Toeplitz Operators, Kähler Manifolds, and Line Bundles
This is a survey paper. We discuss Toeplitz operators in Kähler geometry, with applications to geometric quantization, and review some recent developments.
math.SG↗
arXiv subjects
Tatyana Foth
Publications and source records attributed to Tatyana Foth.
Spanning sets for automorphic forms and dynamics of the frame flow on complex hyperbolic spaces
Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and $Γ$ a lattice in G. We study automorphic forms for $Γ$ if G is of real rank one with some additional assumptions, using dynamical approach based on properties of the homogeneous flow on $Γ\backslash G$ and a Livshitz type theorem we prove for such a flow. In the Hermitian case G=SU(n,1) we construct relative Poincare series associated to closed geodesics on $Γ\backslash G/K$ for one-dimensional representations of K, and prove that they span the corresponding spaces of cusp forms.
math.CV↗
Bohr-Sommerfeld tori and relative Poincare series on a complex hyperbolic space
Automorphic forms on a bounded symmetric domain D=G/K can be viewed as holomorphic sections of $L^{\otimes k}$, where L is a quantizing line bundle on a compact quotient of D and k is a positive integer. Let $Γ$ be a cocompact discrete subgroup of SU(n,1) which acts freely on SU(n,1)/U(n). We suggest a construction of relative Poincaré series associated to loxodromic elements in $Γ$. In complex dimension 2 we describe Bohr-Sommerfeld tori in $Γ\backslash SU(n,1)/U(n)$ associated to hyperbolic elements of $Γ$ and prove that the relative Poincaré series associated to the hyperbolic elements of $Γ$ are not identically zero for large k.
math.DG↗