Workshop

On the Berezin Transform for Compact Kähler Manifolds

  • Martin Schlichenmaier
ITP - Hörsaal 1 Universität Leipzig, ITP (Leipzig)

Abstract

For quantizable compact Kähler manifolds (M,ω) with associated quantum line bundle (L,h,) the Berezin transform I for C functions is introduced. This is a generalization of the original transform between contravariant and covariant Berezin symbols. If one considers all positve tensor powers Lm of L and the Berezin transform I(m) then it admits a complete asymptotic expansion in powers of 1/m , e.g. I(m)f(x)k=0(1/m)kIkf(x) with differential operators Ik. It turns out that I0=id and I1=Δ, the Laplace-Beltrami operator. Consequences of this expansion for the Berezin-Toeplitz operator quantization and the Berezin-Toeplitz deformation quantization are discussed.

H. Grosse

G. Rudolph

Klaus Sibold

Julius Wess

Eberhard Zeidler