arXiv subjects

Matthew Stover

Publications and source records attributed to Matthew Stover.

At least 19 recordsLinked to original sources

arXiv2026

Automorphisms of the moduli space of smooth cubic surfaces and its fundamental group

Let $\mathcal{C}$ be the moduli space of smooth complex cubic surfaces and let $\pi_1(\mathcal{C})$ be its (orbifold) fundamental group. We prove that the ``divisor subgroup'' of $\pi_1(\mathcal{C})$ is characteristic. This can be interpreted as saying that the group theory of $\pi_1(\mathcal{C})$ ``remembers'' the divisor of nodal cubic surfaces. We deduce from this group-theoretic result and some basic complex analysis that $\mathcal{C}$ has no nontrivial biholomorphic automorphisms as complex analytic orbifold.

math.AG↗

arXiv2025

Finite groups and complex projective surfaces

In response to a question raised by Belolipetsky and the first author, we prove that for every finite group $G$ there are infinitely many isomorphism classes of compact complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.

math.GT↗

arXiv2025

Cohomological nonvanishing for algebraic fundamental groups of ball quotients

Suppose $\Gamma < \mathrm{PU}(n,1)$ is a cocompact arithmetic lattice of simplest type with profinite completion $\widehat{\Gamma}$. This paper proves there is an open subgroup $\widehat{\Gamma}_0 \le \widehat{\Gamma}$ such that $H^j(\widehat{\Delta}, \mathbb{F}_p)$ is nontrivial for every open subgroup ${\widehat{\Delta} \le \widehat{\Gamma}_0}$, $j \le 2n$, and sufficiently large prime $p$. If $n \ge 2$, nonvanishing is new for all $j \ge 2$. Consequently, the virtual cohomological dimension of $\widehat{\Gamma}$ is at least $2n$, improving the previous lower bound of $1$. The proof shows there is a profinite fundamental class for the associated ball quotient and that its canonical class is profinite modulo torsion. For congruence $\Gamma$ and $j < \frac{n+1}{2}$, restriction ${H^j(\widehat{\Gamma}, \mathbb{F}_p) \to H^j(\Gamma, \mathbb{F}_p)}$ is shown to be almost surjective in a precise sense; this is related to whether lattices in $\mathrm{PU}(n,1)$ are good in the sense of Serre, which is only known to hold for $n=1$.

math.AG↗

arXiv2025

A hyperbolic $4$-orbifold with underlying space $\mathbb{P}^2$

This paper shows that the complex projective plane $\mathbb{P}^2$ can be realized as the underlying space for a closed hyperbolic $4$-orbifold. This is the first example of a closed hyperbolic $4$-orbifold whose underlying space is symplectic, which is related to the open question as to whether or not closed hyperbolic $4$-manifolds can admit symplectic structures.

math.GT↗

arXiv2025

Complex hyperbolic 2-orbifolds with isolated singularities

For each prime $p$, this paper constructs compact complex hyperbolic $2$-manifolds with an isometric action of $\mathbb{Z} / p \mathbb{Z}$ that is not free and has only isolated fixed points. The case $p = 2$ is special, and finding general examples for $p=2$ is related to whether or not complex hyperbolic lattices are conjugacy separable on torsion.

math.GT↗

arXiv2025

Cocompact Fuchsian groups with a modular embedding

A Fuchsian group $\Gamma$ has a modular embedding if its adjoint trace field is a totally real number field and every unbounded Galois conjugate $\Gamma^\sigma$ comes equipped with a holomorphic (or conjugate holomorphic) map ${\phi^\sigma : \mathbb{B}^1 \to \mathbb{B}^1}$ intertwining the actions of $\Gamma$ and $\Gamma^\sigma$ on the Poincar\'e disk $\mathbb{B}^1$. This paper provides the first cocompact nonarithmetic Fuchsian groups with a modular embedding that are not commensurable with a triangle group. The main result, proved using period domains, is that any immersed totally geodesic complex curve on a complex hyperbolic $2$-orbifold has a modular embedding. Another consequence is arithmeticity of totally geodesic curves on finite-volume complex hyperbolic surfaces that are commensurable with quotients of $\mathbb{B}^1$ by the group generated by reflections in quadrilaterals satisfying certain angle conditions.

math.GT↗

arXiv2025

High dimensional hyperbolic Coxeter groups that virtually fiber

This paper provides an iterative procedure for constructing hyperbolic Coxeter groups that virtually fiber over $\mathbb{Z}$ that is flexible enough to yield infinitely many isomorphism classes in each virtual cohomological dimension (vcd) $n\geq 2$. Our procedure combines results of Jankiewicz, Norin, and Wise with a generalization of a construction due to Osajda involving a new simplicial thickening process. We also give a topological argument showing that the vcd of the right-angled Coxeter groups produced by our construction increases by exactly one with each iteration, guaranteeing that our process produces examples of every vcd.

math.GT↗

arXiv2025

Profinite rigidity of K\"ahler groups: Riemann surfaces and subdirect products

This paper establishes strong profinite rigidity results for K\"ahler groups, showing that certain groups are determined within the class of residually finite K\"ahler groups by their profinite completion. Examples include products of surface groups and certain groups with exotic finiteness properties studied earlier by Dimca-Papadima-Suciu and Llosa Isenrich. Consequently, there are aspherical smooth projective varieties that are determined up to homeomorphism by their algebraic fundamental group. The main tool is the following: the holomorphic fibrations of a closed K\"ahler manifold over hyperbolic 2-orbifolds can be recovered from the profinite completion of its fundamental group. We also prove profinite invariance of the BNS invariant.

math.GT↗

arXiv2024

One-cusped complex hyperbolic 2-manifolds

This paper builds one-cusped complex hyperbolic $2$-manifolds by an explicit geometric construction. Specifically, for each odd $d \ge 1$ there is a smooth projective surface $Z_d$ with $c_1^2(Z_d) = c_2(Z_d) = 6d$ and a smooth irreducible curve $E_d$ on $Z_d$ of genus one so that $Z_d \smallsetminus E_d$ admits a finite volume uniformization by the unit ball $\mathbb{B}^2$ in $\mathbb{C}^2$. This produces one-cusped complex hyperbolic $2$-manifolds of arbitrarily large volume. As a consequence, the $3$-dimensional nilmanifold of Euler number $12d$ bounds geometrically for all odd $d \ge 1$.

math.GT↗

arXiv2024

Residual finiteness and discrete subgroups of Lie groups

Let $G$ be a real Lie group and $\Gamma < G$ be a discrete subgroup of $G$. Is $\Gamma$ residually finite? This paper describes known positive and negative results then poses some questions whose answers will lead to a fairly complete answer for lattices.

math.GR↗

arXiv2024

Rich representations and superrigidity

We investigate and compare applications of the Zilber-Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu's immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.

math.AG↗

arXiv2023

Products of curves as ball quotients

For any $g_1, g_2 \ge 0$, this paper shows that there is a cocompact lattice $\Gamma < \mathrm{PU}(2,1)$ such that the ball quotient $\Gamma \backslash \mathbb{B}^2$ is birational to a product $C_1 \times C_2$ of smooth projective curves $C_j$ of genus $g_j$. The only prior examples were $\mathbb{P}^1 \times \mathbb{P}^1$, due to Deligne--Mostow and rediscovered by many others, and a lesser-known product of elliptic curves whose existence follows from work of Hirzebruch. Combined with related new examples, this answers the rational variant of a question of Gromov in the positive for surfaces of Kodaira dimension $\kappa \le 0$, namely that they admit deformations $V^\prime$ such that there is a compact ball quotient $\Gamma \backslash \mathbb{B}^2$ with a rational map $\Gamma \backslash \mathbb{B}^2 \dashrightarrow V^\prime$. Often the proof gives the stronger conclusion that $V^\prime$ is birational to a ball quotient orbifold. It also follows that every simply connected $4$-manifold is dominated by a complex hyperbolic manifold. All examples considered in this paper are shown to be arithmetic, and even arithmeticity of Hirzebruch's example appears to be new.

math.GT↗

arXiv2022

Algebraic fundamental groups of fake projective planes

Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are $\mathrm{Aut}(\mathbb{C})$-equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite \'etale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.

math.AG↗

arXiv2021

Residual finiteness for central extensions of lattices in $\mathrm{PU}(n,1)$ and negatively curved projective varieties

We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices $\Gamma < \mathrm{PU}(n,1)$ simple type. We prove that the preimage of $\Gamma$ in any connected cover of $\mathrm{PU}(n,1)$, in particular the universal cover, is residually finite. This follows from a more general theorem on residual finiteness of extensions whose characteristic class is contained in the span in $H^2(\Gamma, \mathbb{Z})$ of the Poincar\'e duals to totally geodesic divisors on the ball quotient $\Gamma \backslash \mathbb{B}^n$. For $n \ge 4$, if $\Gamma$ is a congruence lattice, we prove residual finiteness of the central extension associated with any element of $H^2(\Gamma, \mathbb{Z})$. Our main application is to existence of cyclic covers of ball quotients branched over totally geodesic divisors. This gives examples of smooth projective varieties admitting a metric of negative sectional curvature that are not homotopy equivalent to a locally symmetric manifold. The existence of such examples is new for all dimensions $n \ge 4$.

math.AG↗

arXiv2021

Residually finite lattices in $\widetilde{\mathrm{PU}(2,1)}$ and fundamental groups of smooth projective surfaces

This paper studies residual finiteness of lattices in the universal cover of $\mathrm{PU}(2,1)$ and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in $\mathrm{PU}(2,1)$ or a finite covering of it. First, we prove that certain lattices in the universal cover of $\mathrm{PU}(2,1)$ are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in $\mathrm{PU}(2,1)$ to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in $\mathrm{PU}(2,1)$.

math.AG↗

arXiv2020

Geometry of the Wiman-Edge monodromy

The Wiman-Edge pencil is a pencil of genus $6$ curves for which the generic member has automorphism group the alternating group $\mathfrak{A}_5$. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group $\mathfrak{S}_5$. Farb and Looijenga proved that the monodromy of the Wiman-Edge pencil is commensurable with the Hilbert modular group $\mathrm{SL}_2(\mathbb{Z}[\sqrt{5}])$. In this note, we give a complete description of the monodromy by congruence conditions modulo $4$ and $5$. The congruence condition modulo $4$ is new, and this answers a question of Farb-Looijenga. We also show that the smooth resolution of the Baily-Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.

math.AG↗

arXiv2020

Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

For $n \ge 2$, we prove that a finite volume complex hyperbolic $n$-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps between complex hyperbolic manifolds, which is related to a question of Siu, that certain hyperbolic $3$-manifolds cannot be totally geodesic submanifolds of complex hyperbolic manifolds, and that arithmeticity of complex hyperbolic manifolds is detected purely by the topology of the underlying complex variety, which is related to a question of Margulis. Our results also provide some evidence for a conjecture of Klingler that is a broad generalization of the Zilber--Pink conjecture.

math.DS↗

arXiv2019

Congruence RFRS towers

We describe a criterion for a real or complex hyperbolic lattice to admit a RFRS tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups $\mathrm{PSL}(\mathcal{O}_d)$ are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS K\"ahler groups that are not a subgroup of a product of surface groups and abelian groups.

math.GT↗