1,046 research outputs found
Existence of holomorphic sections and perturbation of positive line bundles over --concave manifolds
By using asymptotic Morse inequalities we give a lower bound for the space of
holomorphic sections of high tensor powers in a positive line bundle over a
q-concave domain. The curvature of the positive bundle induces a hermitian
metric on the manifold. The bound is given explicitely in terms of the volume
of the domain in this metric and a certain integral on the boundary involving
the defining function and its Levi form. As application we study the
perturbattion of the complex structure of a q-concave manifold.Comment: 18 pages, AmsTe
Equidistribution results for singular metrics on line bundles
Let L be a holomorphic line bundle with a positively curved singular
Hermitian metric over a complex manifold X. One can define naturally the
sequence of Fubini-Study currents associated to the space of square integrable
holomorphic sections of the p-th tensor powers of L. Assuming that the singular
set of the metric is contained in a compact analytic subset of X and that the
logarithm of the Bergman kernel function associated to the p-th tensor power of
L (defined outside the singular set) grows like o(p) as p tends to infinity, we
prove the following:
1) the k-th power of the Fubini-Study currents converge weakly on the whole X
to the k-th power of the curvature current of L.
2) the expectations of the common zeros of a random k-tuple of square
integrable holomorphic sections converge weakly in the sense of currents to to
the k-th power of the curvature current of L.
Here k is so that the codimension of the singular set of the metric is
greater or equal as k. Our weak asymptotic condition on the Bergman kernel
function is known to hold in many cases, as it is a consequence of its
asymptotic expansion. We also prove it here in a quite general setting. We then
show that many important geometric situations (singular metrics on big line
bundles, Kaehler-Einstein metrics on Zariski-open sets, artihmetic quotients)
fit into our framework.Comment: 40 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
Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface
We generalize the results of Montgomery for the Bochner Laplacian on high
tensor powers of a line bundle. When specialized to Riemann surfaces, this
leads to the Bergman kernel expansion and geometric quantization results for
semi-positive line bundles whose curvature vanishes at finite order. The proof
exploits the relation of the Bochner Laplacian on tensor powers with the
sub-Riemannian (sR) Laplacian
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
On the compactification of hyperconcave ends and the theorems of Siu-Yau and Nadel
We show that the pseudoconcave holes of some naturally arising class of
manifolds, called hyperconcave ends, can be filled in, including the case of
complex dimension 2 . As a consequence we obtain a stronger version of the
compactification theorem of Siu-Yau and extend Nadel's theorems to dimension 2.Comment: 13 pages, AMSLaTeX, short version accepted for publication in
Inventiones Mat
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