arXiv subjects

Domingo Toledo

Publications and source records attributed to Domingo Toledo.

17 recordsLinked to original sources

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↗

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 $Γ< \mathrm{PU}(n,1)$ simple type. We prove that the preimage of $Γ$ 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 $H^2(Γ, \mathbb{Z})$ of the Poincaré duals to totally geodesic divisors on the ball quotient $Γ\backslash \mathbb{B}^n$. For $n \ge 4$, if $Γ$ is a congruence lattice, we prove residual finiteness of the central extension associated with any element of $H^2(Γ, \mathbb{Z})$. 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

Arakelov-Milnor inequalities and maximal variations of Hodge structure

In this paper we study the $\mathbb{C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. A main result of this paper is a bound on this invariant which generalizes both the Milnor-Wood inequality of the Hermitian case and the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case.

math.AG↗

arXiv2017

Non-geodesic variations of Hodge structure of maximum dimension

There are a number of examples of variations of Hodge structure of maximum dimension. However, to our knowledge, those that are global on the level of the period domain are totally geodesic subspaces that arise from an orbit of a subgroup of the group of the period domain. That is, they are defined by Lie theory rather than by algebraic geometry. In this note, we give an example of a variation of maximum dimension which is nowhere tangent to a geodesic variation. The period domain in question, which classifies weight two Hodge structures with $h^{2,0} = 2$ and $h^{1,1} = 28$, is of dimension $57$. The horizontal tangent bundle has codimension one, thus it is an example of a holomorphic contact structure, with local integral manifolds of dimension 28. The group of the period domain is $SO(4,28)$, and one can produce global integral manifolds as orbits of the action of subgroups isomorphic to $SU(2,14)$. Our example is given by the variation of Hodge structure on the second cohomology of weighted projective hypersurfaces of degree $10$ in a weighted projective three-space with weights $1, 1, 2, 5$

math.AG↗

arXiv2014

Bounded negativity of self-intersection numbers of Shimura curves on Shimura surfaces

Shimura curves on Shimura surfaces have been a candidate for counterexamples to the bounded negativity conjecture. We prove that they do not serve this purpose: there are only finitely many whose self-intersection number lies below a given bound. Previously, this result has been shown in [BHK+13] for compact Hilbert modular surfaces using the Bogomolov-Miyaoka-Yau inequality. Our approach uses equidistribution and works uniformly for all Shimura surfaces.

math.AG↗

arXiv2013

Quotients of non-classical flag domains are not algebraic

A flag domain D = G/V for G a simple real non-compact group G with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that any two points in a non-classical domain D can be joined by a finite chain of compact subvarieties of D. Then we prove that for F an infinite, finitely generated discrete subgroup of G, the analytic space F\D does not have an algebraic structure.

math.AG↗

arXiv2012

Hermitian Curvature and Plurisubharmonicity of Energy on Teichmüller Space

Let $M$ be a closed Riemann surface, $N$ a Riemannian manifold of Hermitian non-positive curvature, $f:M\to N$ a continuous map, and $E$ the function on the Teichmüller space of $M$ that assigns to a complex structure on $M$ the energy of the harmonic map homotopic to $f$. We show that $E$ is a plurisubharmonic function on the Teichmüller space of $M$. If $N$ has strictly negative Hermitian curvature, we characterize the directions in which the complex Hessian of $E$ vanishes.

math.DG↗

arXiv2011

Cubic Surfaces with Special Periods

We show that the vector of period ratios of a cubic surface is rational over $Q(ω)$, where $ω= \exp(2πi/3)$ if and only if the associate abelian variety is isogeneous to a product of Fermat elliptic curves. We also show how to construct cubic surfaces from a suitable totally real quintic number field $K_0$. The ring of rational endomorphisms of the associated abelian variety is $K = K_0(ω)$.

math.AG↗

arXiv2011

Affine cubic surfaces and relative SL(2)-character varieties of compact surfaces

A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic surface which is smooth at infinity and whose ideal locus is a generic tritangent plane arises as a relative SL(2)-character variety of the 4-holed sphere. Every such affine cubic for which all the periodic automorphisms of the tritangent plane extend to automorphisms of the cubic arises as a relative SL(2)-character variety of a 1-holed torus.

math.GT↗

arXiv2010

A Milnor-Wood inequality for complex hyperbolic lattices in quaternionic space

We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodesic representations, thereby establishing a global rigidity theorem for totally geodesic representations.

math.DG↗

arXiv2009

Hyperbolic geometry and moduli of real cubic surfaces

Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4 and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in PO(4,1). We also derive several new and several old results on the topology of M_0^R. Let M_s^R be the moduli space of real cubic surfaces that are stable in the sense of geometric invariant theory. We show that this space carries a hyperbolic structure whose restriction to M_0^R is that just mentioned. The corresponding lattice in PO(4,1), for which we find an explicit fundamental domain, is nonarithmetic.

math.AG↗

arXiv2006

The Moduli Space of Cubic Threefolds as a Ball Quotient

The moduli space of cubic threefolds in CP4, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. We describe both the birational modifications and the discrete group explicitly.

math.AG↗

arXiv2005

Generic Integral Manifolds for Weight Two Period Domains

We define the notion of a generic integral element for the Griffiths distribution on a weight two period domain, draw the analogy with the classical contact distribution, and then show how to explicitly construct an infinite-dimensional family of integral manifolds tangent to a given element.

math.AG↗

arXiv2003

Real Cubic Surfaces and Real Hyperbolic Geometry

The moduli space of stable real cubic surfaces is the quotient of real hyperbolic four-space by a discrete, nonarithmetic group. The volume of the moduli space is 37π^2/1080 in the metric of constant curvature -1. Each of the five connected components of the moduli space can be described as the quotient of real hyperbolic four-space by a specific arithmetic group. We compute the volumes of these components.

math.AG↗

arXiv2000

The Complex Hyperbolic Geometry of the Moduli Space of Cubic Surfaces

Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for stable cubic surfaces: the moduli space is biholomorphic to a quotient of the compex 4-ball by an explict arithmetic group generated by complex reflections. This identification gives interesting structural information on the moduli space and allows one to locate the points in complex hyperbolic 4-space corresponding to cubic surfaces with symmetry, e.g., the Fermat cubic surface. Related results, not quite as extensive, were announced in alg-geom/9709016.

math.AG↗

arXiv1998

Discriminant Complements and Kernels of Monodromy Representations

We show that the kernel of the monodromy representation for hypersurfaces of degree d and dimension n is large for d at least three with the exception of the cases (d,n) = (3,0) and (3,1). For these the kernel is finite. By "large" we mean a group that admits a homomorphism to a semisimple Lie group of noncompact type with Zariski-dense image. By the Tits alternative a large group contains a free subgroup of rank two.

alg-geom↗

arXiv1997

A Complex Hyperbolic Structure for Moduli of Cubic Surfaces

We show that the moduli space M of marked cubic surfaces is biholomorphic to the quotient by a discrete group generated by complex reflections of the complex four-ball minus the reflection hyperplanes of the group. Thus M carries a complex hyperbolic structure: an (incomplete) metric of constant holomorphic sectional curvature.

alg-geom↗