Holomorphic Jacobi structures enrich the theory of Poisson manifolds.
problem Holomorphic Poisson structures are limited; holomorphic Jacobi structures offer more.
method Developed holomorphic Jacobi structures and their relationship with other structures.
result Holomorphic Jacobi structures provide a broader framework than holomorphic Poisson structures.
Study holomorphic last multipliers on complex manifolds.
problem Equivalence between holomorphic and real ODE systems.
method Analyzing last multipliers in complex manifold context.
result Relate holomorphic last multipliers to real last multipliers.
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.
Extends holomorphic Cartan geometry to Sasakian manifolds.
problem No specific problem stated; extends geometry to new context.
method Extends holomorphic Cartan geometry to Sasakian manifolds.
result Investigated branched holomorphic Cartan geometries on Sasakian Calabi-Yau manifolds.
Holomorphic Jacobi manifolds integrate to complex contact groupoids.
problem Integrating holomorphic Jacobi manifolds.
method Homogenization scheme to identify and integrate holomorphic Jacobi manifolds to complex contact groupoids.
result Holomorphic Jacobi manifolds integrate to complex contact groupoids.
Flat holomorphic connections on stable bundles over LVMB manifolds are always flat.
problem Characterizing LVMB manifolds and their holomorphic connections.
method Analyzing LVMB manifolds and their tangent bundles, deducing properties of holomorphic connections.
result Holomorphic connections on semi-stable bundles over LVMB manifolds are always flat.
We introduce holomorphic Riemannian maps between almost Hermitian manifolds as a generalization of holomorphic submanifolds and holomorphic submersions, give examples and obtain a geometric characterization of harmonic holomorphic Riemannian maps from almost Hermitian manifolds to Kaehler manifolds.
Holomorphic structures on Oeljeklaus-Toma manifolds are shown to be locally homogeneous.
problem Characterizing holomorphic structures on Oeljeklaus-Toma manifolds.
method Proving local homogeneity for various holomorphic geometric structures.
result Holomorphic geometric structures on Oeljeklaus-Toma manifolds are locally homogeneous.
The paper solves a long-standing problem by providing a symplectic realization for holomorphic Poisson manifolds.
problem Symplectic realization of holomorphic Poisson manifolds.
method Explicit construction of a holomorphic symplectic structure in a neighborhood of the zero section of T∗X. result There exists a holomorphic symplectic structure in a neighborhood of the zero section of T∗X such that the projection map is a symplectic realization of the given Poisson manifold. Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold extends uniquely to the envelope of holomorphy of the domain. This result completes the open problems of my earlier paper on extension of holomorphic geometric structures on complex manifolds. We use this result to classify th…
Holomorphic metrics on complex manifolds imply infinite fundamental groups.
problem Understanding fundamental groups of complex manifolds with holomorphic metrics.
method Analyzing properties of holomorphic Riemannian metrics on compact complex manifolds.
result Compact complex manifolds with holomorphic Riemannian metrics have infinite fundamental groups.
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.
Study compact Kähler manifolds with nonpositive holomorphic sectional curvature and their canonical bundles.
problem Characterizing compact Kähler manifolds with nonpositive holomorphic sectional curvature and properties of their canonical bundles.
method Analyzing properties of Hermitian metrics and canonical bundles on compact Kähler manifolds.
result Proves nefness of canonical bundle and ampleness in complex dimension two for negative holomorphic sectional curvature.
Holomorphic connections on Calabi-Yau manifolds are flat.
problem Existence of holomorphic connections on Calabi-Yau manifolds.
method Proving the existence of flat holomorphic connections for holomorphic vector bundles.
result Holomorphic vector bundles over compact Kähler Calabi-Yau manifolds admit flat holomorphic connections.
A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with b2≥7 admits only finitely…
The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
problem Diameter rigidity of Kähler manifolds with positive holomorphic sectional curvature.
method Establishing diameter rigidity for Kähler manifolds with positive holomorphic sectional curvature.
result Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.
problem Characterizing maps between almost Hermitian manifolds.
method Introducing Hermitian pluriharmonic maps and proving their properties.
result Holomorphic or anti-holomorphic maps are Hermitian pluriharmonic.
Introduces new holomorphic contact structures and proves unobstructedness theorems.
problem Generalizing classical holomorphic contact and symplectic structures.
method Introducing new classes of holomorphic p-contact and s-symplectic manifolds, observing their properties, and proving structure and unobstructedness theorems. result Generalizes classical results on small deformations of complex structures.
The study characterizes symmetries in Kaehler manifolds.
problem Understanding symmetries in Kaehler manifolds.
method Analyzing specific types of Kaehler manifolds: constant holomorphic sectional curvature, semisymmetric, and holomorphically pseudosymmetric.
result Characterization results and geometric interpretation of the complex Tachibana tensor.
Holomorphic maps of degree one are biholomorphic, confirming a partial order.
problem Understanding the partial order of holomorphic maps.
method Analyzing holomorphic maps of positive degree between compact complex manifolds.
result Holomorphic maps of degree one are biholomorphic.
Study holomorphic affine connections on non-Kähler manifolds.
problem Investigate geometric structures on non-Kähler compact complex manifolds.
method Prove properties of holomorphic affine connections on Calabi-Yau manifolds and compact complex manifolds of algebraic dimension one.
result Holomorphic affine structures on Calabi-Yau manifolds with polystable tangent bundles are locally homogeneous.
The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
problem Proving Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
method Analyzing maps between different classes of pseudo-Hermitian manifolds, using curvature assumptions.
result Holomorphic maps are constant under certain curvature conditions.
Study on non-Kähler Calabi-Yau manifolds and their geometric structures.
problem Holomorphic geometric structures on non-Kähler compact complex manifolds.
method Beauville-Bogomolov decomposition and weak Bochner principle.
result Rigidity of Vaisman Calabi-Yau manifolds implies they are Kodaira manifolds.
Study anti-invariant submersions from holomorphic statistical manifolds.
problem Understanding submersions in statistical manifolds.
method Introduced and analyzed anti-invariant holomorphic statistical submersions.
result Supported results with examples.
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.
We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular hol…
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.
New framework generalizes complex projective structures, proving non-existence of certain structures on Calabi-Yau manifolds.
problem Proving non-existence of certain holomorphic structures on Calabi-Yau manifolds.
method Introducing branched holomorphic Cartan geometries, proving independence of results, and using vector bundle properties.
result Non-projective compact simply connected Kähler Calabi-Yau manifolds do not admit branched holomorphic projective structures.
We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex which computes the holomorphic Poisson cohomology. A …
The paper explores constant holomorphic d-scalar curvature on specific manifolds.
problem Existence and prescription of constant holomorphic d-scalar curvature.
method Study of closed, connected almost Hermitian manifolds of dimension n≥6. result Obtained an application and variation formula for a conformal invariant.
Compact lcK manifolds with holomorphic Lee field are Vaisman under certain conditions.
problem Characterizing compact locally conformally Kähler manifolds with holomorphic Lee fields.
method Analyzing conditions for a compact lcK manifold to be Vaisman when it has a holomorphic Lee vector field.
result Compact lcK manifolds with holomorphic Lee field are Vaisman if the Lee field has constant norm or the metric is Gauduchon.
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.
New proofs for complex Hopf manifolds using geometric structures.
problem Proving properties of complex Hopf manifolds.
method Constructing integrable holomorphic G-structures and flat holomorphic Cartan geometries.
result Provided a new proof of flat holomorphic Cartan geometries on complex Hopf manifolds.
Study on holomorphic structures on complex manifolds with specific properties.
problem Holomorphic geometric structures on complex manifolds with vanishing first Chern class.
method Proves properties of holomorphic geometric structures on compact complex manifolds.
result Holomorphic geometric structures are locally homogeneous for certain manifolds.
Paper derives formulas for holomorphic maps between Hermitian manifolds and proves related theorems.
problem Holomorphic maps between Hermitian manifolds and their properties.
method Derives ∂∂-Bochner formulas and proves Schwarz lemma type estimates. result Generalizes Ni's results to Hermitian manifolds, proving rigidity and degeneracy theorems.
Study of holomorphic submanifolds in hypercomplex manifolds with specific metrics.
problem Characterizing holomorphic submanifolds in hypercomplex manifolds with Hermitian and Norden metrics.
method Investigation of necessary and sufficient conditions for total umbilicity and geodesicity.
result Conditions for holomorphic submanifolds to be totally umbilical or totally geodesic are derived.
Optimizes dimension estimate for holomorphic functions on Kähler manifolds.
problem Determining the optimal dimension for holomorphic functions with polynomial growth.
method Analyzes Kähler manifolds with non-negative holomorphic bisectional curvature.
result Identifies the specific gap and optimal dimension for maximal volume growth.
New findings on Kähler structures on specific types of manifolds.
problem Classifying manifolds with p-Kähler structures. method Analyzing nilmanifolds and holomorphically parallelizable solvmanifolds.
result Classification of manifolds with p-Kähler structures for lower values of p. Holomorphic structures on quantum flag manifolds uniquely defined.
problem Defining unique holomorphic structures on quantum flag manifolds.
method Constructing covariant q-deformed holomorphic structures. result Holomorphic structures are unique for simple relative Hopf modules.
Projectivized bundles inherit positive curvature from base Kähler manifolds.
problem Generalizing Hitchin's construction to projectivized bundles.
method Constructing Kähler metrics on projectivized bundles.
result Projectivized bundles P(E) inherit positive holomorphic sectional curvature. Classifies holomorphic parabolic geometries on complex manifolds.
problem Classifying holomorphic parabolic geometries on complex manifolds.
method Bounding numerical dimension and using geometric invariants.
result Uncovering foliations and fibrations on smooth projective varieties.
The paper finds charts for Legendrian curves in complex contact manifolds.
problem Finding charts for Legendrian curves in complex contact manifolds.
method Holomorphic Darboux charts and approximations of Legendrian immersions.
result Holomorphic Legendrian immersions can be approximated by embeddings.
New class of complex manifolds defined, properties studied.
problem Understanding properties of complex manifolds.
method Introduced and studied wHHR manifolds, proved metric equivalence.
result Bergman and Kobayashi metrics are biLipschitz equivalent for wHHR Stein manifolds.
Study extends holomorphic forms on noncompact Kahler manifolds.
problem Extension of holomorphic canonical forms on noncompact Kahler manifolds.
method L2 analytic methods and L2 Hodge theory.
result Generalizes classical results to noncompact cases.
The paper discusses properties of holomorphic one-forms on certain complex manifolds.
problem Analyzing holomorphic one-forms on weakly 1-complete manifolds.
method Examining connectivity of pairs and criteria for proper holomorphic mappings.
result Criteria for proper holomorphic mappings onto Riemann surfaces.
The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…
A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety which is calibrated by a form associated with the holomorphic volume form; this …
The paper proves a Schwarz lemma for mappings between specific types of manifolds.
problem Establishing a Schwarz lemma for mappings between Kähler and complex Finsler manifolds.
method Using properties of holomorphic sectional curvature and radial sectional curvature.
result A Schwarz lemma for holomorphic mappings between Kähler and complex Finsler manifolds.