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On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold
This paper divides into two parts. Let be a compact Hermitian
manifold. Firstly, if the Hermitian metric satisfies the assumption
that for all , we generalize the
volume of the cohomology class in the K\"{a}hler setting to the Hermitian
setting, and prove that the volume is always finite and the
Grauert-Riemenschneider type criterion holds true, which is a partial answer to
a conjecture posed by Boucksom. Secondly, we observe that if the anticanonical
bundle is nef, then for any , there is a smooth
function on such that
and
Ricci. Furthermore, if
satisfies the assumption as above, we prove that for a
Harder-Narasimhan filtration of with respect to , the slopes
for all , which
generalizes a result of Cao which plays a very important role in his studying
of the structures of K\"{a}hler manifolds
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
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