I have a question about natural assignments of $K$-theory classes to open varieties. The question has turned out rather long, for which I apologize.
All varieties I consider are smooth over $k=\mathbf{C}$. The sorts of natural classes I want are things like $\sum_i [\,\Omega^i_X\,]y^i\in K_0(X)[\,y\,]$. Of course this is perfectly well defined regardless of whether $X$ is smooth and proper, but in the non-proper case I will lose the pushforward to a point: $\chi:K_0(X)\rightarrow K_0(*)$.
If I want to preserve this in the case of $U$ open, I might try to consider compactifications $(X,D)$ of $U$, ie $U\subset X$ is open inside a smooth and proper $X$ and $D$ is a divisor with simple normal crossings and irreducible components $(D_i)_{i\in I}$. Now let us define a sort of renormalized $K$-group:$$K_{0}^{\operatorname{ren}}(U):=\operatorname{lim}_{(X,D)}K_0(X)$$ where the limit is taken over pairs $(X,D)$ and $K$-theoretic pushforwards of maps which are the identity on $X\setminus D=U$. A class $\gamma$ in this renormalized $K$-theory is then a compatible system of classes $\gamma(X,D)$ so that whenever we have a map $\pi:(X',D')\rightarrow (X,D)$ which is the identity on the open piece $U$, we have $\pi_*\gamma(X',D')=\gamma(X,D).$ As we are taking a limit with respect to pushforward maps we obtain a well defined pushforward to a point: $\chi^{\operatorname{ren}}:K_0(U)^{\operatorname{ren}}\rightarrow\mathbf{Z}$.
Now I believe a simple argument will imply that any $K$-class on $U$ has some lift to renormalized $K$-theory, as I can choose some cofinal system of blow-ups and just lift inductively. At any rate this is not what I want. I am interested in natural lifts of $K$-classes on $U$ to classes in $K_0^{\operatorname{ren}}(U)$.
Definition. Here $X$ varies over smooth varieties. We call an assignment $X\mapsto \nu_X\in K_0(X)$ natural if it is compatible with etale pullbacks.
Example. $X\mapsto [\Omega^i_X]$ is a natural assignment of $K$-theory classes. So is for example the $K$ theory class associated to the sheaf of differential operators of order at most $8$ or whatever.
Definition. Let $\nu$ be a natural assignment of $K$ theory classes and let $U$ be an open variety. A natural renormalization is a lift of $\nu_{U}\in K_0(U)$ to an element $\nu^{\operatorname{ren}}_U\in K_0^{\operatorname{ren}}(U)$ of the form $$(X,D)\mapsto \nu_X+F(\delta_1,\delta_2,...),$$ where $F$ is some fixed integral symmetric polynomial in infinitely many variables and $\delta_i$ are the classes $[\,\mathcal{O}_{D_i}\,]$ of the boundary divisors $D_i$.
Example. I believe we can find natural lifts of the classes $\Omega^i_U$, given by log de Rham forms $\Omega^i_X(\log D)$.
Question. Are there any other examples of classes admitting natural lifts? I have a feeling that the only such should be differential forms, and some sketched computations with eg tangent sheaves supports this.
