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How do sheaves arise in studying solutions to ordinary differential equations?

EDIT: Is it possible to construct non-isomorphic sheaves on a domain $D \subset \mathbb{R}^n$ using solution sets to differential equations?

EDIT: Is the sheaf of vector spaces arising from the solution set of a linear ODE necessarily a vector bundle?

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This is just a WAG, but I would suspect that exotic $\mathbb{R}^4$s would allow non-isomorphic sheaves. But even if true this is probably not what you have in mind. – Steve Huntsman Dec 26 '09 at 3:27
This is just a WAG, but I think he means "wild-ass guess." – Qiaochu Yuan Jan 12 '10 at 5:48
up vote 19 down vote accepted

Let $U$ be an open subset of $\mathbb R^n$, and let $X$ be a vector field on $U$. You can construct a sheaf $\mathcal F$ of solutions of the ODE $Xf=0$ by letting $\mathcal F(U)$, for each open subset $V\subseteq U$, be the vector space of all $C^\infty$ functions $f$ on $V$ such that $Xf=0$.

By changing the field $X$ you can certainly change the isomorphism clas of $\mathcal F$.

Let $U=\mathbb R^2\setminus\{(0,0)\}$, define fields $X_1(x,y)=\Bigl((\frac1r-1)\frac xr-y,(\frac1r-1)\frac yr+x\Bigr)$ and $X_2(x,y)=(y,-x)$ and consider the corresponding sheaves $\mathcal F_1$ and $\mathcal F_2$. It is not difficult show show that $\mathcal F_1(U)$ is one-dimensional as a real vector space, while $\mathcal F_2(U)$ is infinite dimensional. It follows that $\mathcal F_1\not\cong\mathcal F_2$.

vector fields

Notice that $\mathcal F_1$ and $\mathcal F_2$ are locally isomorphic. This follows easily from the fact that the fields $X_1$ and $X_2$ are non-zero on their domain.

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Sorry, I left a comment here a minute ago that was complete nonsense. The reason I was completely confused: both the OP and you are using the term "ODE"... Shouldn't you call them PDE's when the function has several variables? I realize that you consider a single vector field (that is a single linear PDE of first order)... is it customary to call them ODE's in such settings? – t3suji Dec 26 '09 at 16:15
I think of these things as ODEs because for the most part they reduce to ODEs. Strictly, from an ODE one can construct sheaves on open sets of $\mathbb R$, but those are not very interesting! On the other hand, for ODEs on $\mathbb C$ things are considerably more interesting... – Mariano Suárez-Alvarez Dec 26 '09 at 17:11
Of course, I assumed that OP was talking about sheaves on the complex plane (and I think it is not just me: rajamanikkam answer seems to work better in complex settings). It is just that when I saw the statement `the space of solutions of this ODE is infinite-dimensional' I was somewhat confused. – t3suji Dec 26 '09 at 21:27
jvp, I think that "The sheaf of solutions of the ODE $Xf=0$" is quite clear and consistent with current terminology... – Mariano Suárez-Alvarez Jan 12 '10 at 3:06
Mariano, could you recommend some good references (books, papers, or just lecture notes) on sheaves and differential equations more or less in the spirit of what you said here? If possible, I'd like something less abstract and more down-to-earth than the Hotta, Takeuchi & Tanisaki book mentioned below. Many thanks in advance! – mathphysicist Jan 19 '10 at 23:32

I will start commenting on Mariano's answer. I believe it is a perfect answer for the question

How do sheaves arise in studying solutions of differential equations ?

but not for the question

How do sheaves arise in studying solutions to ordinary differential equations ?

According to the current terminology a function $f$ satisfying $X(f)=0$ is not a solution of the vector field $X$ but a first integral. Moreover, if $X = a(x,y) \partial_x + b(x,y) \partial_y$ then $$ X(f) = a \partial_x f + b \partial_y f . $$ Thus $X(f)=0$ is a PDE and not an ODE. Indeed t3suji made the same point at a comment on Mariano's answer. I understand the solutions of (the ODE determined by) $X$ as functions $\gamma : V \subset \mathbb R \to U$ satisfying $X(\gamma(t))=\gamma'(t)$ for every $t \in V$. Notice that here indeed we have a system of ODEs.

A vector field can be thought as autonomous differential equation and I do not see clearly how to consider the sheaf of its solutions.

On the other hand when we have a non-autonomous ordinary differential equation then there is its sheaf of solutions. This sheaf is a sheaf over the time variable only and not the whole space. ( At this point it is natural to talk about connections and/or jet bundles but I will try to keep things as elementary as possible. )

Note that in general the sheaf of solutions will not be a sheaf of vector spaces: the sum of two solutions, or the multiplication of a solution by a constant need not to be a solution. This will occur only when the differential equation is linear.

The differential equations $y'(t) = y$ and $y'(t) = y^2$, both defined over the whole real line, are examples of differential equations with non-isomorphic sheaves of solutions. The solutions of the first ODE are the multiples of $\exp t $ and define a sheaf of $\mathbb R$-modules. The solutions of the second ODE are zero and $\frac{1}{\lambda - t}$ with $ \lambda \in \mathbb R$. They do define a sheaf of sets, but not a sheaf of $\mathbb R$-modules.

To obtain examples of linear differential equations with non-isomorphic sheaves, one has to have nontrivial fundamental group on the time-variable of the differential equation. Thus it is natural to consider complex differential equations over $\mathbb C^{\ast}$.

The equations $y'(z) = \frac{ \lambda y(z)}{z}$ parametrized by $\lambda \in \mathbb C$ have non-isomorphic sheaves of solutions. More precisely,

  • if $\lambda \in \mathbb Z$ then the solution sheaf is the free $\mathbb C$-sheaf of rank one (solutions of the ODE are complex multiples of $z^{ \lambda }$);
  • if $\lambda \in \mathbb Q - \mathbb Z $ then the solution sheaf has no global sections but some tensor power of it does;
  • if $\lambda \in \mathbb C - \mathbb Q$ then the solution sheaf has no global sections nor any of its powers does.
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Being a solution to a differential equation is a local condition, so solutions to a differential equation are naturally a sheaf.

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One way is through $D$-modules, perverse sheaves, and the Riemann-Hilbert correspondence. A good reference is: "D-Modules, Perverse Sheaves, and Representation Theory", by Hotta, Takeuchi & Tanisaki.

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