In this short post I want to note Deligne’s classic result on vector bundles with flat connections (let’s call them “differential equations”: terminology justified by the `cyclic vector theorem”) and give a stupid example illustrating the basic shape of things. I should thank Ananth Shankar for helping unconfuse me faster than I could unconfuse myself.

 

Let X be a smooth algebraic variety over \mathbb{C}. A vector bundle on X is a locally free coherent (Zariski) sheaf \mathcal{F}, and a connection is an additive map of sheaves

\nabla: \mathcal{F} \rightarrow \mathcal{F} \otimes_{\mathcal{O}_X} \Omega^1_{X/\mathbb{C}} 

satisfying the Leibniz law. There is the usual notion of flatness of a connection. A morphism between two such objects is an \mathcal{O}_X-linear map between the underlying sheaves making the obvious square commute.

 

One can also make the exactly parallel construction over the complex manifold X^{an}. Given an algebraic differential equation, we can analytify it and obtain an analytic differential equation: (-)^{an}: DE(X) \rightarrow DE(X^{an}).

Deligne’s result gives a functor in the other direction (let’s call it RH for “Riemann-Hilbert”)

RH: DE(X^{an}) \rightarrow DE(X).

Theorem (Deligne): The functor RH (exists and) is fully faithful, and the essential image RSDE(X) is characterised by taking a good compactification of X and restricting to those differential equations with regular singularities at the boundary.

This is a little counterintuitive at first. One imagines there being “more” objects in DE(X^{an}) than in DE(X) because one can write down more connections, but actually the extra freedom causes more stuff to become isomorphic, resulting in a “smaller” analytic category. Of course, one has \mathcal{F} \cong RH(\mathcal{F})^{an} but not (unless X is proper) an identification in the other direction.

 

Let’s see what happens in a stupid example. Let X=\mathbb{A}^1 and \mathcal{F} the trivial line bundle. Let’s take our favourite non-vanishing algebraic section v, and observe that any connection is automatically flat and determined by \nabla(v)

(since any other section can be written as f(t)v which by Leibniz is given by

\nabla(f(t)v) = v f'(t) \otimes dt + f(t) \nabla(v)).

If g(t) \in \mathbb{C}[t] let us denote by \nabla_g the connection taking v \mapsto g(t) v \otimes dt.

 

Let’s suppose we have a map \alpha: \nabla_g \rightarrow \nabla_h between two such differential equations. Let \alpha(v) = u(t) v, and the compatibility of \alpha with the connections gives the relation

u'(t) + u(t)h(t) = u(t)g(t).

In other words, u must satisfy a first order differential equation, and in fact we see that up to a constant factor

u(t) = e^{\int (g(t)-h(t))dt}

which is never algebraic unless g(t)-h(t)=0.

 

The upshot is that in the algebraic category each \nabla_g gives a distinct differential equation, while in the analytic category one can use the above recipe to construct isomorphisms between each of the \nabla_g and in particular they are all equal to \nabla_0, which is the unique algebraic connection with regular singularities at \infty.

 

 

Advertisements