The paper proves a blow-up formula for Bott-Chern cohomology and shows the bimeromorphic invariance of non-Kählerness degrees on threefolds.
problem The bimeromorphic invariance of the ∂∂ˉ-Lemma on threefolds. method Proves a blow-up formula for Bott-Chern cohomology and applies it to show bimeromorphic invariance of non-Kählerness degrees.
result The non-Kählerness degrees are bimeromorphic invariants on compact complex threefolds, implying the bimeromorphic invariance of the ∂∂ˉ-Lemma. Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Holomorphic families lead to Moishezon manifolds and bimeromorphic embeddings.
problem Characterizing and embedding Moishezon manifolds in projective space.
method Holomorphic families, local deformation invariance, strongly Gauduchon metrics, Monge-Ampère equations.
result Moishezon manifolds in a family are still Moishezon and admit a bimeromorphic embedding.
Study new invariants in complex geometry using Bott-Chern hypercohomology.
problem Understanding geometry through Bott-Chern hypercohomology and bimeromorphic invariants.
method Construct new invariants involving sheaf cohomology, establish blow-up formula and canonical morphism.
result Compute invariants for specific complex threefolds like Iwasawa manifolds and quintic threefolds.
Formula for Dolbeault cohomologies of complex manifolds via relative cohomology.
problem Calculating Dolbeault cohomologies of complex manifolds after blowing up.
method Introducing relative Dolbeault cohomology, proving blow-up formula, and using bimeromorphic invariance.
result Uniform proof of bimeromorphic invariance of Dolbeault cohomology numbers.
Unique minimal model for LCK manifolds proved.
problem Characterizing unique minimal models for LCK manifolds.
method Proving bimeromorphic maps are holomorphic.
result LCK manifolds have a unique minimal model.
We show that the non pluripolar product of positive currents is a bimeromorphic invariant. Under some natural assumptions, we show that the (weighted) energy associated to big cohomology classes are also bimeromorphic invariants. We compare the weighted energy functionals of currents with respect to different cohomolog…
Simplified proof of complex manifold Todd genera invariance.
problem Invariance of higher Todd genera under bimeromorphic transformations.
method Simple bimeromorphic proof using algebraic methods.
result Bimeromorphic invariance of higher Todd genera of complex manifolds.
Researchers prove no unexpected relations between complex manifold numbers.
problem Proving no unexpected universal linear relations between Hodge, Betti, and Chern numbers of compact complex manifolds.
method Developed a framework to tackle more general questions involving all cohomological invariants, solved specific construction problems.
result Obtained full answers to general questions about universal relations and bimeromorphic invariants in low dimensions.
The paper proves a property of complex geometry under transformations.
problem The ∂∂-lemma property in complex geometry. method Simple proof using a blow-up formula and morphism.
result Heredity and bimeromorphic invariance of the ∂∂-lemma property. The paper explores the structure of double complexes and their applications.
problem Understanding the structure and properties of double complexes.
method Investigates the folklore statement about double complexes and their decomposition into squares and zigzags, and computes the Grothendieck ring of double complexes.
result Obtained a Poincaré duality for higher pages of the Frölicher spectral sequence and constructed a functorial three-space decomposition of the middle cohomology.
We investigate probability measures with finite pluricomplex energy. We give criteria insuring that a given measure has finite energy and test these on various examples. We show that this notion is a biholomorphic but not a bimeromorphic invariant.
Study on complex manifolds and their cohomology properties.
problem Stability of Dolbeault cohomology under modifications of compact complex manifolds.
method Using Čech cohomology theory to study Dolbeault cohomology of blow-ups.
result Construction of a compact complex non-Kähler manifold satisfying the ∂∂ˉ-Lemma. The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.
problem Understanding the Lee-Gauduchon cone for complex manifolds.
method Analyzing the Lee-Gauduchon cone as a convex cone of cohomology classes.
result The Lee-Gauduchon cone is a bimeromorphic invariant.
Study compares Kähler quotients of torus actions under varying moment maps.
problem Comparing Kähler quotients of torus actions under varying moment maps.
method Analyzes the transformation of Kähler quotients as moment maps change, proving bimeromorphic transformations and desingularizations.
result Each nondegenerate singular Kähler quotient has a partial and rational desingularization.
Hyperkahler manifolds with round Kahler cones have unique bimeromorphic models.
problem Existence of round Kahler cones in hyperkahler manifolds.
method Analyzing the Kahler cone and its relation to the Bogomolov-Beauville-Fujiki form.
result Maximal holonomy hyperkahler manifolds with b2>4 have deformations with round Kahler cones. Establishes Kobayashi-Hitchin correspondence for nef and big classes.
problem Analyzing stability and positivity in algebraic geometry.
method Introducing adapted currents and metrics to establish correspondence.
result Equality cases of Bogomolov-Gieseker and Miyaoka-Yau inequalities.
Study shows volumes of complex classes can be represented by convex bodies.
problem Understanding volumes of complex classes on Kähler manifolds.
method Approximation by partial Okounkov bodies, restricted volume properties, and bimeromorphic behavior of currents.
result Volume of transcendental big (1,1)-classes can be realized by convex bodies. We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa. As an application, we show that finite time non-collapsing singu…
Paper defines new stability and metrics for complex spaces.
problem Stability and metrics for complex spaces with big cohomology classes.
method Introduces slope stability and Hermitian-Einstein metrics for big cohomology classes.
result Kobayashi Hitchin correspondence and Bogomolov Gieseker inequality proved.
The paper studies cohomology of complex manifolds using Morse-Novikov and Dolbeault-Morse-Novikov theories.
problem Analyzing cohomology of complex manifolds using Morse-Novikov theory.
method Establishing invariants, the Leray-Hirsch theorem, and blow-up formula for Dolbeault-Morse-Novikov cohomology.
result Established relations and stabilities of dimensions under complex structure deformations.
The study classifies Kähler threefolds with special fiber bundles.
problem Classifying Kähler threefolds with specific fiber bundles.
method Generalized from projective case, using fiber bundles over the circle and étale covers.
result Proves Kotschick's conjecture in dimension 3.
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.
We prove that for a compact Kähler threefold with canonical singularities and vanishing first Chern class, the projective fibres are dense in the semiuniversal deformation space. This implies that every Kähler threefold of Kodaira dimension zero admits small projective deformations after a suitable bimeromorphic modifi…
Study of Kato manifolds and their locally conformally Kähler properties.
problem Characterize Kato manifolds and their locally conformally Kähler metrics.
method Revisit Brunella's proof and construct new examples of Kato manifolds.
result Found a class of Kato manifolds that admit locally conformally Kähler metrics and another class that do not.
Degenerate twistor deformations of Kähler manifolds are also Kähler.
problem Understanding the Kähler structure of degenerate twistor deformations.
method Using positive currents, Hahn–Banach theorem, and Huybrechts's theorem.
result Degenerate twistor deformations of compact holomorphically symplectic Kähler manifolds are Kähler.
The study explores geometric properties of hyperbolic cohomology classes on Kähler manifolds.
problem Understanding the geometric effects of hyperbolic cohomology classes on Kähler manifolds.
method Introducing Kähler topologically hyperbolic manifolds and proving spectral gap theorems for positive holomorphic Hermitian vector bundles.
result Kähler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact Kähler manifolds with trivial first real Chern class.
A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold M, showing that it is commensurable to an arithmetic subgroup in SO(3, b_2-3). A Teichmuller space of M is a space of complex structures on M up to is…
This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact Kähler manifolds, known as Fujiki {\it class} C manifolds. Our main idea is to exp…
In this paper, we will give a complete geometric background for the geometry of Painlevé VI and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space $M_n^{\balpha}(\bt, \blambda, L) $ of stable parabolic connection on $\BP^1$ with logarithmic poles at $D(\bt) = t_1 + ... + …
In this paper we explicitly construct Moishezon twistor spaces on nCP^2 for arbitrary n>1 which admit a holomorphic C*-action. When n=2, they coincide with Y. Poon's twistor spaces. When n=3, they coincide with the one studied by the author in math.DG/0403528. When n>3, they are new twistor spaces, to the best of the a…
Study Moishezon twistor spaces using quartic hypersurfaces and Del Pezzo fibrations.
problem Classify Moishezon twistor spaces with specific half-anti-canonical systems.
method Utilize pluri-half-anti-canonical maps and quartic hypersurfaces to investigate twistor spaces.
result Complete classification of Moishezon twistor spaces with half-anti-canonical systems as pencils.