Eigenvalue problem for Kähler metrics on compact manifolds.
problem Eigenvalue problem for the Laplacian on Kähler manifolds.
method Introducing λk-extremal Kähler metrics and deducing conditions for extremality. result Conditions for a Kähler metric to be λk-extremal. Study describes conformal product structures on compact Kähler manifolds.
problem Characterizing compact Kähler manifolds with conformal product structures.
method Geometric description of conformal product structures on compact Kähler manifolds.
result Geometric characterization of compact Kähler manifolds with conformal product structures.
Study deformations of compact Kähler hyperbolic manifolds.
problem Investigate the deformation openness of Kähler hyperbolicity.
method Propose modified versions of Kähler hyperbolicity as a tool.
result Provide a first step towards understanding deformations.
Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.
problem Compact Kähler manifolds with pseudo-effective tangent bundles.
method Smooth or locally constant rationally connected fibration onto a quotient of a compact complex torus.
result Compact Kähler manifolds with pseudo-effective tangent bundles admit a fibration structure.
The paper proves conditions for compact complex manifolds to be Kahler outside analytic subsets.
problem Conditions for compact complex manifolds to be Kahler outside an analytic subset.
method Analyzes balanced manifolds and uses Hironaka's examples to prove theorems.
result Compact complex manifolds that are Kahler outside an analytic subset are balanced.
Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.
problem Characterizing compact complex manifolds with holomorphic GL(2)-geometry.
method Analyzing Kähler-Einstein and Fano manifolds, using GL(2) and SL(2) geometries.
result Only compact Kähler-Einstein manifolds with holomorphic GL(2)-geometry are covered by compact complex tori, three dimensional quadric, or three dimensional Lie ball.
Study stability of pseudo-Kähler and neutral Calabi-Yau manifolds, finding stability in 2D but failing in higher dimensions.
problem Stability of compact pseudo-Kähler and neutral Calabi-Yau manifolds.
method Analysis of stability through deformation theory and construction of counterexamples.
result Stability of compact pseudo-Kähler surfaces but failure in higher dimensions.
Paper extends Kobayashi-Hitchin correspondence to non-Kähler manifolds.
problem Applying Kobayashi-Hitchin correspondence to non-Kähler manifolds.
method Continuity method for vortex equation, Kobayashi-Hitchin correspondence for holomorphic pairs.
result Proved solvability of vortex equation on holomorphic vector bundles over compact Hermitian manifolds.
The paper studies rigidity results for harmonic forms on Kähler manifolds.
problem Understanding harmonic forms on Kähler manifolds.
method Analyzes rigidity results for harmonic (p,q)-forms in complete Kähler manifolds. result Shows several rigidity results and applications to non-compact Kähler manifolds.
The Hodge theorem connects cohomology groups on compact Kähler manifolds.
problem Establishing a relationship between cohomology groups on compact Kähler manifolds.
method Proving the Hodge decomposition theorem on compact d-Kähler manifolds.
result Hodge decomposition theorem on compact d-Kähler manifolds.
Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.
problem Chern version of the constant holomorphic sectional curvature conjecture for compact locally conformal Kähler manifolds
method Prove the conjecture using curvature identities and properties of Kähler metrics
result Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler
Study describes global sections of chiral de Rham complex on compact Ricci-flat Kähler manifolds.
problem None explicitly stated; focuses on description of global sections.
method Complete description of vertex algebra of global sections.
result Complete description of vertex algebra of global sections of chiral de Rham complex.
Killing fields on compact pseudo-Kähler manifolds are holomorphic.
problem Characterizing Killing fields on compact pseudo-Kähler manifolds.
method Detailed explanation and argumentation of why a previous claim was incomplete.
result Killing fields on compact pseudo-Kähler manifolds are holomorphic.
Compact Kähler manifolds with positive curvature are projective and rationally connected.
problem Characterizing compact Kähler manifolds with positive curvature.
method Proving properties of compact Kähler manifolds with quasi-positive second Chern-Ricci curvature.
result Compact Kähler manifolds with quasi-positive second Chern-Ricci curvature are projective and rationally connected.
A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive dimensional fibers at least one of which is of Kähler type, then X is globally conformall…
Survey on Kähler-Ricci flow solutions.
problem Existence of solutions for Kähler-Ricci flow.
method Survey of recent developments.
result Discussion of solutions existing for all positive times.
Compact Kähler manifolds with positive curvature have contractible covers.
problem Compact Kähler manifolds with positive sectional curvature.
method Proves contractibility of universal cover.
result The universal covering of a compact Kähler manifold with small positive sectional curvature is contractible.
Strong formal properties for toric and homogeneous Kähler manifolds.
problem Understanding formal properties of Kähler manifolds.
method Analyzing rationally and strongly formal properties of toric and homogeneous Kähler manifolds.
result Toric and homogeneous Kähler manifolds are both rationally and strongly formal.
Vanishing theorem for certain tensor fields on compact Hermitian manifolds.
problem Vanishing theorem for holomorphic tensor fields on compact Hermitian manifolds.
method Inspired by X. Yang and L. Ni-F. Zheng's ideas, the proof uses the definiteness of holomorphic sectional curvature.
result Spaces of certain holomorphic tensor fields are trivial under the definiteness of holomorphic sectional curvature.
The study finds conditions for almost-Kähler 4-manifolds to be Kähler.
problem Conditions for almost-Kähler 4-manifolds to be Kähler.
method Analyzes harmonic self-dual Weyl curvature and constant scalar curvature.
result Compact almost-Kähler 4-manifolds with harmonic self-dual Weyl curvature and constant scalar curvature are Kähler if c1⋅ω≥0. The paper explores p-Kähler structures on complex manifolds and their properties.
problem Addressing conjectures on specific types of complex manifolds and their structures.
method Analyzing (n−2)-Kähler nilmanifolds and holomorphically parallelizable nilmanifolds, deriving necessary conditions for smooth curves of p-Kähler structures, studying cohomology classes, and providing examples. result Derives necessary conditions for the existence of smooth curves of p-Kähler structures and provides examples of compact complex manifolds with p-Kähler structures. We investigate degenerate special-Hermitian metrics on compact complex manifolds, in particular, degenerate Kähler and locally conformally Kähler metrics on special classes of non-Kähler manifolds.
Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
Study complex Monge-Ampère equations on compact Kähler manifolds.
problem Finite energy range of complex Monge-Ampère operator.
method Survey and general answer to Guedj-Zeriahi's question.
result General answer to Guedj-Zeriahi's question about finite energy range.
New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.
problem Existence of essential conformal transformations in pseudo-Riemannian manifolds.
method Construction of compact locally conformally pseudo-Kähler manifolds with essential conformal transformations.
result Found compact examples of pseudo-Kähler manifolds with essential conformal transformations that are not conformally flat.
We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.
problem The topology of Kähler manifolds is largely determined by the geometry due to its rigidity.
method We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.
result We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.
A locally conformally Kähler (lcK) manifold is a complex manifold (M,J) together with a Hermitian metric g which is conformal to a Kähler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kähler geometry. The first one is the classification of con…
Mabuchi introduced multiplier Hermitian structures on compact Kahler manifolds and defined metrics similar to Kahler-Einstein metrics under these structures. In this note we generalize the inequality of Moser-Trudinger type on Kahler-Einstein manifolds to this case.
Optimal geometric estimates for Kähler manifolds with bounded Nash entropy
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded q-Nash entropy New compact K-E manifolds with negative curvature found.
problem Constructing compact Kähler-Einstein manifolds with negative curvature.
method Created compact Kähler-Einstein manifolds of dimension n with negative sectional curvature.
result Found compact Kähler-Einstein manifolds of negative curvature not covered by the ball.
A product of Kähler manifolds also carries a Kähler metric. In this short note we would like to study the product of generalized p−Kähler manifolds, compact or not. The results we get extend the known results (balanced, SKT, sG manifolds), and are optimal in the compact case. Hence we can give new non-trivial example…
The paper studies Kähler manifolds with partially semi-positive curvature and rational connectedness.
problem Analyzing compact Kähler manifolds with partially semi-positive curvature and rational connectedness.
method Proving rational connectedness for manifolds with BC-p positive tangent bundles, and applying these results to curvature conditions. result Confirming a conjecture and generalizing results on rational connectedness and curvature conditions.
We show that for n>2 a compact locally conformally Kähler manifold (M2n,g,J) carrying a non-trivial parallel vector field is either Vaisman, or globally conformally Kähler, determined in an explicit way by some compact Kähler manifold of dimension 2n−2.
Nontrivial Massey products found on compact Kähler manifolds.
problem Understanding the cohomology structure of compact Kähler manifolds.
method Analyzing the bigraded quasi-isomorphism type of forms on compact Kähler manifolds.
result Nontrivial ABC-Massey products exist on compact Kähler manifolds, including on surfaces and higher-dimensional manifolds.
New criterion found for Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
problem Existence of Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
method Algebraic criterion and stability condition introduced.
result New stability condition is both sufficient and necessary for the existence of Hermitian-Yang-Mills metrics.
In this article we study the limiting behavior of the Kähler Ricci flow on complete non-compact Kähler manifolds. We provide sufficient conditions under which a complete non-compact gradient Kähler-Ricci soliton is biholomorphic to $\ce^n$. We also discuss the uniformization conjecture by Yau \cite{Y} for complete non-…
The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.
problem Characterizing compact Kähler manifolds with nonnegative holomorphic sectional curvature.
method Holonomy principle and geometric properties.
result Compact Kähler manifolds with nonnegative holomorphic sectional curvature are projective and rationally connected.
The paper proves a structure theorem for compact Kähler manifolds with semi-positive holomorphic sectional curvature.
problem Understanding the fundamental groups of compact Kähler manifolds with specific curvature properties.
method Analyzing foliations and topological properties to prove a locally trivial fibration structure.
result Compact Kähler manifolds with semi-positive holomorphic sectional curvature admit a locally trivial fibration structure.
Let (Mn,g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold Nk with c1<0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive b…
We construct a compact Kähler manifold of nonnegative quadratic bisectional curvature, which does not admit any Kähler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional Kähler C-space with second Betti number equal to 1, and its canonical metric is a Kähler-Einstein metric of posit…
A locally conformally Kahler (LCK) manifold is a manifold which is covered by a Kahler manifold, with the deck transform group acting by homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We …
The paper proves a function extension on Kähler manifolds.
problem Proving a function extension on Kähler manifolds.
method Analyzing strictly psh functions on compact Kähler submanifolds.
result A strictly psh function on the whole manifold can be extended from a submanifold.
New approach finds Kähler metrics on compact complex manifolds.
problem Finding Kähler metrics on compact complex manifolds.
method Defining a new functional whose critical points are Kähler metrics.
result Critical points of the new functional are precisely the Kähler metrics.
Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension
problem Convergence of scalar curvature in Kähler-Ricci flow
method Uniform μ-entropy or uniform Sobolev inequality result Scalar curvature converges to negative Kodaira dimension
In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler mani…
The paper shows how compact Kähler manifolds with a special bundle can be broken down into simpler parts.
problem Understanding compact Kähler manifolds with a specific type of anti-canonical bundle.
method Introduced a new approach to extend a strategy from smooth projective varieties to singular Kähler spaces, using a flatness criterion for pseudo-effective sheaves.
result Compact Kähler manifolds with a nef anti-canonical bundle admit a locally trivial fibration with rational connected fibers and a Calabi-Yau base.
Study compact Kähler manifolds with pseudo-effective tangent bundles.
problem Characterize compact Kähler manifolds with strongly pseudo-effective tangent bundles.
method New proofs and characterizations of properties of vector bundles.
result Only projective spaces have big tangent bundles.
We give a classification of compact conformally Kahler Einstein-Weyl manifolds whose Ricci tensor is hermitian.