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

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.