Cohomology classes of complex approximable algebras

TL;DR

This paper proves that complex approximable graded algebras are associated with infinite Weil divisors having finite cohomology class, advancing understanding of their geometric structure.

Contribution

It demonstrates that over the complex numbers, the infinite Weil divisor linked to an approximable algebra has a finite cohomology class, refining previous results.

Findings

Infinite Weil divisors associated with approximable algebras have finite cohomology class over complex numbers.

The result clarifies the geometric nature of approximable algebras in complex geometry.

The paper extends the understanding of the structure of graded algebras in the arithmetic and complex setting.

Abstract

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.