133 research outputs found
The Chern-Ricci flow and holomorphic bisectional curvature
In this note, we show that on Hopf manifold , the non-negativity of the holomorphic bisectional curvature is not
preserved along the Chern-Ricci flow.Comment: Sci. China Math. 201
Scalar curvature on compact complex manifolds
In this paper, we prove that, a compact complex manifold admits a smooth
Hermitian metric with positive (resp. negative) scalar curvature if and only if
(resp. ) is not pseudo-effective. On the contrary, we also show
that on an arbitrary compact complex manifold with complex dimension , there exist smooth Hermitian metrics with positive total scalar curvature
and one of the key ingredients in the proof relies on a recent solution to the
Gauduchon conjecture by G. Sz\'{e}kelyhidi, V. Tosatti and B. Weinkove.Comment: To appear in Tran. Amer. Math. So
RC-positivity, rational connectedness and Yau's conjecture
In this paper, we introduce a concept of RC-positivity for Hermitian
holomorphic vector bundles and prove that, if is an RC-positive vector
bundle over a compact complex manifold , then for any vector bundle ,
there exists a positive integer such that
for and . Moreover, we obtain that, on a compact K\"ahler
manifold , if is RC-positive for every ,
then is projective and rationally connected. As applications, we show that
if a compact K\"ahler manifold has positive holomorphic sectional
curvature, then is RC-positive and
for every , and in
particular, we establish that is a projective and rationally connected
manifold, which confirms a conjecture of Yau([57, Problem 47]).Comment: Accepted by Cambridge Journal of Mathematic
A partial converse to the Andreotti-Grauert theorem
Let be a smooth projective manifold with . We show
that if a line bundle is -ample, then it is -positive. This
is a partial converse to the Andreotti-Grauert theorem. As an application, we
show that a projective manifold is uniruled if and only if there exists a
Hermitian metric on such that its Ricci curvature
has at least one positive eigenvalue everywhere
Big vector bundles and compact complex manifolds with semi-positive tangent bundles
We classify compact K\"ahler manifolds with semi-positive holomorphic
bisectional and big tangent bundles. We also classify compact complex surfaces
with semi-positive tangent bundles and compact complex -folds of the form
whose tangent bundles are nef. Moreover, we show that if is a
Fano manifold such that has nef tangent bundle, then
RC-positivity and the generalized energy density I: Rigidity
In this paper, we introduce a new energy density function on the
projective bundle for a smooth map
between Riemannian manifolds We get new
Hessian estimates to this energy density and obtain various new Liouville type
theorems for holomorphic maps, harmonic maps and pluri-harmonic maps. For
instance, we show that there is no non-constant holomorphic map from a compact
\emph{Hermitian manifold} with positive (resp. non-negative) holomorphic
sectional curvature to a \emph{Hermitian manifold} with non-positive (resp.
negative) holomorphic sectional curvature.Comment: Preliminary version and comments are welcom
RC-positive metrics on rationally connected manifolds
In this paper, we prove that if a compact K\"ahler manifold has a smooth
Hermitian metric such that is uniformly RC-positive,
then is projective and rationally connected. Conversely, we show that, if a
projective manifold is rationally connected, then the tautological line
bundle is uniformly RC-positive (which is equivalent
to the existence of some RC-positive complex Finlser metric on ). As an
application, we prove that if is a compact K\"ahler manifold with
certain quasi-positive holomorphic sectional curvature, then is projective
and rationally connected
Scalar curvature, Kodaira dimension and -genus
Let be a compact Riemannian manifold with quasi-positive Riemannian
scalar curvature. If there exists a complex structure compatible with ,
then the canonical bundle is not pseudo-effective and the Kodaira
dimension . We also introduce the complex Yamabe number
for compact complex manifold , and show that if
, then ; moreover, if is also spin, then
the Hirzebruch -hat genus
Ricci Curvatures on Hermitian manifolds
In this paper, we introduce the first Aeppli-Chern class for complex
manifolds and show that the - component of the curvature -form of the
Levi-Civita connection on the anti-canonical line bundle represents this class.
We systematically investigate the relationship between a variety of Ricci
curvatures on Hermitian manifolds and the background Riemannian manifolds.
Moreover, we study non-K\"ahler Calabi-Yau manifolds by using the first
Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we
construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds
. We also construct a smooth family of Gauduchon metrics on
a compact Hermitian manifolds such that the metrics are in the same first
Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and
nonnegative, but their Riemannian scalar curvatures are constant and vary
smoothly between negative infinity and a positive number. In particular, it
shows that Hermitian manifolds with nonnegative first Chern class can admit
Hermitian metrics with strictly negative Riemannian scalar curvature
RC-positivity and rigidity of harmonic maps into Riemannian manifolds
In this paper, we show that every harmonic map from a compact K\"ahler
manifold with uniformly RC-positive curvature to a Riemannian manifold with
non-positive complex sectional curvature is constant. In particular, there is
no non-constant harmonic map from a compact K\"ahler manifold with positive
holomorphic sectional curvature to a Riemannian manifold with non-positive
complex sectional curvature
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