This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the Kähler class, is used to define and characterize a special class of totally geodesic embeddings, called "tight embeddings". In addition, special isometric actions of finitely generated groups on Hermitian symmetric spaces are studied. Results of a joint work with M. Burger and A. Iozzi about surface group representations are also discussed.
We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles.
We study the barycentric straightening of simplices in irreducible symmetric spaces of non-compact type. We show that, for an n-dimensional symmetric space of rank r>1, the p-Jacobian has uniformly bounded norm, as soon as p is at least n-r+2. As a consequence, for a non-compact, connected, semisimple real Lie group G, in degrees n-r+2 and higher, every cohomology class has a bounded representative. This answers Dupont's problem in small codimension. We also give examples of symmetric spaces where the barycentrically straightened simplices of dimension n-r have unbounded volume, showing that t…
Two main theorems are proved in this paper. Theorem 1: There is a constant C(n, D) depending only on n and D such that for a closed Riemannian n-manifold satisfying Ric > -(n-1) and Diam < D, the ith bounded Betti number is bounded by C(n, D). Here the ith bounded Betti number is defined as the dimension of the image of the ith bounded cohomology in the ith real cohomology. Theorem 2: There are only finitely many isometric isomorphism types of bounded cohomology groups among closed Riemannian n-manifolds satisfying K > -1 and Diam < D. The dimension of the ith bounded cohomology might be infin…
We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the representation theoretic generalized Milnor-Wood inequality and the various different versions of Milnor-Wood inequalities for Higgs bundles that are known in the literature.
We relate $L^{q,p}$-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of $\ell^{q,p}$-cohomology, packing cohomology. This implies quasi-isometry invariance of $L^{q,p}$-cohomology together with its multiplicative structure. The result partially extends to the Rumin $L^{q,p}$-cohomology of bounded geometry contact manifolds.
This is the second paper in a series devoted to developing an arithmetic PDE analogue of Riemannian geometry. In Part 1 arithmetic PDE analogues of Levi-Civita and Chern connections were introduced and studied. In this paper arithmetic analogues of curvature and characteristic classes are developed.
Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and $θ$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space $\mathcal{E}^p(X,θ)$, making it a complete geodesic metric space.
We show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most of the higher rank symmetric spaces. As a consequence, we prove that the locally finite simplicial volume of most $\mathbb Q$-rank $1$ locally symmetric spaces is positive, which has been open for many years. Finally we improve the degree theorem for $\mathbb Q$-rank $1$ locally symmetric spaces of Connell and Farb. We also address the issue of surjectivity of the comparison map in real rank $2$ case.
We study possible real structures in the space of solutions to the quantum differential equation. We show that, under mild conditions, a real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry near the large radius limit. We compute an example of P^1 which is pure and polarized over the whole Kaehler moduli space H^2(P^1,C^*).
In this paper, we study a certain cohomology attached to a smooth function, which arose naturally in Poisson geometry. We explain how this cohomology depends on the function, and we prove that it satisfies both the excision and the Mayer-Vietoris axioms. For a regular function we show that the cohomology is related to the de Rham cohomology. Finally, we use it to give a new proof of a well-known result of A. Dimca in complex analytic geometry.
Let $M$ be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential $2$-form $ξ\inΩ^2(M)$ defines a bounded cocycle $c_ξ\in C_b^2(M)$ by integrating $ξ$ over straightened $2$-simplices. In particular Barge and Ghys proved that, when $M$ is a closed hyperbolic surface, $Ω^2(M)$ injects this way in $H_b^2(M)$ as an infinite dimensional subspace. We show that any class of the form $[c_ξ]$, where $ξ$ is an exact differential 2-form, belongs to the radical of the cup product on the graded algebra $H_b^\bullet(M)$.
In this paper we study the reduced and unreduced $L^{q,p}$-cohomology groups of manifolds of bounded geometry and their behavior under uniform maps. A \textit{uniform map} is a uniformly continuous map such that the diameter of the preimage is bounded in terms of the diameter of the subset. In general, the pullback map along a uniformly proper lipschitz map doesn't induce a morphism in, reduced or not, $L^{q,p}$-cohomology. Then, our goal is to introduce some contravariant functors between the category of manifolds of bounded geometry and uniform maps and the category of complex vector spaces …
In this paper we study the minimal and maximal $L^{2}$-cohomology of oriented, possibly not complete, Riemannian manifolds. Our focus will be on both the reduced and the unreduced $L^{2}$-cohomology groups. In particular we will prove that these groups are invariant under uniform homotopy equivalence quasi-isometric on the unbounded ends. A uniform map is a uniformly continuous map such that the diameter of the preimage of a subset is bounded in terms of the diameter of the subset itself. A map $f$ between two Riemannian manifolds $(X,g)$ and $(Y,h)$ is \textit{quasi-isometric on the unbounded…
We focus our attention on the de Rham operators' underlying properties which are specified by intrinsic effects of differential geometry structures. And then we apply the procedure of regularization in the context of Lipschitz version of de Rham calculus on metric simplicial complexes with bounded geometry.
We study the cohomology of the complexes of differential, integral and pseudo forms on odd symplectic manifolds taking the wedge product with the symplectic form as differential. We show that the cohomology classes are in correspondence with inequivalent Lagrangian submanifolds and that they all define semidensities on them. Further, we introduce new operators that move from one Lagragian submanifold to another and we investigate their relation with the so-called picture changing operators for the de Rham differential. Finally, we prove the isomorphism between the cohomology of the de Rham dif…
In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compact set has nonnegative Ricci curvature and nonnegative mean curvature (of the boundary) then its first relative L^2-cohomology vanishes (this in particular answers a question of Roe). We prove the Hodge-de Rham-theorem for L^2-cohomology of oriented boundary-manifolds of bounded g…
We show there exists a closed locally symmetric manifold $M$ modeled on $SL_n(\mathbb R)/SO(n)$, and a non-trivial homology class in degree $dim(M)-rank(M)$ represented by a totally geodesic submanifold that contains a circle factor. As a result, the comparison map $c^k:H_b^k(M,\mathbb R)\rightarrow H^k(M,\mathbb R)$ is not surjective in degree $k=dim(M)-rank(M)$. This provides a counterpart to a result of Lafont-Wang which states that $c^k$ is always surjective in degree $k\geq dim(M)-rank(M)+2$.
Let $X=G/K$ be a higher rank symmetric space of non-compact type, where $G$ is the connected component of the isometry group of $X$. We define the splitting rank of $X$, denoted by $\text{srk}(X)$, to be the maximal dimension of a totally geodesic submanifold $Y\subset X$ which splits off an isometric $\mathbb R$-factor. We compute explicitly the splitting rank for each irreducible symmetric space. For an arbitrary (not necessarily irreducible) symmetric space, we show that the comparison map $η:H^{*}_{c,b}(G,\mathbb{R})\rightarrow H^{*}_c(G,\mathbb{R})$ is surjective in degrees $*\geq \text{s…
We prove a cyclic cohomological analogue of Haefliger's van Est-type theorem for the groupoid of germs of diffeomorphisms of a manifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential operators on that groupoid, which is shown to carry an intrinsic Hopf algebraic structure. We establish a canonical isomorphism between the periodic Hopf cyclic cohomology of this extended Hopf algebra and the Gelfand-Fuchs cohomology of the Lie algebra of formal vector fields. We then show that this isomorphism can be explicitly implemented at the cochain …
We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.
Bounded cohomology classes from differential forms
Istituto Nazionale di Alta Matematica “Francesco Severi”, Research Unit Scuola Normale Superiore（高等数学国家研究所“弗朗切斯科·塞韦里”，比萨高等师范学院研究单元）；Scuola Normale Superiore（比萨高等师范学院）
