Proves K-stability of cubic threefolds and calculates their Kähler-Einstein metrics.
problem K-stability of cubic threefolds and explicit calculation of Kähler-Einstein metrics.
method Detailed study of three-dimensional canonical and terminal singularities, estimate of Kawamata log terminal volumes.
result All smooth cubic threefolds admit Kähler-Einstein metrics, and a precise list of singular KE ones is provided.
We study the variety of Poisson structures and compute Poisson cohomology for two families of Fano threefolds - smooth cubic threefolds and the del Pezzo quintic threefold. Along the way we reobtain by a different method earlier results of Loray, Pereira and Touzet in the special case we are considering.
Computes the monodromy of cubic surfaces branching over smooth cubic curves.
problem Understanding the monodromy of cubic surfaces.
method Computational approach using the relationship between inflection points and lines on cubic surfaces.
result The monodromy map is surjective onto the centralizer of the image of a generator of the deck group.
Planar pure braids form a group that acts on a CAT(0) cubical complex.
problem Characterizing the structure of planar pure braids as a group.
method Demonstrated that the planar pure braid group is a diagram group.
result The planar pure braid group Γn is a diagram group. Extends results on Futaki invariant for Kaehler metrics on algebraic manifolds.
problem Existence of Kaehler metrics with constant scalar curvature.
method Revisits and extends classical and recent results on Futaki invariant.
result Extensions and applications of Futaki invariant in Kaehler geometry.
Develops a new non-abelian framework for Riemann surfaces and differential equations.
problem Analyzing second-order differential equations on Riemann surfaces.
method Gauge-theoretic framework and non-abelian approach.
result Extends Dedekind's Schwarzian approach to generic one-parameter families of curves of genus g.
The paper classifies diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.
problem Classifying diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.
method Using differential-geometric gluing method and classifications of simply-connected 6-manifolds.
result Any two doubling Calabi-Yau threefolds with Picard number two are not diffeomorphic to each other when the underlying Fano threefolds are distinct families.
Optimizes bounds for threefold singularity volumes.
problem Bounding local volumes of threefold singularities.
method Analyzes Gorenstein canonical non-hypersurface threefold singularities.
result Establishes optimal upper bound for local volumes.
Cubic predicts stock market indices by fusing stock latent embeddings and converting to binary classification.
problem Challenges in predicting stock market indices due to isolated time series treatment and simple regression.
method Fusion of stock latent embeddings, binary encoding classification, and confidence-guided prediction.
result Cubic outperforms state-of-the-art baselines in stock index prediction tasks.
Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.
problem Understanding non-Gorenstein involutions on Calabi-Yau threefolds.
method Classification of Calabi-Yau threefolds with specific properties.
result Classification of Calabi-Yau threefolds with Picard rank one and non-Gorenstein involutions.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
K-stability proven for a specific type of Fano threefold.
problem Proving K-stability of Fano threefolds.
method Analyzing double covers of blow-ups with specific branch divisors.
result Proven K-stability of Fano threefolds of rank 2 and degree 14.
Study proves most Fano threefolds are Kähler-Einstein.
problem Existence of Kähler-Einstein metrics on smooth Fano threefolds.
method Investigated smooth Fano threefolds of Picard rank one and anticanonical degree 22 with C∗ action. result Proved existence of Kähler-Einstein metrics on most such threefolds, except possibly two cases.
Study noncommutative deformations of Calabi-Yau threefolds.
problem Understanding the geometry of Calabi-Yau threefolds under noncommutative deformations.
method Analyzing the influence of Poisson structures on quantum moduli spaces.
result The choice of Poisson structure significantly affects the geometry of quantum moduli spaces.
The study examines non-Kähler threefolds with specific metrics and finds they are quasi-bundles over surfaces.
problem Understanding non-Kähler threefolds with algebraic dimension two.
method Examining compact complex non-Kähler threefolds with locally conformally Kähler metrics and proving they are quasi-bundles over projective surfaces under certain assumptions.
result They are blown-up quasi-bundles over a projective surface.
Study properties of algebraic threefolds with specific singularities.
problem Characterize and classify algebraic threefolds with certain singularities.
method Use topological invariants, rational homology, Poincaré duality, and Lie algebras.
result Relate topological invariants to Lie algebras and representations.
New research confirms Kähler-Einstein metrics for all Fano threefolds of degree 22.
problem Identifying Fano threefolds of degree 22 with Kähler-Einstein metrics.
method Analyzing the structure and properties of Fano threefolds of Picard rank one with anti-canonical degree 22.
result All remaining Fano threefolds of degree 22 admit Kähler-Einstein metrics.
The authors give a complete classification of projective threefolds admitting a holomorphic conformal structure. A Corollary is the complete list of projective threefolds, whose tangent bundle is a symmetric square.
Constructs Lagrangians in Calabi-Yau threefolds using tropical curves.
problem Constructing Lagrangian structures in Calabi-Yau threefolds.
method Tropical curves and toric degeneration techniques.
result Constructs multiple Lagrangian rational homology spheres with specific weights.
We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact Kähler threefolds. The equation could be thought of as a generalization of the Hitchin equation on Riemann surfaces to Kähler threefolds. In the appendix of arXiv:0805.2192,…
Holomorphic forms found on 3D spaces without zeros.
problem Existence of holomorphic one-forms without zeros on threefolds.
method Classification of real 6-manifolds fibering over the circle.
result Proves a conjecture of Kotschick in dimension three.
The paper bounds Chern numbers of threefolds, generalizing previous work.
problem Bounding Chern numbers of threefolds, especially those with negative Kodaira dimension.
method Analyzing smooth Mori fibre spaces and applying topological bounds.
result Chern numbers of threefolds are bounded by underlying topological manifolds.
We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds …
Study on balanced Hermitian threefolds with parallel Bismut torsion.
problem Characterizing compact, balanced BTP threefolds.
method Detailed description of all compact, balanced BTP threefolds.
result Characterization of all compact, balanced BTP threefolds.
Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.
problem Understanding non-Kähler complex threefolds and conifold transitions.
method Review of basics, description of topological features, survey of recent developments.
result Survey of recent developments on the geometrization of conifold transitions.
We investigate geometric invariants of the one parameter family of Mukai threefolds that admit C∗ action. In particular we find the invariant divisors in the anticanonical system, and thus establish a bound on the log canonical thresholds. Furthermore we find an explicit description of such threefolds in t…
Kähler-Ricci flow shows type II singularity on Fano threefolds.
problem Understanding the behavior of Kähler-Ricci flow on Fano threefolds.
method Analyzing the Kähler-Ricci flow on Fano threefolds from a specific family.
result Kähler-Ricci flow develops type II singularity on Fano threefolds from the specified family.
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. Study shows Futaki invariant vanishes on most Fano threefolds.
problem Analyzing the Futaki invariant on K-polystable Fano threefolds.
method Examining the Futaki invariant as a map from the Kähler cone to the dual of the Lie algebra of the reduced automorphism group.
result Futaki invariant vanishes identically on most Fano threefolds.
Found a stable 3D shape with specific properties.
problem Finding K-stable Fano threefolds.
method Analyzing specific Fano threefolds with given properties.
result Identified a K-stable Fano threefold with Picard rank 3 and anti-canonical degree 28.
Complex manifolds can only map to curves, restricting Clemens threefolds and S6.
problem Restricting the mapping properties of complex manifolds and threefolds.
method Analyzing the properties of complex manifolds and their mappings.
result No holomorphic mapping from a specific class of threefolds onto a complex space.
Constructs models for Fano threefolds using Lagrangian torus fibrations.
problem Models for Fano threefolds using Lagrangian torus fibrations.
method Toric degeneration for affine manifolds with singularities, correspondence between polytopes and Fano manifolds.
result Total space of each fibration is homeomorphic to the expected Fano threefold and numerical invariants coincide.
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.
Study corrects previous computation of Lagrangian topology in Calabi-Yau threefold.
problem Correcting a previous computation of the topology of a real Lagrangian in Schoen's Calabi-Yau threefold.
method Used two methods to compute mod 2 cohomology: Mayer-Vietoris calculation and an exact sequence.
result Agreement between the two methods confirms the topology of the real Lagrangian.
We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP…
Study on Calabi-Yau threefolds' diffeomorphism classes.
problem Investigating Calabi-Yau threefolds' classification.
method Focus on embedded Calabi-Yau threefolds in toric Fano manifolds.
result Curious remark on mirror symmetry.
We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.
We construct balanced metrics on the family of non-Kähler Calabi-Yau threefolds that are obtained by smoothing after contracting (−1,−1)-rational curves on Kähler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to connected sum of k≥2 copies of $S^3\time…
Algorithm computes eigenvalues and eigenforms on Calabi-Yau threefolds.
problem Computing spectra of Laplace-de Rham operator on Calabi-Yau manifolds.
method Numerical algorithm using approximate Calabi-Yau metric.
result Computed eigenvalues and eigenforms for the Fermat quintic threefold.
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 K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.
problem Understanding the boundary of K-moduli of prime Fano threefolds of genus twelve.
method Developed a modular relation between Fano threefolds and their anticanonical K3 surfaces, proving forgetful morphism is an open immersion.
result Proved the boundary of K-moduli of V22 is purely divisorial and consists of four irreducible components. Study special Lagrangian sections in Calabi-Yau threefolds, showing stability conditions imply isomorphism to special Lagrangians.
problem Understanding stability conditions on Fukaya-Seidel categories of Calabi-Yau threefolds.
method Analyzing sections of special Lagrangian fibrations, constructing Bridgeland stability conditions, and relating to deformed Hermitian Yang-Mills connections.
result Semistability of L[2] implies isomorphism to special Lagrangian sections. Constructs special Lagrangian 3-spheres in non-Kähler compact threefolds.
problem Understanding transitions between Kähler and non-Kähler geometries.
method Analyzes topological transitions of Calabi-Yau threefolds to construct special Lagrangian cycles.
result Special Lagrangian 3-spheres emerge from non-Kähler geometries, exchanging holomorphic 2-cycles for 3-cycles.
In their papers published in 1993 and 1994, by expressing certain physical quantity in two distinct ways, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable equivalence between Ray-Singer analytic torsion and elliptic instanton numbers for Calabi-Yau threefolds. After their discovery, in a paper published in 2008, …
In this paper, we give an expression and some estimates of the curvature tensor of the Hodge metric over the moduli space of a polarized Calabi-Yau threefold. The symmetricity of the Yukawa coupling is also studied. In the last section of this paper, an extra restriction of the limiting Hodge structure for the degenera…
After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corol…
The paper connects algebraic geometry to threefold theories.
problem Enumerative geometry of nested Hilbert schemes.
method Study of threefold theories including Donaldson-Thomas, Vafa-Witten, and Seiberg-Witten.
result Connections between algebraic geometry and threefold theories.
Study on surfaces in flag threefold with constraints on twistor fibers.
problem Understanding the arrangement and existence of twistor fibers in surfaces of specific bidegree.
method Analyzing surfaces of bidegree (1,d) in the flag threefold, proving existence and non-existence of twistor fibers.
result Existence and non-existence of surfaces containing specific numbers of twistor fibers, with improved results for d=2 and d=3.