Uniform RC-positivity results for direct image bundles.
problem Understanding the relation between rational connectedness and RC-positivity.
method Analyzing vector bundles and their direct images, using weak RC-positivity as a starting point.
result Uniform RC-positivity of direct image bundles under weak RC-positivity conditions.
The paper shows how uniform RC-positivity on manifolds implies projectivity and rational connectedness.
problem Understanding the conditions for projectivity and rational connectedness in Kähler manifolds.
method Analyzing the properties of uniformly RC-positive metrics and their relationship to projectivity and rational connectedness.
result Uniformly RC-positive metrics on rationally connected manifolds imply projectivity and rational connectedness.
The paper proves rational connectedness for certain Kähler manifolds.
problem Rational connectedness of compact Kähler manifolds.
method Uniform weak RC-positivity of the tangent bundle.
result Compact Kähler manifolds with uniformly weakly RC-positive tangent bundles are projective and rationally connected.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
In this paper, we introduce a concept of RC-positivity for Hermitian holomorphic vector bundles and prove that, if E is an RC-positive vector bundle over a compact complex manifold X, then for any vector bundle A, there exists a positive integer cA=c(A,E) such that $$H^0(X,\mathrm{Sym}^{\otimes \ell}E^*\otimes…
New characterizations of partial positivity using Hörmander's L2-estimate.
problem Characterizing partial positivity in complex geometry.
method Using a twisted version of Hörmander's L2-estimate. result New characterizations of partial positivity, including uniform q-positivity and RC-positivity. No non-constant holomorphic maps between certain complex manifolds with specific properties.
problem Existence of non-constant holomorphic maps between complex manifolds.
method Analyzing properties of tangent and cotangent bundles, pseudo-effectiveness, and nefness.
result Holomorphic maps between certain complex manifolds are constant.
Paper shows no non-constant harmonic maps under certain curvature conditions.
problem Existence of non-constant harmonic maps between specific manifolds.
method Analyzes curvature conditions and applies rigidity theorems.
result No non-constant harmonic maps exist under specified conditions.
The study finds conditions for scalar-flat metrics on ruled surfaces.
problem Conditions for the existence of scalar-flat metrics on ruled surfaces.
method Analyzes intrinsic number and complex structure of ruled surfaces.
result Scalar-flat metrics exist only for ruled surfaces with genus g≥2 and m(X)>2−2g. In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…
The paper introduces a new energy density function and proves Liouville type theorems for various maps.
problem Proving Liouville type theorems for holomorphic, harmonic, and pluri-harmonic maps.
method Introducing a new energy density function and deriving Hessian estimates.
result No non-constant holomorphic map exists between certain Hermitian manifolds with specific curvature conditions.