The well-known correspondence between vector bundles with flat connection on a smooth complex algebraic variety $X$ and complex representations of $\pi_1(X^{an})$, the fundamental group of the analytification. This extends to the famous Riemann-Hilbert correspondence between regular holonomic $D_X$-modules on $X$ and perverse sheaves on $X^{an}$. Is there any way in which one can view perverse sheaves as representations of some topological object associated to $X^{an}$?

## 1 Answer

You're not going to get representations of a group (or groupoid, or even category), even in simple examples. On the other hand, it is probably true for formal reasons that perverse sheaves for a fixed stratification (by algebraic subvarieties) are representations of some finitely generated (non-commutative) algebra. In many easy examples, you can actually write down a quiver with relations whose representations give the category you want.

**Example.** Perverse sheaves on $\mathbf C$ for the stratification $U \amalg Z$ with $U = \mathbf C \setminus \{0\}$ and $Z = \{0\}$ are equivalent to representations of the quiver $u\!:\bullet \rightleftarrows \bullet :\!v$ such that the action of $vu+1$ is invertible.

This example, and a generalisation to perverse sheaves on $\mathbf C^n$ for the product stratification, is due to Galligo, Granger, and Maisonobe [GGM85].

There are many similar results in the literature for other stratified spaces, e.g. hyperplane arrangements or (recently) symmetric powers of $\mathbf C^2$. Representation theorists seem to really like these descriptions.

My understanding is that experts believe something similar to be true in general, related to the exodromy theorem of Treumann [Tre09], Lurie [HA, Appendix A], and Barwick, Glasman, and Haine [BGH18]. I am not aware of a method to compute this quiver in general, nor is it clear to me what you could use it for. That said, I believe that there are people thinking about these questions as we speak (I may or may not turn out to be one of them).

**References.**

[BGH18] C. Barwick, S. Glasman, P. Haine, *Exodromy*. Preprint/book in preparation, 2018.

[GGM85] A. Galligo, M. Granger, Ph. Maisonobe, *${\mathcal D}$-modules et faisceaux pervers dont le support singulier est un croisement normal*. Ann. Inst. Fourier **35**.1 (1985), p. 1-48.

[HA] J. Lurie, *Higher algebra*. Preprint/book in preparation, version from 2017.

[Tre09] D. Treumann, *Exit paths and constructible stacks*. Compos. Math. **145**.6 (2009), p. 1504-1532.

`ag.algebraic-geometry`

or`rt.representation-theory`

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