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Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of $\mathcal{D}_X$-modules with holonomic cohomology.

The category $\mathcal{C}$ is triangulated (/stable $\infty$-) symmetric monoidal and so it makes sense to ask about its prime spectrum (the collection of all prime ideal thick subcategories).

What is the prime spectrum of $\mathcal{C}$? If this is too difficult/general, are there any specific $X$ for which this is known?

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    $\begingroup$ If you restrict to regular holonomic D-modules, then the Riemann-Hilbert correspondence turns this into a question about constructible sheaves. Fix a stratification S. Restriction to strata gives you a map D^{cstr}_S(X)---> \prod D(Loc(X_s)). I claim the induced map on spectra is a bijection. Up to topology, we're now reduced to computing the spectra of derived cats of representation rings over C. I guess I dunno what that is off-hand... my first guess would be homogeneous primes in the coh of \pi_1, but these \pi_1 won't be finite so I'm less confident about that $\endgroup$
    – Dylan Wilson
    Commented Sep 8, 2017 at 13:36

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