# Comptes Rendus
## Mathématique

Nero Budur, Leonardo A. Lerer and Haopeng Wang

Note on absolute sets of rigid local systems

Volume 360 (2022), p. 467-474

<https://doi.org/10.5802/crmath.315>

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Algebraic geometry / Géométrie algébrique

# Note on absolute sets of rigid local systems

Nero Budur<sup>a, b</sup>, Leonardo A. Lerer<sup>*, c</sup> and Haopeng Wang<sup>a</sup>

<sup>a</sup> Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven,
Belgium

<sup>b</sup> BCAM, Mazarredo 14, 48009 Bilbao, Spain

<sup>c</sup> Département de Mathématiques d'Orsay, Université Paris-Saclay, F-91405 Orsay,
France

E-mails: nero.budur@kuleuven.be (N. Budur), a.leonardo.lerer@gmail.com (L. Lerer), haopeng.wang@kuleuven.be (H. Wang)

**Abstract.** In this note we give a description up to a quasi-finite morphism of the absolute sets of simple cohomologically rigid local systems on a smooth complex quasi-projective algebraic variety. In dimension one or rank two, this proves a conjecture of Budur-Wang on the structure of these sets.

**2020 Mathematics Subject Classification.** 14M35, 32S40.

**Funding.** The authors were supported by the grants STRT/13/005 and Methusalem METH/15/026 from KU Leuven, G097819N and G0F4216N from FWO. The third author was also supported by a CSC PhD scholarship.

Manuscript received 13 April 2021, accepted 9 December 2021.

## 1. Introduction

The aim of this note is to state the results of [2], give brief indications on their proofs, and provide the motivation behind them. Let $X$ be a smooth complex algebraic variety and $r > 0$ an integer. The Betti moduli space $M_B(X, r)$ is an affine scheme defined over $\mathbb{Q}$ whose complex points parametrize semisimple local systems on $X^{\text{an}}$ of rank $r$, cf. Section 2.1. Certain points in $M_B(X, r)(\mathbb{C})$ are of particular interest: they correspond to the local systems of *geometric origin*. For instance, local systems that are direct summands of higher pushforwards $R^i f_* \mathbb{C}$, where $i \ge 0$ and $f: Y \to X$ is a smooth projective morphism, are of geometric origin (the precise definition of geometric origin is given in [1, 6.2.4]).

It is a subtle problem to describe the distribution of these “special” points in $M_B(X, r)$. A natural question, analogous to the André-Oort problem in the case of Shimura varieties, is to describe the Zariski closure $\overline{\Sigma}_{\text{Zar}}$ where $\Sigma \subset M_B(X, r)(\mathbb{C})$ is a subset of points of geometric origin. We recall here the corresponding picture for the Shimura variety $\mathcal{A}_g$ parametrizing principally polarized complex abelian varieties. The role of the points of geometric origin in $M_B(X, r)$

<footnote>* Corresponding author.</footnote>
is played by the CM points in $\mathcal{A}_g$, i.e. points in $\mathcal{A}_g(\mathbb{C})$ parametrizing abelian varieties with complex multiplication. If $\Sigma \subset \mathcal{A}_g(\mathbb{C})$ is a subset of CM points then the André-Oort conjecture (now a theorem) states that the irreducible components of $\Sigma^{\text{Zar}}$ are connected Shimura subvarieties of $\mathcal{A}_g$. Roughly speaking, Shimura subvarieties of $\mathcal{A}_g$ parametrize abelian varieties whose associated weight one Hodge structure has more tensor Hodge classes than a very general abelian variety. Shimura subvarieties of $\mathcal{A}_g$ also contain a dense set of CM points for the Euclidean topology. The counterparts in $M_B(X, r)$ of the Shimura subvarieties of $\mathcal{A}_g$ are the absolute $\overline{\mathbb{Q}}$-constructible subsets, that we define in Section 2. They were first defined by Simpson in [12] when $X$ is a projective variety. We follow a less constrained definition of [3] available also in the non-projective case. The essential ingredient needed for their definition is the Riemann-Hilbert correspondence between complex local systems on $X^{\text{an}}$ and regular algebraic flat connections on $X$. The absolute $\overline{\mathbb{Q}}$-constructible subsets of dimension 0 give yet another collection of interesting points: as conjectured by Simpson, they should correspond exactly to the local systems of geometric origin (see point (A3) below). One can formulate a set of conjectures in analogy with the Shimura case. This is done in [3]:

**Conjecture A.** Let $X$ be a smooth complex algebraic variety, and $r > 0$ an integer. The following hold in $M_B(X, r)$:

(A1) The collection of absolute $\overline{\mathbb{Q}}$-constructible subsets is generated from the absolute $\overline{\mathbb{Q}}$-closed subsets via finite sequences of taking unions, intersections, and complements; the Euclidean (or equivalently, the Zariski) closure of an absolute $\overline{\mathbb{Q}}$-constructible set is absolute $\overline{\mathbb{Q}}$-closed; and an irreducible component of an absolute $\overline{\mathbb{Q}}$-closed subset is absolute $\overline{\mathbb{Q}}$-closed.

(A2) Any non-empty absolute $\overline{\mathbb{Q}}$-constructible subset contains a Zariski dense subset of absolute $\overline{\mathbb{Q}}$-points.

(A3) A point is an absolute $\overline{\mathbb{Q}}$-point if and only if it is a local system of geometric origin.

(A4) The Zariski closure of a set of absolute $\overline{\mathbb{Q}}$-points is $\overline{\mathbb{Q}}$-pseudo-isomorphic to an absolute $\overline{\mathbb{Q}}$-constructible subset of some $M_B(X', r')$, for possibly a different smooth complex algebraic variety $X'$ and rank $r'$. Moreover, the $\overline{\mathbb{Q}}$-pseudo-isomorphism restricts to a bijection between the sets of absolute $\overline{\mathbb{Q}}$-points. (See Definition 3 for the notion of $\overline{\mathbb{Q}}$-pseudo-morphisms.)

In this note we address a modification of Conjecture A for certain subvarieties of $M_B(X, r)$, which are themselves absolute $\overline{\mathbb{Q}}$-constructible subsets $M_B(X, r)$. They correspond to the submoduli of simple local systems (Section 3) and cohomologically rigid local systems (Section 4). We give a description up to a quasi-finite morphism of the absolute $\overline{\mathbb{Q}}$-constructible sets of simple cohomologically rigid local systems. Conjecture A for this kind of absolute sets is then: reduced to (A3) (Theorem 7), and shown to hold if $\dim X = 1$ or if $r = 2$ (Theorem 8). If $r = 1$, or if $X$ is a complex affine torus or an abelian variety, Conjecture A is known by [3]. One can find in [5] an arithmetic analog of our results.

## 2. The Betti moduli space and absolute sets

### 2.1. Betti moduli

Let $X$ be a smooth complex algebraic variety with a base point $x_0 \in X(\mathbb{C})$. Let $r > 0$ be an integer. The space of representations of rank $r$ of the (topological) fundamental group $\pi_1(X, x_0)$ is

$$R_B(X, x_0, r) := \operatorname{Hom}(\pi_1(X, x_0), \operatorname{GL}_r),$$

the $\overline{\mathbb{Q}}$-scheme representing the functor associating to a $\overline{\mathbb{Q}}$-scheme $T$ the set $\operatorname{Hom}(\pi_1(X, x_0), \operatorname{GL}_r(\Gamma(T, \mathcal{O}_T)))$. There is an action of $\operatorname{GL}_r$ on $R_B(X, x_0, r)$ by conjugation: on $T$-valued points $R_B(X, x_0, r)(T)$, the group $\operatorname{GL}_r(\Gamma(T, \mathcal{O}_T))$ acts by conjugation on the target.
The moduli space of local systems of rank $r$ is defined as the affine GIT quotient of $R_B(X, x_0, r)$ with respect to this action,

$$M_B(X, r) := R_B(X, x_0, r) // \mathrm{GL}_r.$$

It is an affine scheme defined over $\mathbb{Q}$. The complex points of $M_B(X, r)$ are in one-to-one correspondence with the isomorphism classes of complex semisimple local systems of rank $r$ on $X^{\mathrm{an}}$. For a representation $\rho: \pi_1(X, x_0) \to \mathrm{GL}_r(\mathbb{C})$ we denote the associated local system on $X^{\mathrm{an}}$ by $L_{\rho}$. We denote by

$$q_B : R_B(X, x_0, r) \to M_B(X, r)$$

the quotient morphism. The fiber over a complex point $L \in M_B(X, r)(\mathbb{C})$ is the set of representations $\rho$ such that $L_{\rho}^{ss} \cong L$, where $\rho^{ss}$ is the semisimplification of $\rho$. By $R_B(X, x_0)$ we will mean the disjoint union over all $r$ of $R_B(X, x_0, r)$ and similarly for $M_B(X)$.

## 2.2. Unispaces

Unispaces, defined in [3], enlarge the class of spaces for which constructibility can be defined in a useful way. Let $Alg_{ft,reg}(\mathbb{C})$ be the category of finite type regular $\mathbb{C}$-algebras and Set the category of sets. A unispace $\mathcal{C}$ is a functor

$$\mathcal{C} : Alg_{ft,reg}(\mathbb{C}) \to Set.$$

Given $R \in Alg_{ft,reg}(\mathbb{C})$ elements of $\mathcal{C}(R)$ will be denoted by $\mathcal{F}_R$. If $R \to R'$ is a morphism in $Alg_{ft,reg}(\mathbb{C})$ and $\mathcal{F}_R \in \mathcal{C}(R)$, the image of $\mathcal{F}_R$ in $\mathcal{C}(R')$ under the morphism $\mathcal{C}(R) \to \mathcal{C}(R')$ will be denoted $\mathcal{F}_R \star_R R'$. We think of an element of $\mathcal{C}(R)$ as a family of objects in $\mathcal{C}(\mathbb{C})$ parametrized by $\mathrm{Spec}(R)$. A subset $S \subset \mathcal{C}(\mathbb{C})$ is called $\mathbb{C}$-constructible (respectively, $\mathbb{C}$-closed) with respect to the unispace $\mathcal{C}$ if for every $R \in Alg_{ft,reg}(\mathbb{C})$ and every $\mathcal{F}_R \in \mathcal{C}(R)$, the subset of maximal ideals

$$\{m \in \mathrm{Spec}(R)(\mathbb{C}) | \mathcal{F}_R \star_R R / m \in S\}$$

is the set of complex points of a constructible subscheme of $\mathrm{Spec}(R)$.

### Examples

Let $X$ be a complex algebraic variety.

(1) We denote by $\mathrm{LocSys}_{\mathrm{free}}(X, \_)$ the unispace associating to $R \in Alg_{ft,reg}(\mathbb{C})$ the set of isomorphism classes of local systems of free $R$-modules of finite rank on $X^{\mathrm{an}}$.

(2) We denote by $D_c^b(X, \_)$ the unispace associating to $R \in Alg_{ft,sreg}(\mathbb{C})$ the set of isomorphism classes of objects of $D_c^b(X, R)$, the bounded derived category of constructible sheaves of finite type $R$-modules on $X^{\mathrm{an}}$.

In the notation of the previous example, there is a natural transformation of functors

$$\mathrm{LocSys}_{\mathrm{free}}(X, \_) \to D_c^b(X, \_). \qquad (1)$$

For $R \in Alg_{ft,reg}(\mathbb{C})$, it sends a local system $\mathcal{L}_R \in \mathrm{LocSys}_{\mathrm{free}}(X, R)$ to the complex in $D_c^b(X, R)$ whose only nonzero term is $\mathcal{L}_R$ placed in degree zero.

## 2.3. $\overline{\mathbb{Q}}$-constructibility

Consider the unispace $\mathrm{LocSys}_{\mathrm{free}}(X, \_)$ (the same discussion applies verbatim to $D_c^b(X, \_)$). We can modify its definition in the obvious way to define a functor $Alg_{ft,reg}(K) \to Set$, where $K$ is a field of characteristic zero and $Alg_{ft,reg}(K)$ is the category of finite type regular $K$-algebras. Fix an embedding $i: \overline{\mathbb{Q}} \hookrightarrow \mathbb{C}$. We write $\overline{\mathbb{Q}} \subset L \subset \mathbb{C}$ to denote a tower of field extensions whose composite is $i$. A subset $S \subset \mathrm{LocSys}_{\mathrm{free}}(X, \mathbb{C})$ is $\overline{\mathbb{Q}}$-constructible (resp. $\overline{\mathbb{Q}}$-closed) if it is $\mathbb{C}$-constructible (resp.
**Definition 2.** A subset $S \subset M_B(X)(\mathbb{C})$ is absolute K-constructible (resp. absolute K-closed) if $S = q_B(l^{-1}(T))$ where $T \subset \text{LocSys}(X, \mathbb{C})$ is absolute K-constructible (resp. absolute K-closed) with respect to the unispace $\text{LocSys}_{\text{free}}(X, \mathbb{C})$. If $\sigma \in \text{Gal}(\mathbb{C}/\mathbb{Q})$ the set $S^\sigma \subset M_B(X^\sigma)(\mathbb{C})$ is defined as in (3) with the points of $S$ corresponding to isomorphism classes of semisimple local systems.

Notice that an absolute K-constructible subset of $M_B(X, r)(\mathbb{C})$ is indeed the set of complex points of a constructible K-subscheme of $M_B(X, r)$. For equivalent definitions of absoluteness and some basic properties, see [2, Section 4.3]. In Conjecture A we use the notion of “$\overline{\mathbb{Q}}$-pseudo-isomorphism”. The definition is as follows and it is related with the theory of motivic measures and the Grothendieck group of varieties, see [8, 9]:

**Definition 3.** Let $K$ be a field, $Y$ (resp. $Y'$) a $K$-scheme, and $S$ (resp. $S'$) a constructible subscheme of $Y$ (resp. $Y'$). The sets $S(\mathbb{C})$ and $S'(\mathbb{C})$ are $K$-pseudo-isomorphic if there exist $n \in \mathbb{N}$, constructible $K$-subschemes $S_i$ of $S$ (resp. $S'_i$ of $S'$), for $i = 1, \dots, n$, such that $S(\mathbb{C}) = \coprod_{i=1}^n S_i(\mathbb{C})$ (resp. $S'(\mathbb{C}) = \coprod_{i=1}^n S'_i(\mathbb{C})$) and such that there exist $K$-isomorphisms $S_i \to S'_i$, $i = 1, \dots, n$. The ambient schemes $Y$ and $Y'$ will be clear from the context.

## 2.5. Restricting Conjecture A to submoduli

Let $M \subset M_B(X)(\mathbb{C})$ be an absolute K-constructible subset. Let $\tilde{M} = l(q_B^{-1}(M)) \subset \text{LocSys}(X, \mathbb{C})$. Define the unispace

$$ \mathcal{M} : R \mapsto \{ \mathcal{L}_R \in \text{LocSys}_{\text{free}}(X, R) \mid \mathcal{L}_R \otimes_R R / m \in \tilde{M}, \forall m \in \text{Spec}(R)(\mathbb{C}) \} $$

together with the usual base change. Clearly $\mathcal{M}(\mathbb{C}) = \tilde{M}$. There is a natural transformation

$$ \mathcal{M} \to \text{LocSys}_{\text{free}}(X, \underline{\phantom{X}}) \quad (4) $$

of unispaces defined over $K$ given by inclusion. For $\sigma \in \text{Gal}(\mathbb{C}/\mathbb{Q})$, we let $\mathcal{M}_{X^\sigma}$ be the unispace associated to $M^\sigma$ (the latter set is defined in Definition 2). Note that $(\underline{\phantom{\sigma}})$ and $(\underline{\phantom{\sigma}})^\sigma$ commute, and $\mathcal{M} = \mathcal{M}_X$. We adapt Definition 1 to the collection of unispaces $\mathcal{M}_{X^\sigma}$. We say that a subset $S \subset \tilde{M}$ is absolute K-constructible (resp. absolute K-closed) with respect to $\mathcal{M}$ if $S^\sigma$ is K-constructible (resp. K-closed) with respect to $\mathcal{M}_{X^\sigma}$ for all $\sigma \in \text{Gal}(\mathbb{C}/\mathbb{Q})$. At the level of the Betti moduli space, the corresponding definition is:

**Definition 4.** A subset $S$ of $M$ is called absolute K-constructible (resp. absolute K-closed) in $M$ if $S = q_B(l^{-1}(T))$ for some subset $T$ of $\tilde{M}$ that is absolute K-constructible (resp. absolute K-closed) with respect to the unispace $\mathcal{M}$.

One can then restrict the formulation of Conjecture A to $M$: we replace $M_B(X)$ by $M$ and replace all instances of the phrase “absolute $\overline{\mathbb{Q}}$-constructible subset” (resp. “absolute $\overline{\mathbb{Q}}$-closed subset”) by “absolute $\overline{\mathbb{Q}}$-constructible subset of $M$” (resp. “absolute $\overline{\mathbb{Q}}$-closed subset of $M$”). Conjecture (A3) will deal only with the equivalence for absolute $\overline{\mathbb{Q}}$-points in $M$. We shall refer to the restricted version of the conjecture as “Conjecture A for $M$”.

## 3. Simple local systems

Let $M_B^s(X, r)$ be the Zariski open subscheme of $M_B(X, r)$ whose complex points parametrize simple local systems. It is easy to see that $M_B^s(X, r)(\mathbb{C})$ is absolute $\overline{\mathbb{Q}}$-constructible. Our main result concerning the locus of simple local systems is:

**Theorem 5.** Conjecture (A1) for $M_B^s(X, r)(\mathbb{C})$ holds.
Let us say some words on the proof. The notion of absolute $\overline{\mathbb{Q}}$-constructibility given in Definition 2 has a more geometric version, which however is less flexible. Embed $X$ in a smooth projective variety $\overline{X}$ with $D := \overline{X} \setminus X$ a simple normal crossing divisor. It is proved by Nitsure [10, Theorem 3.5] and Simpson [13, Theorem 4.10] that there exists a coarse moduli space $M_{DR}(\overline{X}/D, r)$ for Jordan equivalence classes of rank $r$ semistable logarithmic connections on $\overline{X}$ with poles on $D$. It is a countable disjoint union of finite type $\mathbb{C}$-schemes. The underlying $\mathcal{O}_{\overline{X}}$-module of a logarithmic connection is, by definition, only a torsion-free $\mathcal{O}_{\overline{X}}$-module, locally free over $X$. The Riemann-Hilbert correspondence induces a complex analytic map

$$RH^{\text{an}} : M_{DR}(\overline{X}/D, r)^{\text{an}} \to M_B(X, r)^{\text{an}}. \qquad (5)$$

One can then give a definition of absoluteness as in 2 but using $M_B(X, r)$ (resp. $M_{DR}(\overline{X}/D, r)$) in place of $D_c^b(X, \mathbb{C})$ (resp. $D_{rh}^b(\mathcal{D}_X)$), see [2, Section 5.3]. The advantage of this definition is the possibility of working with geometric objects rather than abstract sets of isomorphism classes. On the other hand, there are some drawbacks. First, it is much easier to define functors at the categorical level rather than at the level of coarse moduli spaces. Second, the condition of semistability on connections, needed to ensure the existence of a coarse moduli space, leads to some complications: for example, it is not known whether $RH^{\text{an}}$ is surjective (notice that without semistability the surjectivity is clear, e.g. using Deligne extensions). However, many difficulties disappear, and the two definitions of absoluteness agree, if we restrict $RH^{\text{an}}$ to the analytic open subpace $M_{DR}^{\text{good}}(\overline{X}/D, r) \subset M_{DR}(\overline{X}/D, r)^{\text{an}}$ which parametrizes simple logarithmic connections such that no two distinct eigenvalues of the residue along a component of $D$ differ by an integer (it is the complement of a locally finite, countable collection of Zariski closed subchemes). Then, the map $RH_0 := RH^{\text{an}} : M_{DR}^{\text{good}}(\overline{X}/D, r) \to M_B^s(X, r)^{\text{an}}$ is surjective. The key ingredient for the proof of Theorem 5 is a result of Nitsure-Sabbah [11, Section 8] stating that $RH_0$ is a local analytic isomorphism.

## 4. Cohomologically rigid local systems

We continue to consider a smooth complex variety $X$ with a good compatification $j : X \to \overline{X}$.

**Definition 6.** Let $L$ be a simple complex local system on $X$. We say that $L$ is cohomologically rigid if $\mathbb{H}^1(\overline{X}, j_* End^0(L)) = 0$ where $End^0(L)$ denotes the local system of traceless endomorphisms of $L$ and $j_*$ is the intermediate extension functor.

In general the vector space $\mathbb{H}^1(\overline{X}, j_* End^0(L))$ is the Zariski tangent space at $L$ of the Betti moduli space of complex local systems with prescribed determinant and prescribed local monodromies at infinity. Thus a cohomologically rigid complex local system is rigid, in the sense that $L$ is an isolated point of this moduli space, and in addition it is a smooth point. Without fixing the determinant nor the monodromy at infinity, the set of simple cohomologically rigid local systems in $M_B(X, r)(\mathbb{C})$ is the set of complex points of a possibly high-dimensional constructible subscheme $M_B^{\text{cohrig}}(X, r) \subset M_B(X, r)$. From the results of [3] and the cohomological characterization $\mathbb{H}^1(\overline{X}, j_* End^0(L)) = 0$, one deduces that $M_B^{\text{cohrig}}(X, r)(\mathbb{C})$ is an absolute $\overline{\mathbb{Q}}$-constructible subset, see [2, Proposition 6.2 (2)]. Our second main result is:

**Theorem 7.** Conjectures (A1), (A2), (A4) for $M_B^{\text{cohrig}}(X, r)$ hold. Conjecture (A3) is equivalent to Simpson's conjecture that cohomologically rigid simple local systems with quasi-unipotent determinant and conjugacy classes at infinity are of geometric origin.

Simpson's conjecture is proved when $X$ is a curve and the rank is arbitrary in [7], and when $X$ is arbitrary and the rank is 2 in [4]. Therefore we obtain:

**Theorem 8.** Conjecture A for $M_B^{\text{cohrig}}(X, r)$ holds if $X$ is a curve or if the rank $r = 2$.
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