We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$.

We study the arithmeticity of $\mathbb {C}$-Fuchsian subgroups of some nonarithmetic lattices constructed by Deraux et al. [‘New non-arithmetic complex hyperbolic lattices’, Invent. Math. 203 (2016), 681–771]. Our results give an answer to a question raised by Wells [Hybrid Subgroups of Complex Hyperbolic Isometries, Doctoral thesis, Arizona State University, 2019].

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

Let ${\mathcal{D}}$ be the irreducible Hermitian symmetric domain of type $D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian variation of real Hodge structure ${\mathcal{V}}_{\mathbb{R}}$ of Calabi–Yau type over ${\mathcal{D}}$. This short note concerns the problem of giving motivic realizations for ${\mathcal{V}}_{\mathbb{R}}$. Namely, we specify a descent of ${\mathcal{V}}_{\mathbb{R}}$ from $\mathbb{R}$ to $\mathbb{Q}$ and ask whether the $\mathbb{Q}$-descent of ${\mathcal{V}}_{\mathbb{R}}$ can be realized as sub-variation of rational Hodge structure of those coming from families of algebraic varieties. When $n=2$, we give a motivic realization for ${\mathcal{V}}_{\mathbb{R}}$. When $n\geqslant 3$, we show that the unique irreducible factor of Calabi–Yau type in $\text{Sym}^{2}{\mathcal{V}}_{\mathbb{R}}$ can be realized motivically.

We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $\left( V,\,Q \right)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi $ is an orbit of a covering of Conf$\left( V,\,Q \right)$, the conformal group of the pair $\left( V,\,Q \right)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak{g}$, and furthermore a real form ${{\mathfrak{g}}_{\mathbb{R}}}$. The connected and simply connected Lie group ${{G}_{\mathbb{R}}}$ with $\text{Lie}\left( {{G}_{\mathbb{R}}} \right)\,=\,{{\mathfrak{g}}_{\mathbb{R}}}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi $.

A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type $(n,\,n,\,\infty ;\,k)$.

We prove that the Eisenstein–Picard modular group SU(2,1;ℤ[ω3]) can be generated by four given transformations.

In this work, we investigate how to decompose a pair $\left( A,\,B \right)$ of loxodromic isometries of the complex hyperbolic plane $\mathbf{H}_{\mathbb{C}}^{2}$ under the form $A\,=\,{{I}_{1}}{{I}_{2}}$ and $B\,=\,{{I}_{3}}{{I}_{2}}$, where the ${{I}_{k}}$'s are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group $\left\langle A,\,B \right\rangle $.

We give an explicit upper bound on the number of non-constant holomorphic maps from a quasi-projective manifold into a complex hyperbolic manifold of finite volume. This gives an effective version of the results of Sunada and Noguchi.

Starting from a Whitney decomposition of a symmetric cone $\Omega$, analogous to the dyadic partition $[2^j, 2^{ j + 1})$ of the positive real line, in this paper we develop an adapted Littlewood–Paley theory for functions with spectrum in $\Omega$. In particular, we define a natural class of Besov spaces of such functions, $B^{p, q}_\nu$, where the role of the usual derivation is now played by the generalized wave operator of the cone $\Delta(\frac{\partial}{\partial x})$. We show that $B^{p, q}_\nu$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p, q}_\nu (T_\Omega)$, at least in a 'good range' of indices $1 \leq q < q_{\nu, p}$. We obtain the sharp $q_{\nu, p}$ when $p \leq 2$, and conjecture a critical index for $p > 2$. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors $P_\nu \colon L^{p, q}_\nu \to A^{p, q}_\nu$, for which our result implies a positive answer when $q_{\nu, p}' < q < q_{\nu, p}$. This extends, to general cones, previous work of the authors on the light-cone.

Finally, we focus on light-cones and introduce new necessary and sufficient conditions for our conjecture to hold in terms of inequalities related to the cone multiplier problem. In particular, using recent work by Laba and Wolff, we establish the validity of our conjecture for light-cones when $p$ is sufficiently large.

We show that every Hilbert C$^*$-module $E$ is a JB$^*$-triple in a canonical way, establish an explicit expression for the holomorphic automorphisms of the unit ball of $E$, discuss the existence of fixed points for these automorphisms and give sufficient conditions for $E$ to have the density property.

Let $Y$ be an infinite covering space of a projective manifold $M$ in ${{\mathbb{P}}^{N}}$ of dimension $n\ge 2$. Let $C$ be the intersection with $M$ of at most $n-1$ generic hypersurfaces of degree $d$ in ${{\mathbb{P}}^{N}}$. The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$ be the smoothed distance from a fixed point in $Y$ in a metric pulled up from $M$. Let ${{\mathcal{O}}_{\phi }}(X)$ be the Hilbert space of holomorphic functions $f$ on $X$ such that ${{f}^{2}}{{e}^{-\phi }}$ is integrable on $X$, and define ${{\mathcal{O}}_{\phi }}(X)$ similarly. Our main result is that (under more general hypotheses than described here) the restriction ${{\mathcal{O}}_{\phi }}(Y)\to {{\mathcal{O}}_{\phi }}(X)$ is an isomorphism for $d$ large enough.

This yields new examples of Riemann surfaces and domains of holomorphy in ${{\mathbb{C}}^{n}}$ with corona. We consider the important special case when $Y$ is the unit ball $\mathbb{B}$ in ${{\mathbb{C}}^{n}}$, and show that for $d$ large enough, every bounded holomorphic function on $X$ extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on $\mathbb{B}$. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from $X$ to $\mathbb{B}$.

We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of SU(2, 2) with its conjugate.