No Arabic abstract
For a noncompact complex hyperbolic space form of finite volume $X=mathbb{B}^n/Gamma$, we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification $overline{X}$ of $X$ similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity $overline{X}-X$ called `canonical radius of a cusp of $Gamma$. The main result in the article is that there is a constant $r^*=r^*(n)$ depending only on the dimension, so that if the canonical radii of all cusps of $Gamma$ are larger than $r^*$, then there exist symmetric differentials of $overline{X}$ vanishing at infinity. As a corollary, we show that the cotangent bundle $T_{overline{X}}$ is ample modulo the infinity if moreover the injectivity radius in the interior of $overline{X}$ is larger than some constant $d^*=d^*(n)$ which depends only on the dimension.
We give a necessary and sufficient condition for a non-degenerate symmetric 3-differential with nonzero Blaschke curvature on a complex surface to be locally representable as a product of three closed holomorphic 1-forms. We give t
We investigate the $CR$ geometry of the orbits $M$ of a real form $G_0$ of a complex simple group $G$ in a complex flag manifold $X=G/Q$. We are mainly concerned with finite type, Levi non-degeneracy conditions, canonical $G_0$-equivariant and Mostow fibrations, and topological properties of the orbits.
We study, from the point of view of CR geometry, the orbits M of a real form G of a complex semisimple Lie group G in a complex flag manifold G/Q. In particular we characterize those that are of finite type and satisfy some Levi nondegeneracy conditions. These properties are also graphically described by attaching to them some cross-marked diagrams that generalize those for minimal orbits that we introduced in a previous paper. By constructing canonical fibrations over real flag manifolds, with simply connected complex fibers, we are also able to compute their fundamental group.
The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $mathbb{D}_d$ in $mathbb{C}^d$. For super-critical weighted Bergman spaces, the interpolation is uniform when the functions range over the unit ball of the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and Theorem 4.11); optimal simultaneous uniform interpolation for weighted Bergman spaces (cf. Theorem 1.8, Proposition 1.9 and Theorem 4.13); strong simultaneous uniform interpolation for weighted harmonic Hardy spaces (cf. Theorem 1.11 and Theorem 4.15); and establish the impossibility of the uniform simultaneous interpolation for the Bergman space $A^2(mathbb{D}_d)$ on $mathbb{D}_d$ (cf. Theorem 1.12 and Theorem 6.7).
We construct stable vector bundles on the space of symmetric forms of degree d in n+1 variables which are equivariant for the action of SL_{n+1}(C), and admit an equivariant free resolution of length 2. For n=1, we obtain new examples of stable vector bundles of rank d-1 on P^d, which are moreover equivariant for SL_2(C). The presentation matrix of these bundles attains Westwicks upper bound for the dimension of vector spaces of matrices of constant rank and fixed size.