490 research outputs found
Superconnection and family Bergman kernels
We establish an asymptotic expansion for families of Bergman kernels. The key
idea is to use the superconnection as in the local family index theorem.Comment: C. R. Math. Acad. Sci. Pari
Bergman kernels and symplectic reduction
We generalize several recent results concerning the asymptotic expansions of
Bergman kernels to the framework of geometric quantization and establish an
asymptotic symplectic identification property. More precisely, we study the
asymptotic expansion of the -invariant Bergman kernel of the spin^c Dirac
operator associated with high tensor powers of a positive line bundle on a
symplectic manifold. We also develop a way to compute the coefficients of the
expansion, and compute the first few of them, especially, we obtain the scalar
curvature of the reduction space from the -invariant Bergman kernel on the
total space. These results generalize the corresponding results in the
non-equivariant setting, which has played a crucial role in the recent work of
Donaldson on stability of projective manifolds, to the geometric quantization
setting. As another kind of application, we generalize some Toeplitz operator
type properties in semi-classical analysis to the framework of geometric
quantization. The method we use is inspired by Local Index Theory, especially
by the analytic localization techniques developed by Bismut and Lebeau.Comment: 132 page
Berezin-Toeplitz quantization and its kernel expansion
We survey recent results about the asymptotic expansion of Toeplitz operators
and their kernels, as well as Berezin-Toeplitz quantization. We deal in
particular with calculation of the first coefficients of these expansions.Comment: 34 page
Berezin-Toeplitz quantization on Kaehler manifolds
We study the Berezin-Toeplitz quantization on Kaehler manifolds. We explain
first how to compute various associated asymptotic expansions, then we compute
explicitly the first terms of the expansion of the kernel of the
Berezin-Toeplitz operators, and of the composition of two Berezin-Toeplitz
operators. As application we estimate the norm of Donaldson's Q-operator.Comment: 45 pages, footnote at page 3 and Remark 0.5 added; v.3 is a final
update to agree with the published pape
Characteristic Laplacian in sub-Riemannian geometry
We study a Laplacian operator related to the characteristic cohomology of a
smooth manifold endowed with a distribution. We prove that this Laplacian does
not behave very well: it is not hypoelliptic in general and does not respect
the bigrading on forms in a complex setting. We also discuss the consequences
of these negative results for a conjecture of P. Griffiths, concerning the
characteristic cohomology of period domains
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