I learned very recently that in the early-mid 90s Gromov and Shubin proved a generalization of the Riemann-Roch theorem. Let $X$ be a compact closed ${\cal C}^\infty$-variety of dimension $n\geq2$. Suppose that $\cal E$ and $\cal F$ are complex vector bundles on $X$ of rank $q$ and let $A:{\cal E}\rightarrow{\cal F}$ be an elliptic differential operator of order $d$. Then for a divisor $D$ on $X$ one can define the space $L(D,A)$ of sections $f$ of $\cal E$ such that $Af=0$ and the zeroes and poles of $f$ are subordinated to $D$ in the usual sense. Then their basic result is that $\dim(L(D,A))-\dim(L(-D,A^t))={\rm ind}(A)+q\deg(D)$ where ${\rm ind}(A)$ is the index of $A$ and $\deg(D)$ is the degree of $D$ (whose definition is slightly different then the usual "algebraic" one).

Not being really an expert of elliptic operators, I wonder how much one can push analogies with the holomorphic (or algebraic) situation. For instance, suppose (to make things simple) that $X$ is a compact Riemann surface ($n=2$) and that $\cal E$ and $\cal F$ are line bundles. Let $f$ be a global ${\cal C}^\infty$-section of $\cal E$ such that $Af=0$. Is there any hope that the degree of the divisor of $f$ (possibly suitably modified) is 0?