Maximal representations of complex hyperbolic lattices in SU(m,n)

Let $\Gamma$ denote a lattice in $SU(1,p)$, with $p$ greater than 1. We show that there exists no Zariski dense maximal representation with target $SU(m,n)$ if $n>m>1$. The proof is geometric and is based on the study of the rigidity properties of the geometry whose points are isotropic $m$-subspaces of a complex vector space $V$ endowed with a Hermitian metric $h$ of signature $(m,n)$ and whose lines correspond to the $2m$ dimensional subspaces of $V$ on which the restriction of $h$ has signature $(m,m)$.Comment: 41 pages, 2 figures, accepted for pubblication in GAF

Intersection cohomology of higher rank Teichm\"uller components

Exploiting the non-abelian Hodge correspondence, together with the Cayley correspondence, in this paper, we compute the intersection cohomology of certain singular higher rank Teichm\"uller components of character varieties of the fundamental group of a compact Riemann surface.Comment: 19 page

Stacky Formulations of Einstein Gravity

This is an investigation of "stacky" structures for Einstein gravity together with an alternative reformulation in the language of formal moduli problems. In the first part of the paper, we first revisit the aspects of (vacuum) Einstein gravity on a Lorentzian 3-manifold $M$ with cosmological constant $\Lambda=0$. Next, we shall provide a realization of the moduli space of Einstein's field equations as a certain stack. We indeed construct the stack of (vacuum) Einstein gravity in $n$-dimensional set-up with vanishing cosmological constant by using the homotopy theoretical formulation of stacks. With this new formulation, we also upgrade the equivalence of certain 2+1 quantum gravities with gauge theory to the isomorphism between the corresponding moduli stacks. The second part of the paper, on the other hand, is designed as a detailed survey on formal moduli problems. It is in particular devoted to formalize Einstein gravity in the language of formal moduli problems and to study the algebraic structure of observables in terms of factorization algebras.Comment: 54 page