816 research outputs found
Strong openness of multiplier ideal sheaves and optimal extension
In this note, we reveal that our solution of Demailly's strong openness
conjecture implies a matrix version of the conjecture; our solutions of two
conjectures of Demailly-Koll\'{a}r and Jonsson-Mustat\u{a} implies the truth of
twisted versions of the strong openness conjecture; our optimal
extension implies Berndtsson's positivity of vector bundles associated to
holomorphic fibrations over a unit disc.Comment: 11 pages,it has been accepted for publication in SCIENCE CHINA
Mathematic
CR eigenvalue estimate and Kohn-Rossi cohomology
Let be a compact connected CR manifold with a transversal CR -action
of dimension , which is only assumed to be weakly pseudoconvex. Let
be the -Laplacian. Eigenvalue estimate of
is a fundamental issue both in CR geometry and analysis. In this
paper, we are able to obtain a sharp estimate of the number of eigenvalues
smaller than or equal to of acting on the -th Fourier
components of smooth -forms on , where and
. Here the sharp means the growth order with respect to
is sharp. In particular, when , we obtain the asymptotic estimate of
the growth for -th Fourier components of
as . Furthermore, we establish a Serre
type duality theorem for Fourier components of Kohn-Rossi cohomology which is
of independent interest. As a byproduct, the asymptotic growth of the
dimensions of the Fourier components for is established. Compared with previous results in this field, the
estimate for already improves very much the corresponding estimate
of Hsiao and Li . We also give appilcations of our main results, including
Morse type inequalities, asymptotic Riemann-Roch type theorem,
Grauert-Riemenscheider type criterion, and an orbifold version of our main
results which answers an open problem.Comment: 39 pages, submitted on January 17, 2018. Comments welcome! arXiv
admin note: text overlap with arXiv:1506.06459, arXiv:1502.02365 by other
author
Effectiveness of Demailly's strong openness conjecture and related problems
In this article, stimulated by the effectiveness in Berndtsson's solution of
the openness conjecture and continuing our solution of Demailly's strong
openness conjecture, we discuss conditions to guarantee the effectiveness of
the conjecture and establish such an effectiveness result. We explicitly point
out a lower semicontinuity property of plurisubharmonic functions with a
multiplier, which is implicitly contained in our paper arXiv:1401.7158. We also
obtain optimal effectiveness of the conjectures of Demailly-Koll\'{a}r and
Jonsson-Mustat\u{a} respectively.Comment: 31 pages, 0 figures. arXiv admin note: substantial text overlap with
arXiv:1401.715
Characterization of multiplier ideal sheaves with weights of Lelong number one
In this article, we characterize plurisubharmonic functions of Lelong number
one at the origin, such that the germ of the associated multiplier ideal sheaf
is nontrivial: in arbitrary complex dimension, their singularity must be the
sum of a germ of smooth divisor and of a plurisubharmonic function with zero
Lelong number. We also present a new proof of the related well known
integrability criterion due to Skoda.Comment: 14 pages, 0 figures. Revised versio
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