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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?

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The analogy with the theory of holomorphic line bundles on Riemann surfaces is very precise. J. J. Duistermaat, in his paper ``On solutions of first order elliptic equations for sections of complex line bundles'', proves that every first order elliptic operator between complex line bundles over an oriented surface can be written as $Lu=\bar\partial u + a \bar{u}$, where $a$ is a $C^{\infty}$ section of a certain complex line bundle. This result has a purely local proof. But then he proves a more surprising result: out of the object $a$, one can canonically construct a new complex structure on the line bundle that $u$ lives in, which changes the equation from $\bar\partial u = -a\bar{u}$ to $\bar\partial v=0$. So in other words, the solutions of the original equation are precisely the space of holomorphic sections of some holomorphic line bundle. The degree of the divisor of a section is just given by the Chern class of the line bundle, which is purely topological, so you can calculate it straight from the original elliptic equation, and it doesn't change under the change of complex structure.

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Ben, I've been away from MO for a while and I read your answer just now. The result you're pointing at seems very relevant for a possible application I have in mind. Thanks a lot! –  Andrea Mori Jun 19 '11 at 14:41

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