Let G be either SU(p,2) with p>=2, Sp(2,R) or SO(p,2) with p>=3. The symmetric spaces associated to these G's are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of uniform lattices of SU(m,1), m>1, into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G=SU(p,2) with p>=2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p,2)/S(U(p)xU(2)), on which it acts cocompactly.
This paper generalizes a rigidity result of complex hyperbolic spaces by M. Herzlich. We prove that an almost Hermitian spin manifold $(M,g)$ of real dimension $4n+2$ which is strongly asymptotic to $\hyp{\C}^{2n+1}$ and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space. The fact that we do not assume $g$ to be Kähler reflects in the inequality for the scalar curvature.
We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic projective space or to the quaternionic hyperbolic space. We describe locally the hyper-Hermitian quaternionic Kaehler manifolds with closed Lee form and show that the only complete simply connected such manifold is the quaternionic hyperbolic space.
We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with "good" hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to b…
We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $ρ$ is a maximal representation of a uniform complex hyperbolic lattice $Γ\subset{\rm SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G$, then $n=2$ and $G={\rm E}_6$, and we describe completely the representation $ρ$. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author (arxiv:1506.07274). However we do not focus immediately on the ex…
Let $L$ be a negative holomorphic line bundle over an $(n-1)$-dimensional complex torus $D$. Let $h$ be a Hermitian metric on $L$ such that the curvature form of the dual Hermitian metric defines a flat Kähler metric on $D$. Then $h$ is unique up to scaling, and, for some closed tubular neighborhood $V$ of the zero section $D \subset L$, the form $ω_h = -(n+1)i\partial\overline\partial\log(-{\log h})$ defines a complete Kähler-Einstein metric on $V \setminus D$ with ${\rm Ric}(ω_h) = -ω_h$. In fact, $ω_h$ is complex hyperbolic, i.e., the holomorphic sectional curvature of $ω_h$ is constant, an…
A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac14]$, and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenstädt's hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.
We investigate degenerate special-Hermitian metrics on compact complex manifolds, in particular, degenerate Kähler and locally conformally Kähler metrics on special classes of non-Kähler manifolds.
Generalizing results due to Brady and Farb we prove the existence of a bilipschitz embedded manifold of pinched negative curvature and dimension m_1+m_2-1 in the product X:=X_1^{m_1} times X_2^{m_2} of two Hadamard manifolds X_i^{m_i} of dimension m_i with pinched negative curvature. Combining this result with a Theorem by Buyalo and Schroeder we prove the additivity of the hyperbolic rank for products of manifolds with pinched negative curvature.
Let M be a compact irreducible Hermitian symmetric space and write M=G/K, with G the group of holomorphic isometries of M and K the stability group of the point of 0 in M. We determine the maximal dimension of a complex projective space embedded in M as a totally geodesic submanifold.
An $F$-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated hypersurface singularities. This paper reviews the construction of hermitian metrics on $F$-manifolds from $tt^*$ geometry. It clarifies the logic between several notions. It also introduces a new {\it canonical} hermitian metric. Near irreducible points it makes the manifold almost hyperbolic. This holds for the singularity case and will hopefully lead to app…
We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces of dimension greater than two. For the quaternionic hyperbolic spaces of dimension greater than two we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Elie Cartan.
In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we are able to establish several Schwarz type results for both the \emph{CR} maps and the transversally holomorphic maps between pseudo-Hermitian manifolds. Finally, we also discuss the \emph{CR} hyperbolicity problem for pseudo-Hermitian manifolds.
We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.
We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, $SU(np,p), Sp(2n+2,\mathbb R), SO^*(2n+2), SO(2n,2)$. This reproves or generalizes some results in \cite{GM, KKP, Klingler-inv, Pozzetti}.
These notes are an extended version of a talk given by the author in the seminar "Theorie Spectrale et Geometrie" at the Institut Fourier in No- vember 2016. We present here some aspects of a work in collaboration with B. Collier and N. Tholozan (arXiv:1702.08799). We describe how Higgs bundle theory and pseudo-hyperbolic geometry interfere in the study of maximal representations into Hermitian Lie groups of rank 2.
There are two main types of rank 2 Bäcklund transformations relating a pair of hyperbolic Monge-Ampère systems, which we call Type $\mathscr{A}$ and Type $\mathscr{B}$. For Type $\mathscr{A}$, we completely determine a subclass whose local invariants satisfy a specific but simple algebraic constraint; such Bäcklund transformations are parametrized by a finite number of constants, whose cohomogeneity can be either 2, 3 or 4. In addition, we present an invariantly formulated condition that determines whether a generic Type $\mathscr{B}$ Bäcklund transformation is one that, under suitable choices…
This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-Λ^2 < K < -λ^2$ where $λ/Λ> >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank rigidity t…
I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.
In this paper, a geometric process to compare solutions of symmetric hyperbolic systems on (possibly different) globally hyperbolic manifolds is realized via a family of intertwining operators. By fixing a suitable parameter, it is shown that the resulting intertwining operator preserves Hermitian forms naturally defined on the space of homogeneous solutions. As an application, we investigate the action of the intertwining operators in the context of algebraic quantum field theory. In particular, we provide a new geometric proof for the existence of the so-called Hadamard states on globally hy…
