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.
In this article, we consider Cayley deformations of a compact complex surface in a Calabi--Yau four-fold. We will study complex deformations of compact complex submanifolds of Calabi--Yau manifolds with a view to explaining why complex and Cayley deformations of a compact complex surface are the same. We in fact prove …
In this paper, we study strongly Gauduchon metrics on compact complex manifolds. We study the cohomology cones SG in the de Rham cohomology groups generated by all strongly Gauduchon metrics and its direct images under proper modifications. We also study the moduli of strongly Gauduchon manifolds. We prove an existence…
Compact hyperbolic complex manifolds are rigid under deformation.
problem Studying the deformation behavior of compact hyperbolic complex manifolds.
method Analyzing smooth families of compact complex manifolds over the unit disk and compact Riemann surfaces.
result The H-locus is either at most a discrete subset or the whole domain, depending on the family structure. 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.
Study on complex submanifolds in Endo-Pajitnov manifolds.
problem Existence and characterization of complex submanifolds in Endo-Pajitnov manifolds.
method Identification of a class of Endo-Pajitnov manifolds containing compact complex submanifolds and establishment of an algebraic condition for the absence of compact complex curves.
result Established an algebraic condition for the absence of compact complex curves in Endo-Pajitnov manifolds.
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.
Develops Lie algebraic approach for compact complex homogeneous manifolds.
problem Proves important results on compact complex homogeneous manifolds.
method Uses standard results in Lie theory to associate a canonical abelian Lie algebra with a given integrable complex structure.
result Provides a new method of associating a canonical abelian Lie algebra with a given integrable complex structure.
Small Nijenhuis tensor found on compact manifolds.
problem Finding compact manifolds with small Nijenhuis tensor.
method Provided explicit examples of manifolds with small Nijenhuis tensor.
result Examples of manifolds with small Nijenhuis tensor in various dimensions.
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.
In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples (Δ,h,G) of n…
Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.
problem Establishing Hölder continuity of solutions to complex Monge-Ampère equations.
method Stability result for solutions in Lp space, Hölder continuity proof. result Solutions are Hölder continuous with the same exponent as in the Kähler case.
The note provides uniform estimates for complex Hessian equations on compact Hermitian manifolds.
problem Uniform estimates for solutions to degenerate complex Hessian equations on compact Hermitian manifolds.
method The approach relies on corresponding a priori estimates for Monge-Ampère equations.
result Extension and short alternative proof of results for complex Hessian equations.
A Sasaki-like almost contact complex Riemannian manifold is defined as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. Explicit compact and non-compact examples are given. A canonical construction producing a Sasaki-like almost contact complex Riemannian ma…
Study L2 Hilbert complexes on complex manifolds.
problem Analyse L2 Hilbert complexes on complex manifolds. method Define and study L2 Aeppli-Bott-Chern Hilbert complex; examine properties on various manifolds; use self-adjoint extensions of differential operators. result Kernels of operators on compact Hermitian manifolds are isomorphic to Aeppli or Bott-Chern cohomology.
We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections.
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. The study proves conditions for Hermitian metrics on compact almost complex manifolds.
problem Conditions for Hermitian metrics on compact almost complex manifolds.
method Analyzes compact almost complex manifolds with Hermitian metrics and integral conditions involving ∂-harmonic (0,1)-forms. result The integral condition is automatically satisfied for strongly Gauduchon metrics, and equivalent to being strongly Gauduchon for integrable almost complex structures.
Compact complex manifolds with specific group actions are conformally flat.
problem Compact complex manifolds with invariant conformal holomorphic structures.
method Study of manifolds with transitive and essentially acting complex semi-simple Lie groups.
result If a complex semi-simple Lie group acts transitively and essentially, the manifold is conformally flat.
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.
In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Kahler Calabi-Yau surfaces or Hopf surfaces.
The existence of some complex geometrical structures on a compact manifold such as complex structures, Kaehler (pseudo-Kaehler) structures often impose certain restrictions on its underling topological or differentiable manifold. In this article we survey recent developments in the study of the existence, classificatio…
Uniformizes compact complex manifolds via Anosov representations.
problem Uniformization of compact complex manifolds.
method Anosov homomorphisms with small limit sets.
result Local homeomorphism of character variety to Teichmüller space.
We study the stability of compact pseudo-Kähler manifolds, i.e. compact complex manifolds X endowed with a symplectic form compatible with the complex structure of X. When the corresponding metric is positive-definite, X is Kähler and any sufficiently small deformation of X admits a Kähler metric by a well-know…
Study proves correspondence for special bundles on complex surfaces.
problem Proving correspondence for parabolic bundles on complex surfaces.
method Kobayashi-Hitchin correspondence for parabolic bundles over compact non-Kähler surfaces.
result Proved Kobayashi-Hitchin correspondence for specified bundles.
Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.
problem Complex Monge-Ampère equation with positive Radon measure on compact Hermitian manifolds.
method Proves bounded subsolution theorem.
result Establishes bounded subsolution theorem for complex Monge-Ampère equation.
Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.
problem Existence and uniqueness of viscosity solutions to complex Hessian equations.
method Proves existence and uniqueness using viscosity solutions and determinant domination conditions.
result Viscosity solutions exist and are unique under certain conditions.
We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex 3-folds of the form P(T∗X) whose tangent bundles are nef. Moreover, we show that if X is a Fano manifold such t…
Study cohomology of Bigolin complex on complex manifolds.
problem Characterize cohomology of Bigolin complex on compact complex manifolds.
method Analyze the decomposition of the double complex into squares and zigzags, focusing on the zigzags contributing to cohomology.
result In complex dimension 3, multiplicities of zigzags are characterized by Betti, Hodge, Aeppli numbers plus Bigolin numbers.
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
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.
A Theorem of Wang in [Wa] implies that any holomorphic parallelism on a compact complex manifold M is flat with respect to some complex Lie algebra structure whose dimension coincides with that of M. We study here rational parallelisms on complex manifolds. We exhibit rational parallelisms on compact complex manifolds …
Study proposes a functional for LCK metrics on complex manifolds.
problem Existence problem for locally conformally Kähler metrics on compact complex manifolds.
method Introducing and studying a functional that varies by complex dimension.
result Functional approach to existence problem for LCK metrics.
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. Survey on holomorphic structures on complex manifolds.
problem Holomorphic foliations with transverse holomorphic Cartan geometries.
method Analyzes G-structures and Cartan geometries on compact complex manifolds.
result Foliated holomorphic Cartan geometries on compact complex manifolds.
The paper proves strong holomorphic Morse inequalities on complex manifolds with optimal estimates.
problem Holomorphic Morse inequalities on non-compact complex manifolds with optimal fundamental estimates.
method Established strong holomorphic Morse inequalities under optimal fundamental estimates.
result Strong holomorphic Morse inequalities hold true on non-compact complex manifolds with optimal fundamental estimates.
Regularities and stability shown for a specific type of complex parallelizable manifolds.
problem Stability and regularity of Chern-flat metrics on complex parallelizable manifolds.
method Study of Hermitian metrics governed by the second Chern-Ricci form on compact complex manifolds.
result Chern-flat metrics are dynamically stable on compact complex parallelizable manifolds.
We show that if a holomorphic n dimensional compact torus action on a compact connected complex manifold of complex dimension n has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.
We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of (∙,0)- and (0,∙)-Hodge numbers on a compact complex manifold, and obtain the equality for the number…
Paper generalizes sub-slope definition and solves complex equations on compact manifolds.
problem Solving complex equations on compact almost Hermitian manifolds.
method Generalized sub-slope definition and proved existence of solutions for a class of equations.
result Solved complex Hessian quotient and deformed Hermitian-Yang-Mills equations.
We show a general existence theorem to the complex Monge-Ampère type equation on compact Kähler manifolds.
Study of deformed Bott-Chern cohomology on complex manifolds.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
We show that, up to scaling, the complex Monge-Ampere equation on compact Hermitian manifolds always admits a smooth solution.
The paper proves properties of complex surfaces and their curvature.
problem Understanding curvature properties on compact complex surfaces.
method Establishing Chern number identities and applying to curvature conditions.
result Compact complex surfaces with specific curvature conditions are Kähler surfaces.
Study of affine and projective structures on foliated complex manifolds.
problem Formalizing and analyzing affine and projective structures on foliations.
method Formalizing concepts, providing local normal forms, proving index formulae, classifying structures.
result Compact algebraic manifolds of even dimension do not admit foliated projective structures.
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.
The study finds continuous solutions to complex Hessian equations on compact Hermitian manifolds.
problem Finding continuous solutions to complex Hessian equations on compact Hermitian manifolds.
method Deriving an L∞-estimate for bounded solutions to the complex m-th Hessian equations on compact Hermitian manifolds, assuming a positive right-hand side in the Orlicz space Lmn(logL)n(h∘log∘logL)n. result Establishing the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions.
This paper is devoted to give a complete unified study of several weak forms of $\ddb-$Lemma on compact complex manifolds.