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.
The purpose of this paper is to study the bimeromorphic invariants of compact complex manifolds in terms of Bott-Chern cohomology. We prove a blow-up formula for Bott-Chern cohomology. As an application, we show that for compact complex threefolds the non-Kählerness degrees, introduced by Angella-Tomassini [Invent. Mat…
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.
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.
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…
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…
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.
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.
We view Dolbeault-Morse-Novikov cohomology H^{p,q}_η(X) as the cohomology of the sheaf Ω_{X,η}^p of η-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we conside…
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…
We construct a simply-connected compact complex non-Kähler manifold satisfying the ∂∂ˉ-Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂∂ˉ-Lemma under modifications of compact complex m…
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 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.
Embeddings are ubiquitous in machine learning, appearing in recommender systems, NLP, and many other applications. Researchers and developers often need to explore the properties of a specific embedding, and one way to analyze embeddings is to visualize them. We present the Embedding Projector, a tool for interactive v…
Disk Embeddings tackle embedding DAGs with exponential growth.
problem Embedding DAGs with exponentially increasing ancestors and descendants.
method Disk Embeddings framework for quasi-metric spaces, including Hyperbolic Disk Embeddings.
result Disk Embeddings outperform existing methods in complex DAGs.
Proposes QQE for transforming and embedding data distributions.
problem Transforming and embedding data distributions for better representation or visualization.
method Quantile-Quantile Embedding (QQE) using quantile-quantile plot concept.
result QQE allows for better discrimination of classes in some cases.
Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
problem Embedding maps in higher dimensions without self-intersections.
method Lifting maps to embeddings in product spaces.
result Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
A new embedding structure, res-embedding, improves deep CTR models by enhancing generalization performance.
problem Deep CTR models often suffer from poor generalization performance due to learning of embedding parameters.
method Developed a res-embedding structure that combines a central embedding vector from an item-based interest graph with a residual embedding vector.
result Empirical evaluation shows significant improvement in model performance using the res-embedding structure.
New embeddings for manifolds using heat kernels.
problem Constructing canonical conformal embeddings for manifolds.
method Employing heat kernel embedding from Bérard-Besson-Gallot'94 to find canonical conformal embeddings.
result Intrinsic construction of canonical conformal embeddings with dimensions growing exponentially with t. Introduces PELP for graph-enhanced word embeddings.
problem Combining graph side-information into static word embeddings.
method Probabilistic embeddings using Laplacian priors.
result Unified and flexible approach to various embedding methods.
Curvature regularization prevents distortion in graph embeddings.
problem Graph topology patterns distort in Euclidean space, making detection difficult.
method Proposes curvature regularization to enforce flatness in embedding manifolds.
result Significant improvements in five embedding methods on open graph datasets.
Parallelizes graph embedding for large graphs.
problem Large graphs make existing graph embedding techniques inefficient.
method Distributed parallel computation framework using a cluster of compute nodes.
result Parallel computation scales well and maintains embedding quality.
Word embeddings are a powerful approach for unsupervised analysis of language. Recently, Rudolph et al. (2016) developed exponential family embeddings, which cast word embeddings in a probabilistic framework. Here, we develop dynamic embeddings, building on exponential family embeddings to capture how the meanings of w…
dynnode2vec embeds dynamic networks efficiently.
problem Capturing evolving patterns in large dynamic networks.
method dynnode2vec: a random walk based method initialized with previous embedding vectors.
result Demonstrates advantages over static methods on large dynamic network datasets.
Proposes cone embedding for better graph hierarchical structure representation.
problem Lack of natural and interpretable hierarchical indicators in graph embeddings.
method Metric cone embedding method to capture hierarchical structure.
result Extracts hierarchical structure from other graph embedding outputs.
Improves machine learning performance with domain-specific embeddings.
problem Tuning word embeddings for specific use cases and domains.
method Combines multiple domain-specific embeddings using a ranking function and dimensionality reduction.
result Effective domain-specific embeddings improve machine learning performance.
Classifies linear embeddings of grassmannians and ind-grassmannians.
problem Understanding linear embeddings of grassmannians and ind-grassmannians.
method Classification through isomorphism of Picard groups and direct limits.
result Most linear embeddings of grassmannians are equivariant.
DANE adapts network embeddings across multiple domains.
problem Learning embeddings for multiple networks without transferability.
method Graph Convolutional Network with adversarial learning.
result DANE achieves superior performance in cross-network domain adaptation.