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.
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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.
We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds wi…
We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein metric…
Point Singularities and Local Third Chern Classes for Rank-Two Torsion-free Sheaves on Threefoldsmath.DG
Study local third Chern class for point singularities on threefolds.
problem Understanding gauge theory singularity contributions on threefolds.
method Local algebraic data and deformation invariance, K-theoretic interpretation.
result Local third Chern class can be computed from family data and is deformation invariant.
Degenerations of rank-two bundles on threefolds lead to isolated point singularities, with rigidity and bubbling properties.
problem Degenerations of rank-two vector bundles on complex threefolds to a rank-two torsion-free sheaf with an isolated point singularity.
method Proving a rigidity identity and using it to obtain smoothability obstructions and construct local smoothings.
result Smoothability obstructions and local smoothings are obtained, with a rigidity identity linking algebraic bubbling multiplicity and Ext-length.
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 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…
A removal singularity theorem of the Donaldson-Thomas instantons on compact Kähler threefoldsmath.DG
We consider a perturbed Hermitian-Einstein equation, which we call the Donaldson-Thomas equation, on compact Kähler threefolds. In arXiv:0805.2195, we analysed some analytic properties of solutions to the equation, in particular, we proved that a sequence of solutions to the Donaldson-Thomas equation has a subsequence …
Log Conifold Transitionsmath.AG
New non-Kähler 3-folds constructed via log conifold transitions.
problem Constructing new non-Kähler 3-folds from Fano threefold pairs.
method Defining log conifold transitions and studying their deformation theory.
result Local smoothings of nodes can be lifted to global first-order deformations.
The study proves a key inequality for specific types of three-dimensional spaces.
problem Establishing a mathematical inequality for a specific class of three-dimensional spaces.
method Developed the orbifold version of the Bogomolov-Gieseker inequality for stable Q-sheaves on log terminal Kähler threefolds.
result Proved the Bogomolov-Gieseker inequality for log terminal Kähler threefolds.
We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first…
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…
Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that…
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 is purely divisorial and consists of four irreducible components.
Extends results on smoothability of singular Fano and Calabi-Yau varieties.
problem Smoothability of singular Fano and Calabi-Yau varieties under terminal singularities.
method Generalizes deformation theory results for Calabi-Yau and Fano threefolds to higher dimensions, using higher Du Bois and rational singularities.
result Identifies a class of singularities for which smoothing results hold, including generalized Fano and Calabi-Yau varieties.
CR singularities in 3-manifolds can be cancelled by an isotopy supported in an arbitrarily small neighborhood of a Seifert surface.
problem Canceling CR singularities in 3-manifolds
method Isotopy supported in an arbitrarily small neighborhood of a Seifert surface
result CR singularities can be cancelled
Lecture notes on conifold transitions between Calabi-Yau manifolds.
problem Understanding conifold transitions in complex threefolds.
method Differential geometric approach, focusing on explicit calculations and examples.
result Necessary condition for smoothings of nodal Calabi-Yau threefolds proved.
After establishing suitable notions of stability and Chern classes for singular pairs, we use Kähler-Einstein metrics with conical and cuspidal singularities to prove the slope semistability of orbifold tangent sheaves of minimal log-canonical pairs of log general type. We then proceed to prove the Miyaoka-Yau inequali…
The paper constructs flat metrics on orbifolds and resolutions.
problem Finding flat metrics on orbifolds and their resolutions.
method Gluing construction and analysis of singularities.
result All crepant resolutions of non-Kähler Calabi-Yau orbifolds with Chern-Ricci flat balanced metrics admit such metrics.
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.
For a one parameter family of Calabi-Yau threefolds, Green, Griffiths and Kerr have expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV invariants, s…
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 paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.
problem Deformation theory of Calabi-Yau varieties with log canonical singularities.
method Study of higher Du Bois and rational singularities, focusing on 0-liminal singularities.
result Existence of first order smoothings for isolated 0-liminal hypersurface singularities.
Neural networks approximate Calabi-Yau metrics and curvature.
problem Finding Ricci-flat metrics for Calabi-Yau manifolds.
method Use neural networks to approximate metrics within a Kähler class.
result Neural networks can approximate topological characteristics of Calabi-Yau manifolds.
Novel threefold partitioning of -spinor space on cone links.
problem Understanding equivariant Riemann-Roch defects in complex spaces with conic singularities.
method Investigates supertraces over local cohomology groups and spectral asymmetry.
result Novel complex equivariant and invariants defined.
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.
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.
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.
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.
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,…
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.
We investigate geometric invariants of the one parameter family of Mukai threefolds that admit 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…
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 .
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.
We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real 6-manifold fibres smoothly over the circle, and we give a complete classification of all threefolds with that property. Our results prove a conjecture of Kotschick in dimension three.
We study the problem of existence of Kähler--Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree that admit a faithful action of the multiplicative group . We prove that, except possibly two explicitly described cases, all such smooth Fano threefolds are Kähler--…
Paper rigorously defines Feynman graph integrals on Kähler manifolds.
problem Establishing convergence of Feynman graph integrals on Kähler manifolds.
method Using Getzler's rescaling technique, graph integrands are extended to forms with divisorial-type singularities in the compactification of configuration spaces.
result Feynman graph integrals are rigorously defined as Cauchy principal value integrals.
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.
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 -rational curves on Kähler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to connected sum of copies of $S^3\time…
We initiate the study of a new nonlinear parabolic equation on a Riemann surface. The evolution equation arises as a reduction of the Anomaly flow on a fibration. We obtain a criterion for long-time existence for this flow, and give a range of initial data where a singularity forms in finite time, as well as a range of…
Study shows stability of tangent bundle through conifold transitions.
problem Stability of tangent bundle through conifold transitions.
method Hermitian-Yang-Mills metric and conformally balanced metrics.
result Tangent bundle admits a Hermitian-Yang-Mills metric near vanishing cycles of .