Kernel of a non-integrable connection

Question

The Riemann-Hilbert correspondence states that the kernel of an integrable (zero curvature) connection is a local system. Here, a connexion on a vector bundle $E$ over a manifold $X$ is a morphism of sheaves \begin{equation} \nabla : {\cal O}_X(E) \to \Omega_X(E) \end{equation} which satisfies a Liebniz rule.

I wonder what happens if the connexion has curvature. $\ker \nabla$ is a well-defined subsheaf of ${\cal O}_X(E)$ ; but is it still a local system of smaller rank ? Or can it be a skyscraper sheaf ?

Do you have examples of such sheaves ? Can one say something about their general behaviour ? (I know it's vague, I'm really looking for intuition).

I am mostly interested in the holomorphic case, but any example is welcome. The relative case $X/B$ is also of interest to me.

Answer 1

Parallel sections are invariant under parallel transport along all holomorphic disks, so if holomorphic somewhere, they are holomorphic everywhere. The kernel of the connection is a linear subspace of the holomorphic sections, so a linear system. If you start with a nonzero holomorphic section of a holomorphic vector bundle, you can construct local holomorphic connections which preserve it, in some neighborhood of any chosen point.

Edit: each parallel section is uniquely determined by its initial value at a point. As you travel around loops, it undergoes some monodromy. But the set of initial values at a point for which there is a parallel section is a vector subspace $F_{x_0}$ of the fiber $E_{x_0}$ at that point $x_0$. That fiber $F_{x_0}$ goes around in parallel inside $E$, making a vector bundle $F$ with flat connection.

Edit: When the connection is real analytic, we can parallel transport along any path, analytically continuing any given local parallel section, as linear ODE always have solutions. Analyticity ensures that the resulting local extension of a parallel section remains parallel. So all parallel sections extend to the universal covering space. Hence the fibers $F_{x_0}$ are indeed all of the same dimension, and carried invariantly under parallel transport. For $C^{\infty}$ connections, this won't work; you can have local parallel sections which do not extend over "bumps", and $F_{x_0}$ will change dimension.

Answer 2

Indeed, $\ker\nabla$ is a well-defined subsheaf of $\mathcal{E}$. In particular, it is a sheaf of finite-dimensional vector spaces with the stalk-rank $\dim(\ker\nabla)_x$ bounded by the vector bundle rank $\text{rk}$ $E$. But $\ker\nabla$ is not a local system any more if $\nabla$ is not flat. That happens if and only if $$ x\mapsto \dim(\ker\nabla)_x $$ is a locally constant function on $X$.

As far as I know, in the smooth setting, the best we can get is the following: define $$ X^{\leq d}:=\{x\in X | \dim(\ker\nabla)_x\leq d\}\,. $$ Then $\{X^{\leq d}\}_d$ is a locally finite collection of closed sets in $X$, and the restriction of $\ker\nabla$ to the subsets $X^{\leq d}-X^{\leq d-1}$ is locally constant for all $d$ ($\leq \text{rk}$ $E$).

This is proved in Brian Conrad's notes on the Riemann-Hilbert correspondence (link). Maybe there's a better/different description of such sheaves. Edit: They're clearly constructible sheaves on $X$ (equivalently, representations of the exit-path category (see Treumann)).

This is a result that relies on uniqueness of local parallel sections (as a consequence of the initial value problem for first-order ODEs), but tells nothing about existence. In fact, the sheaf $\ker\nabla$ may as well be empty, i.e. concentrated in $X^{\leq 0}$: an element of the fiber $\mathcal{E}_x$ may not extend even locally to a parallel section. The obstruction to the existence of such solution lies exactly on the curvature of the connection. (When the connection is flat, parallel transport is independent of path, and we may use parallel transport of a vector $v\in \mathcal{E}_x$ in a small simply connected neighborhood of $x$ to define a parallel section of $\mathcal{E}$ extending $v$.)

Answer 3

In (real) differential geometry the kernel of a connection $\nabla$ is called the space of parallel sections. These form a bundle whose fiber over a point $p$ is the space of fixed points of the defining representation of the holonomy group $\mathrm{Hol}(\nabla, p)$. (And, in fact, the structure group of the bundle can be reduced to the holonomy group.) By the Ambrose-Singer theorem, the Lie algebra of $\mathrm{Hol}(\nabla, p)$ is given by curvature of $\nabla$ at $p$.