Kähler structures on complex torus

The Direct Integral of Some Weighted Bergman Spaces

“…Let s 0 = e −i(cy2+f ) s. Since s is non-vanishing, s 0 is also non-vanishing. Similar arguments to those after [2,Equation (3.11)] show that we have ξ • s 0 = 0 for all ξ ∈ R n+k . Thus, s 0 is G-invariant.…”

“…Therefore, given h ∈ H, one may attempt to write h as a 'span' of the functions e rz over r ∈Ĝ, namely h = r∈Ĝ φ(r)e rz . (1.6) This works well in the case where n = 0 andĜ = Z k is discrete, as carried out in [2]. But with n > 0, the first component of the parameter r = (r 1 , r 2 ) in (1.6) is no longer discrete under the natural measure of R n .…”

“…The special case G = R k /Z k has been proved in [2]. The main argument here is therefore to deal with the R n -action via the direct integral expression.…”

“…We now prove Theorem 1.2. The toric part R k /Z k of G has been handled in [2], so our main focus is on the Euclidean part R n . For clarity of explanation, we start with the special case G = R n .…”

“…By Lemma 3.1, µ − γ = c dy 2 + df for some c ∈ C k and function f . Using the section e cx2−if s of L and the arguments leading to [2,Equation (3.11)], it follows that c ∈ Z k . Therefore, as explained in (1.2), the expression e −i(cy2+f ) is well defined.…”