Arveson's version of the Gauss-Bonnet-Chern formula for Hilbert modules over the polynomial rings

TL;DR

This paper extends Arveson's Gauss-Bonnet-Chern formula to Hilbert modules over polynomial rings with infinitely many variables, establishing new invariants and solving the finite defect problem for submodules.

Contribution

It proves the equivalence between local algebraicity of submodules and the validity of Arveson's formula, and develops the asymptotic invariants for infinite-variable modules.

Findings

Submodules being locally algebraic are equivalent to the Gauss-Bonnet-Chern formula holding.

Established asymptotic Arveson's curvature invariant and Euler characteristic for infinite-variable modules.

Proved that the Hardy space over infinitely many variables has no nontrivial finite-rank submodules.

Abstract

In this paper, we refine the framework of Arveson's version of the Gauss-Bonnet-Chern formula by proving that a submodule in the Drury-Arveson module being locally algebraic is equivalent to Arveson's version of the Gauss-Bonnet-Chern formula holding true for the associated quotient module. Moreover, we establish the asymptotic Arveson's curvature invariant and the asymptotic Euler characteristic for contractive Hilbert modules over polynomial rings in infinitely many variables, and obtain the infinitely-many-variables analogue of Arveson's version of Gauss-Bonnet-Chern formula. Finally, we solve the finite defect problem for submodules of the Drury-Arveson module $H^2$ in infinitely many variables by proving that $H^2$ has no nontrivial submodules of finite rank.