Local systems with quasi-unipotent monodromy at infinity are dense (Q6584657)

Let \(X\) be a connected normal algebraic variety over \(\mathbb{C}\). For an integer \(r\geq 1\) let \(\mathrm{Ch}_{GL_r, X}\) be the character variety parametrizing conjugacy classes of semi-simple rank \(r\) representations of the fundamental group \(\pi_1(X(\mathbb{C}))\) of \(X\) into \(GL_r(\mathbb{C})\). This is an affine variety over \(\mathbb{C}\) which in fact naturally descends to an affine scheme over \(\mathbb{Z}\).\N\NIf \(\overline{X}\) is a normal compactification of \(X\), for every codimension \(1\) irreducible component \(D_i\) of \(\overline{X}\setminus X\) there is a well-defined conjugacy class \(\langle\gamma_i\rangle\subset \pi_1(X(\mathbb{C}))\) of a loop around that component. A representation \(\rho:\pi_1(X(\mathbb{C}))\to GL_r(\mathbb{C})\) is said to have \textit{quasi-unipotent monodromy at infinity} if all these conjugacy classes get sent by \(\rho\) to matrices whose eigenvalues are roots of unity. This property of \(\rho\) does not depend on the choice of the compactification \(\overline{X}\).\N\NThe authors prove that for any \(X\) and \(r\) the set of \(\mathbb{C}\)-points of the character variety \(\mathrm{Ch}_{GL_r, X}\) corresponding to representations with quasi-unipotent monodromy at infinity is Zariski dense. Note that this is a non-trivial group-theoretic restriction on the finitely presented group \(\pi_1(X(\mathbb{C}))\) together with a distinguished finite set of conjugacy classes \(\langle\gamma_i\rangle\). For example, if \(\gamma\in SL_n(\mathbb{Z})\) is a matrix not all of whose eigenvalues are roots of unity and \(n\geq 3\), then points of the character variety of \(n\)-dimensional representations of \(SL_n(\mathbb{Z})\) in which \(\gamma\) is quasi-unipotent are not Zariski dense.\N\NThe authors also prove the analogous density result for \(G\)-representation varieties of \(\pi_1(X(\mathbb{C}))\) for any linear algebraic group \(G\).\N\NOne motivation for considering representations with quasi-unipotent monodromy at infinity is that all representations of \textit{geometric origin} have this property, by Grothendieck's monodromy theorem. Recall that a representation \(\rho:\pi_1(X(\mathbb{C}))\to GL_r(\mathbb{C})\) is said to be of geometric origin if there exists a dense Zariski open \(U\subset X\) and a smooth proper family \(f:Y\to U\) such that the restriction to \(U\) of the local system of \(\mathbb{C}\)-vector spaces corresponding to \(\rho\) is a direct summand of the local system \(R^if_*\mathbb{C}\) of \(i\)-th cohomology, for some \(i\).\N\NThe authors make a conjecture that motivates their main result: representations of geometric origin are Zariski dense in the character variety. This conjecture follows from the conjectures of \textit{B. Wang} and \textit{N. Budur} [Ann. Sci. Éc. Norm. Supér. (4) 53, No. 2, 469--536 (2020; Zbl 1453.14054)], and immediately implies the main theorem of the present paper, but recently \textit{A. Landesman} and \textit{D. Litt} [J. Am. Math. Soc. 37, No. 3, 683--729 (2024; Zbl 1539.14020)] and \textit{Y. H. J. Lam} [``Motivic local systems on curves and Maeda's conjecture'', Preprint, \url{arXiv:2211.06120}] found counterexamples to the conjecture.\N\NInstead the proof of the main theorem relies on the abundance of points satisfying a weaker property than being of geometric origin, but which still implies quasi-unipotence of the monodromy at inifnity. This collection of points is defined using the following additional arithmetic structure on the character variety. If the variety \(X\) descends to a subfield \(F\subset \mathbb{C}\), the theory of étale fundamental group gives rise to an interesting action of the absolute Galois group \(\mathrm{Gal}(\overline{F}/F)\) on the \(\overline{\mathbb{Z}}_{\ell}\)-points of the character variety, for any prime \(\ell\). The authors use a technique going back to \textit{V. Drinfeld} [Math. Res. Lett. 8, No. 5--6, 713--728 (2001; Zbl 1079.14509)] that relies on de Jong's conjecture [\textit{A. J. de Jong}, Isr. J. Math. 121, 61--84 (2001; Zbl 1054.11032)] that gives sufficiently many, in a precise sense, points in the character variety that are stabilized by one Frobenius element in the Galois group \(\mathrm{Gal}(\overline{F}/F)\). All such points have quasi-unipotent monodromy (if the place of \(F\) corresponding to the Frobenius element is well-chosen with respect to \(X\)) which leads to the proof of the main theorem.\N\NIn accordance with the idea of this proof, in a related paper [\textit{H. Esnault} and \textit{M. Kerz}, Épijournal de Géom. Algébr., EPIGA 6, Article 5, 18 p. (2022; Zbl 1512.11058)] the authors make a weaker conjecture to the effect that points stabilized by a power of a single Frobenius element should be Zariski dense in the generic fiber of the local deformation ring of any irreducible \(\overline{\mathbb{F}}_{\ell}\)-representation of the fundamental group. Despite this conjecture being currently open, the authors are able to prove the main theorem of the present paper thanks to the fact that their setting allows to vary the place at which the Frobenius element is chosen.