Reference: Jacob Lurie’s talk
Hilbert proposed the following problem in 1900:
Can we find Fuchsian differential equations with prescribed monodromy?
Somehow the answer depends on the precise formulation of the question.
A Fuchsian system looks like this:
\[\frac{\mathrm{d} f_i}{\mathrm{d} z} = \sum_{j=1}^n A_{ij}(z) f_j\]
where \(A_{ij}(z)\) are rational functions having at most simple poles at some points \(a_i\), vanishing at \(\infty\) (This is required by Fuchsian).
\[A_{ij}(z) = \frac{P_{ij}(z)}{(z-a_1)(z-a_2)\cdots(z-a_k)}\]
where \(P_{ij}(z)\) are polynomials with degree \(< k\).
\[\frac{\mathrm{d} f}{\mathrm{d} z} = \frac{1}{2z}f\]
This is a Fuchsian system with singularity at \(0\), whose solutions look like \(c\sqrt{z}\), but it is only defined locally not globally.
Analytic continuation around \(0\) will give you a negative sign \(\sqrt{z}\mapsto -\sqrt{z}\). This is a phenomenon of monodromy.
Fix a Fuchsian system with singularity \(\{a_1,\dots,a_k\}\subset \mathbb{C}\), fixing any base point \(x_0\in \mathbb{C}\setminus \{a_1,\dots,a_k\}\), and a path \(\gamma\) from \(x_0\) to \(x_1\). The monodromy is the automorphism of the fiber \(\mathbb{C}^n\) at \(x_0\) induced by the analytic continuation along \(\gamma\), which gives a representation of the fundamental group \(\pi_1(\mathbb{C}\setminus \{a_1,\dots,a_k\},x_0)\) into \(GL(n,\mathbb{C})\)
\[\pi_1(\mathbb{C}\setminus \{a_1,\dots,a_k\},x_0)\to \mathrm{GL}(n,\mathbb{C})\]
called the monodromy representation.
Let us consider the Fuchsian system
\[\frac{\mathrm{d} f}{\mathrm{d} z} = \frac{\alpha}{z}f\]
where \(\alpha\in \mathbb{C}\), the solution looks like \(f=z^\alpha\), and the monodromy looks like
\[\mathbb{Z}=\pi_1(\mathbb{C}\setminus \{0\}) \to \mathrm{GL}(1,\mathbb{C})=\mathbb{C}^\times : n\mapsto e^{2\pi i n \alpha}.\]
Does every representation \(\pi_1(\mathbb{C}\setminus \{a_1,\dots,a_k\})\to \mathrm{GL}(n,\mathbb{C})\) come from a Fuchsian system?
Let \(X=\mathbb{P}^1_\mathbb{C}\) be the Riemann sphere and \(\mathcal{E}\) a trivial homomorphic bundle of rank \(n\). A system of equations of the form \(\frac{\mathrm{d} f_i}{\mathrm{d} z} = \sum_{j=1}^n A_{ij}(z) f_j\) is equivalent to a connection \(\nabla: \mathcal{E}\to \Omega^1_X \otimes_{\mathcal{O}_X} \mathcal{E}\), such that
\[\left\{\text{Solutions of }\frac{\mathrm{d} f_i}{\mathrm{d} z} = \sum_{j=1}^n A_{ij}(z) f_j\right\} \Leftrightarrow \{\text{Flat sections of } (\mathcal{E}, \nabla)\}\]
The Fuchsian condition is equivalent to saying \(\nabla\) is meromorphic with at most simple poles at \(\{a_1,\dots,a_k,\infty\}.\)
Flat sections of \(\mathcal{E}\) determine a local system on \(\mathbb{C}\setminus \{a_1,\dots,a_k\}\), which is a locally constant sheaf of vector spaces on \(\mathbb{C}\setminus \{a_1,\dots,a_k\}\).
In this sense, the Hilbert problem asks: does every local system arise from a Fuchsian system? And this question is generalized to any Riemann surface \(X\) not just \(\mathbb{P}^1_\mathbb{C}\).
Let \(X\) now be a Riemann surface and \(U = X\setminus \{a_1,\dots,a_k\}\) be the complement of \(k\) points.
The representation of \(\pi_1(U)\) are equivalent as local systems on \(U\)
By some differential geometry, that is also equivalent to a holomorphic bundle \(\mathcal{E}_U\) with a connection \(\nabla\).
But the representation of \(\pi_1(U)\) gives you a bundle with connection only on \(U\), can \(\mathcal{E}_U\) be extended to a holomorphic bundle \(\mathcal{E}_X\) with \(\nabla\) has at most simple poles at \(\{a_1,\dots,a_k\}\)?
The problem is local, you only have to solve it around the singularities.
We can assume \(X=B_\mathbb{C}(0,1)\) and \(U=B_\mathbb{C}(0,1)\setminus \{0\}\).
A representation of \(\pi_1(U)\cong \mathbb{Z}\) is given by a single invertible matrix \(B\in \mathrm{GL}_n(\mathbb{C})\), this data determines the local system.
Write \(B = \exp(2\pi i A)\) for \(A\in M_n(\mathbb{C})\).
Take \(\mathcal{E}_X = \mathcal{O}_X^n\) to be trivial bundle of rank \(n\) with connection matrix \(\frac{A}{z}\mathrm{d}z\), it corresponds to the Fuchsian system
\[\frac{\mathrm{d} f}{\mathrm{d} z} = \frac{A}{z} f, \quad f=\begin{pmatrix}f_1 \\ \vdots \\ f_n\end{pmatrix}.\]
When you solve the monodromy of the above system, you will get a factor of \(\exp(2\pi i A)\), which is the same as \(B\).
Conclusion: every local system on \(\mathbb{C}\setminus \{a_1,\dots,a_k\}\) arises from a vector bundle \(\mathcal{E}\) on \(\mathbb{P}_\mathbb{C}^1\) with a connection having at most simple poles at \(\{a_1,\dots,a_k,\infty\}\). If \(\mathcal{E}\) is trivial, it is equivalent to a classical Fuchsian system of differential equations.
The tricky thing is that, in general \(\mathcal{E}\) may not be trivial! But it is always possible if you remove any one point from \(X\), e.g. \(\infty\).
You could ask what happens for manifolds with higher dimension?
Let \(X\) be a connected complex manifold. So that the representations of \(\pi_1(X)\) are identified with local systems on \(X\), which is in turn identified with holomorphic vector bundles \(\mathcal{E}\) with a flat connection (this is automatic in dimension 1).
What about algebraic geometry?
Let \(X\) be a smooth projective variety over \(\mathbb{C}\). Serre’s GAGA theorem says that
Any holomorphic bundle \(\mathcal{E}\) on \(X\) admits a unique algebraic structure, compatible with any holomorphic connection on \(\mathcal{E}\).
Let \(X, Y\) be smooth projective varieties which are isomorphic as abstract schemes, their fundamental groups \(\pi_1(X),\pi_1(Y)\) need not be isomorphic, but they have the same finite dimensional complex representations! (These categories of representations have a purely algebraic description: \(\mathrm{Rep}(\pi_1(X))\mapsto\) holomorphic vector bundle on \(X\) with a flat connection, \(\xrightarrow{\text{GAGA}}\) algebraic vector bundle on \(X\) with a flat connection. This is purely algebraic and cannot see the difference between \(X\) and \(Y\).)
Let \(X\) be a smooth algebraic variety which may not be projective. In this general case, GAGA principle fails, holomorphic bundles are not the same as algebraic bundles!
But the situation is better for vector bundles with a flat connection.
Let \(\mathcal{E}\) be a holomorphic bundle on \(X\) equipped with a flat connection \(\nabla\).
First let’s assume \(X\) embeds into a projective variety \(\overline{X}\) (assume \(X\) is quasi-projective), we are compactifying \(X\)
Using Hironaka’s resolutions of singularities, you can arrange \(\overline{X}\) in a nice way
\(\overline{X}\) is smooth
\(\overline{X}\setminus X\) is a normal crossing divisor
We can extend \(\mathcal{E}\) to a holomorphic bundle \(\overline{\mathcal{E}}\) where \(\nabla\) has logarithmic poles along \(D\).