1,721,084 research outputs found
A Laplace integral, the T-Y-Z expansion and Berezin's transform on a Kaehler manifold
No full textLet M be an n-dimensional complex manifold endowed with a C∞ Kähler metric g. We show that a certain Laplace-type integral Lm (x), when x varies in a sufficiently small open set U ⊂ M, has an asymptotic expansion Lm(x)= 1/mn ∑r≤0 m-rCr(f) (x), where Cr:C∞ (U) → C∞ (U) are smooth differential operators depending on the curvature of g and its covariant derivatives. As a consequence we furnish a different proof of Lu's theorem by computing the lower order terms of Tian-Yau-Zelditch expansion in terms of the operator Cj. Finally, we compute the differential operators Qj of the expansion Berm (f) = ∑r≤0 m-r Qr (f) of Berezin's transform in terms of the operators Cj
Regular quantizations of Kaehler manifolds and constant scalar curvature metrics
No full textIn this paper we prove that if a Ka ̈hler manifold (M, ω) admits a regular quantization then its scalar curvature is constant. Moreover, we apply this result to the two-dimensional complete Reinhardt domains in C2 to show that such domains admit a regular quantization iff they are biholomorphically isometric to the 2-ball in C2 endowed with the hyperbolic metric
Quantization of bounded domains
No full textWe consider the quantization of a complex manifold endowed with the Bergman form following the ideas of Cahen, Gutt and Rawnsley. In particular we give a geometric interpretation for the quantization to be regular in terms of the Hilbert space of square integrable holomorphic n-forms on M and the Hilbert space of holomorphic n-forms on M bounded with respect to the Liouville element
Regular quantizations and covering maps
No full textLet \tilde M-->M be
a holomorphic (unbranched) covering map between two compact
complex manifolds, with b_2(\tilde M)=1.
We prove that if \tilde M and M both admit regular Kaehler forms
\tilde\omega and \omega respectively then, up to homotheties,
(\tilde M, \tilde\omega) and (M, \omega) are biholomorphically
isometric
Bergman and balanced metrics on complex manifolds
No full textIn this paper we find sufficient conditions for a Bergman Einstein metric on a complex manifold to be balanced in terms of its Bochner's coordinates
The function epsilon for complex tori and Riemann surfaces
No full textIn the framework of the quantization of Ka ̈hler manifolds carried out in [3], [4], [5] and [6], one can define a smooth function, called the function epsilon, which is the central object of the theory. The first explicit calculation of this function can be found in [10].
In this paper we calculate the function epsilon in the case of the complex tori and the Riemann surfaces
A Laplace integral on a Kaehler manifold and Calabi's diastasis function
No full textIn this paper we give a different proof of Engliš’s result [J. Reine Angew. Math. 528 (2000) 1–39] about the asymptotic expansion of a Laplace integral on a real analytic Kähler manifold (M,g) by using the link between the metric g and the associated Calabi’s diastasis function D. We also make explicit the connection between the coefficients of Engliš’ expansion and Gray’s invariants [Michigan Math. J. (1973) 329–344]
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