let $X$ denote a smooth complex algebraic variety. Let $D_{rh}(X)$ denote the category of regular holonomic $D$-modules on $X$ and $D_{rh}^b(D(X))$ denote the bounded derived category of $D$-modules on $X$ with regular holonomic homology. Let $X^{an}$ denote the associated complex manifold and let $Mod_c(X^{an})$ denote the category of (algebraically) constructible sheaves on $X^{an}$. Also let $D^b_c(X^{an}, \mathbb{C})$ denote the bounded derived category of complex sheaves on $X$ with constructible homology. There is a (left exact, contravariant) solution functor:
$$Sol: D_{rh}(X) \rightarrow Mod_c(X^{an})$$
given by $Sol(M):= Hom_{D(X^{an})}(M^{an}, \mathcal{O}_{X^{an}}).$ The Riemann-Hilbert Correspondence asserts that this induces an anti-equivalence of categories:
$$RSol: D_{rh}^b(D(X)) \cong D^b_c(X^{an}, \mathbb{C}).$$
Now because the categories of $D$-modules on $X$ and complex sheaves on $X^{an}$ have enough injectives we can think about these derived categories as the homotopy categories of the DG-categories of complexes whose objects are injective with bounded homology. Thus $D_{rh}^b(D(X))$ is the homotopy category of the DG-category $K_{rh}^b(D(X))$, whose objects are injective chain complexes with bounded, regular holonomic homology. Similarly $D^b_c(X^{an}, \mathbb{C})$ is the homotopy category of the DG-category $K^b(X^{an},\mathbb{C})$, whose objects are injective chain complexes with bounded, constructible homology. The solution function naturally gives a functor:
$$Sol_{DG}: K_{rh}^b(D(X)) \rightarrow K^b(X^{an},\mathbb{C}).$$
Passing to the homotopy categories gives the Riemann-Hilbert Correspondence. My question is the following: Can the Riemann-Hilbert Correspondence be lifted to the DG setting? In other words, is $Sol_{DG}$ an equivalence of DG-categories?
Best Answer
The answer is yes, if 'equivalence of dg categories' means the usual thing: given dg categories $D_{1}$, $D_{2}$, a dg equivalence between them is a dg functor $F: D_{1} \rightarrow D_{2}$ such that 1) the induced map on complexes $F_{x,y}:D_{1}(x,y) \rightarrow D_{2}(F(x),F(y))$ is a quasi-isomorphism for every $x,y \in D_{1}$ and 2) the induced functor on homotopy categories $[F]: [D_{1}] \rightarrow [D_{2}]$ is an equivalence. The first condition is the homotopical version of fully faithful and the second condition ensures essential surjectivity up to equivalence. The standard statement of Riemann-Hilbert gives 2), but in fact the proof usually verifies 1) along the way. See for instance 7.2.2 in D-modules, Perverse Sheaves, and Representation Theory by Hotta, Takeuchi, and Tanisaki. (They actually treat the covariant Riemann-Hilbert correspondence, using the de Rham functor, but you can get the contravariant version, involving the solution functor, by duality.)
About Ben's comment. There exists an example in positive characteristic of two dgas whose triangulated module categories are equivalent but this equivalence is not induced by a Quillen equivalence of model categories. See Dugger-Shipley, A curious example of triangulated-equivalent model categories which are not Quillen equivalent. Actually, what they show is that the algebraic K-theory of the two dgas is different, and so is not invariant under triangulated equivalence. I take this as convincing evidence that the notion of triangulated category is deficient.