416 research outputs found
The Toledo invariant on smooth varieties of general type
We propose a definition of the Toledo invariant for representations of
fundamental groups of smooth varieties of general type into semisimple Lie
groups of Hermitian type. This definition allows to generalize the results
known in the classical case of representations of complex hyperbolic lattices
to this new setting: assuming that the rank of the target Lie group is not
greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood
type inequality and we characterize the corresponding maximal representations.Comment: 19 page
On the equidistribution of totally geodesic submanifolds in compact locally symmetric spaces and application to boundedness results for negative curves and exceptional divisors
We prove an equidistribution result for totally geodesic submanifolds in a
compact locally symmetric space. In the case of Hermitian locally symmetric
spaces, this gives a convergence theorem for currents of integration along
totally geodesic subvarieties. As a corollary, we obtain that on a complex
surface which is a compact quotient of the bidisc or of the 2-ball, there is at
most a finite number of totally geodesic curves with negative self
intersection. More generally, we prove that there are only finitely many
exceptional totally geodesic divisors in a compact Hermitian locally symmetric
space of the noncompact type of dimension at least 2.Comment: The paper has been substantially rewritten. Corollary 1.3 in the
previous versions was false as stated. This has been corrected (see Corollary
1.5). The main results are not affecte
Harmonic maps and representations of non-uniform lattices of PU(m,1)
We study representations of lattices of PU(m,1) into PU(n,1). We show that if
a representation is reductive and if m is at least 2, then there exists a
finite energy harmonic equivariant map from complex hyperbolic m-space to
complex hyperbolic n-space. This allows us to give a differential geometric
proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a
new invariant associated to representations into PU(n,1) of non-uniform
lattices in PU(1,1), and more generally of fundamental groups of orientable
surfaces of finite topological type and negative Euler characteristic. We prove
that this invariant is bounded by a constant depending only on the Euler
characteristic of the surface and we give a complete characterization of
representations with maximal invariant, thus generalizing the results of D.
Toledo for uniform lattices.Comment: v2: the case of lattices of PU(1,1) has been rewritten and is now
treated in full generality + other minor modification
Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type
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 K\"{a}hler
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
On the second cohomology of K\"ahler groups
Carlson and Toledo conjectured that any infinite fundamental group
of a compact K\"ahler manifold satisfies . We assume
that admits an unbounded reductive rigid linear representation. This
representation necessarily comes from a complex variation of Hodge structure
(\C-VHS) on the K\"ahler manifold. We prove the conjecture under some
assumption on the \C-VHS. We also study some related geometric/topological
properties of period domains associated to such \C-VHS.Comment: 21 pages. Exposition improved. Final versio
Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groups
We complete the classification of maximal representations of uniform complex
hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional
groups and . We prove that if is a maximal
representation of a uniform complex hyperbolic lattice , , in an exceptional Hermitian group , then and , and we describe completely the representation . 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
exceptional cases and instead we provide a more unified perspective, as
independent as possible of the classification of the simple Hermitian Lie
groups. This relies on the study of the cominuscule representation of the
complexification of the target group. As a by-product of our methods, when the
target Hermitian group has tube type, we obtain an inequality on the Toledo
invariant of the representation which is stronger
than the Milnor-Wood inequality (thereby excluding maximal representations in
such groups).Comment: Comments are welcome
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