Riemann-Hilbert correspondence versus Simpson correspondence

Question

I couple of days ago, I asked extensively the same question on Stack-exchange (see https://math.stackexchange.com/questions/3592151/riemann-hilbert-correspondence-versus-simpson-correspondence)and go no answer.

Let us assume that X is a connected, smooth complex algebraic variety. Then the Riemann-Hilbert correspondence tells us that the functor which sends a flat connection with regular singularities on a vector bundle of X to its asocciated monodromy representation is an equivalence of categories.

Furthermore, the Simpson correspondence tells us that there is an equivalence of categories between the category of complex representations of the fundamental group of curves and the category of semi-stable Higgs bundles with trivial Chern class.

Its seems to me that the Simpson correspondence should be a consequence of the Riemann-Hilbert correspondence, especially since the category of Higgs bundles is roughly the tangent category of the category of vector bundles. However, based on the number of papers written on this, and the fact that Simpson wrote a ICM note on this, this is clearly not the case. So I guess I must miss what the additional content of the Simpson correspondence is. Could someone help me?

Answer 1

From my point of view, Riemann-Hilbert and non-abelian Hodge are really two independent statements - though the statement of the latter may be wound up in the former in some sense.

There are three different types of objects at play:

1) Higgs bundles (Dolbeault),

2) flat connections (de Rham),

3) reps of $\pi_1$ (Betti).

Riemann-Hilbert relates 2) and 3) via the operation of taking a connection to its monodromy. Simpson's non-abelian Hodge correspondence relates 1) with either 2) or 3) depending on your perspective (in any case, one can go between them via Riemann-Hilbert).

It might be helpful to keep in mind how abelian Hodge theory works. Here, if $X$ is a compact Kahler manifold, say, there are three vector spaces:

1) $H^1_{Dol}(X) = H^1(X;\mathcal O) \oplus H^0(X;\Omega^1)$

2) $H^1_{dR}(X)$

3) $H^1_{Betti}(X;\mathbb C) = Hom(\pi_1(X),\mathbb C)$

The analogue of Riemann-Hilbert here is the de Rham theorem identifying 2) and 3). On the other hand, the analogue of non-abelian Hodge is, well, abelian Hodge theory, which says that 1) and 2) may be identified (via yet another object: the harmonic forms).

There is no sense in which Hodge theory follows from the de Rham theorem, and I would say that the former involves much more subtle and intricate ideas.