part of disease management in patients with non-alcoholic fatty liver disease (NAFLD), but adherence to current exercise recommendations is poor. Novel low-volume sprint interval training (SIT) protocols with total training time commitments of ≤30 min per week have been shown to improve cardiometabolic risk and functional capacity in healthy sedentary participants, but the efficacy of such protocols in the management of NAFLD remains unknown. The aim of the present study was to examine whether a low-volume SIT protocol can be used to improve liver function, insulin resistance, body composition, physical fitness, cognitive function and general well-being in patients with NAFLD.Methods: In the present study, 7 men and 2 women with NAFLD (age: 45±8 y, BMI: 28.7±4.1 kg·m−2) completed a 6-week control period followed by 6 weeks of twice-weekly SIT sessions (5-10×6-s ‘all-out’ cycle sprints). Body composition, blood pressure, liver function, metabolic function, functional capacity, cognitive function and quality of life were assessed at baseline, following the control period, and following the SIT intervention.Results: Walking speed during the walk test (+12%), estimated V̇O2max (+8%), verbal fluency (+44%), and blood platelet count (+12%; all p<0.05) significantly increased during the control period. These measures remained significantly raised compared to baseline following the SIT intervention, but did not significantly change any further compared to the post-control time-point. Diastolic blood pressure decreased from 87±10 to 77±8 mm Hg from the end of the control period to the end of the SIT intervention (p<0.05).Conclusion: This study does not support the use of 6 weeks of a low volume SIT protocol involving twice-weekly sessions with 5-10×6-s ‘all-out’ cycle sprints as an intervention for NAFLD disease management