Study deformations of Calabi-Yau foliations using Kuranishi spaces.
problem Deforming Calabi-Yau foliations and understanding their properties.
method Analysis of three types of deformations (unfoldings, holomorphic, transversally holomorphic) using Kuranishi spaces.
result Smoothness of Kf and product structure of Kh. We study Riemannian foliations whose transverse Levi-Civita connection ∇ has special holonomy. In particular, we focus on the case where Hol(∇) is contained either in SU(n) or in Sp(n). We prove a Weitzenbock formula involving complex basic forms on Kähler foliations and we apply this formula for pointing…
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.
New Bochner technique for foliations with non-negative Ricci curvature.
problem Analyzing foliations with non-negative transverse Ricci curvature.
method Generalizing Bochner technique to foliations with non-negative transverse Ricci curvature.
result Obtained a new vanishing theorem for basic cohomology.
We prove a priori estimates for a class of transverse fully nonlinear equations on Sasakian manifolds and give some geometric applications such as the transversion Calabi-Yau theorem for transverse balanced and (strongly) Gauduchon metrics. We also explain that similar results hold on compact oriented, taut, transverse…
Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension d admits, away from a closed analytic subset of positive codimension, …
Researchers found a Calabi-Yau structure and constructed a Bargmann type transformation on the Cayley projective plane.
problem Existence of Calabi-Yau structure and construction of Bargmann type transformation.
method Pairing of polarizations, natural Lagrangian foliation, and Kähler structure.
result Quantization of geodesic flow through elliptic Fourier integral operators.
We construct the moduli space of contact instantons, an analogue of Yang-Mills instantons defined for contact metric 5-manifolds and initiate the study of their structure. In the K-contact case we give sufficient conditions for smoothness of the moduli space away from reducible connections and show the dimension is…
We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special…
For projective varieties with definite first Chern class we have one type of canonical metric which is called Kähler-Einstein metric. But for varieties with an intermidiate Kodaira dimension we can have several different types of canonical metrics. In this paper we introduce a new notion of canonical metric for varieti…
Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.
problem Creating complete Calabi-Yau metrics on non-compact manifolds.
method Extending Székelyhidi's work, constructing metrics with varying complex structures and possible singularities.
result Produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.
New examples of Calabi-Yau 3-folds with unique properties.
problem Finding new Calabi-Yau 3-folds with specific properties.
method Constructing complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones with orbifold singularities.
result Examples of Calabi-Yau 3-folds with maximal volume growth and orbifold singularities.
Uniform estimates for Calabi-Yau degenerations proved.
problem Calabi-Yau degenerations of polarised algebraic manifolds.
method Uniform Skoda and L∞-estimates for Kähler potentials. result Uniform Skoda type estimate and L∞-estimate for Calabi-Yau Kähler potentials proved. 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.
Polyhomogeneous expansions for Calabi-Yau metrics near singularities.
problem Analyzing metrics near conical singularities of Calabi-Yau conifolds.
method Weighted Melrose-type blow-ups, gluing, and solving complex Monge-Ampère equations.
result Polyhomogeneous expansions of smooth Calabi-Yau metrics on resolutions and smoothings.
We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form M×S3/Γ, where M is a smooth manifold and S3/Γ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on M whose quiver bundles are based on the affine ADE Dynkin diagram associ…
New Calabi-Yau metrics constructed with detailed geometry at infinity.
problem Constructing complete Calabi-Yau metrics with specific properties.
method Weighted blow-up and Hölder spaces for Laplacian analysis.
result Examples of Calabi-Yau metrics with conical singularities and non-uniqueness of tangent cones.
We study Calabi-Yau 3-folds M_0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M_0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M_0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on…
In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…
Solves Calabi-Yau equation on complex manifolds with non-positive astheno-Ricci curvature.
problem Solving the form type Calabi-Yau equation on complex manifolds.
method Defined astheno-Ricci curvature and proved existence of solution under non-positive curvature condition.
result Existence of solution for Calabi-Yau equation with non-positive astheno-Ricci curvature.
In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along T4. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group Z. In this paper, we also produce the first examples of simply con…
New Calabi-Yau metrics converge polynomially to Calabi model space.
problem Finding complete Calabi-Yau metrics with polynomial convergence rate.
method Defined new metrics on Calabi-Yau complements with ample normal bundles.
result Uniqueness of these metrics within a cohomology class.
Sharp diameter bounds for Calabi-Yau degenerations proved.
problem Bounding the diameter of Calabi-Yau metrics during degeneration.
method Sharp upper and lower bounds derived for Ricci-flat Kahler metrics.
result Conjecture confirmed by obtaining precise diameter bounds.
Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.
problem Behavior of warped-QAC Calabi-Yau metrics on affine quadrics.
method Gluing construction for collapsing warped-QAC Calabi-Yau manifolds.
result Verification of Yang Li's conjecture on warped QAC Calabi-Yau metrics.
This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.
Study on Calabi-Yau metrics and their degenerations.
problem Understanding degenerations of Ricci-flat Kahler metrics on Calabi-Yau manifolds.
method Survey of recent results on questions about degenerations.
result Survey of recent results on degenerations of metrics.
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.
Constructs special Lagrangian submanifolds in Calabi-Yau 3-folds.
problem Constructing special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds.
method Constructs special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds fibered by K3 surfaces.
result Special Lagrangian submanifolds shrink to 1-dimensional graphs in the base as the 3-folds collapse.
We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki-Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki-Einstein structure, whi…
Study on collapsing Calabi-Yau manifolds and their metrics.
problem Understanding degenerations of Calabi-Yau manifolds with Ricci-flat Kahler metrics.
method Survey of recent developments, focusing on volume collapsing metrics.
result New insights into the behavior of Calabi-Yau manifolds under volume collapse.
A generalized Calabi-Yau structure is a geometrical structure on a manifold which generalizes both the concept of the Calabi-Yau structure and that of the symplectic one. In view of a result of Lin and Tolman in generalized complex cases, we introduce in this paper the notion of a generalized moment map for a compact L…
Complete Calabi-Yau metrics made on special 3D spaces.
problem Creating complete Calabi-Yau metrics on complex 3D spaces.
method Used gluing construction and perturbation argument.
result Produced complete Calabi-Yau metrics with unbounded curvature.
Complete Calabi-Yau metrics on abelian fibrations over complex space.
problem Constructing complete Calabi-Yau metrics on noncompact abelian fibrations.
method Using abelian fibrations over C, we construct complete Calabi-Yau metrics and provide compactification. result We provide a compactification for the abelian fibration X such that the compactified variety has a negative canonical bundle. Proves certain Calabi-Yau varieties are projective.
problem Compact Calabi-Yau varieties with isolated singularities are not always projective.
method Analysis and Ohsawa's degenerate spectral sequence in higher dimensions.
result Proves compact Calabi-Yau varieties with certain isolated singularities are projective.
Study shows Calabi-Yau metrics converge to a specific form under certain conditions.
problem Degeneration of Calabi-Yau metrics and their limits.
method Optimal transport problem and minimisation of Kontorovich functional.
result Limit data of Calabi-Yau metrics can be encoded into a unique minimiser.
Study the limit of Calabi-Yau metrics with degenerate skeletons.
problem Understanding the behavior of Calabi-Yau metrics with degenerate skeletons.
method Using polarised degenerations and optimal transport problems.
result Describe the limiting behaviour of the Calabi-Yau potential.
Study on Calabi-Yau locally conformally Kähler manifolds proving they are Vaisman.
problem Characterizing Calabi-Yau locally conformally Kähler manifolds.
method Analyzing properties of Calabi-Yau locally conformally Kähler manifolds, proving they are Vaisman.
result Calabi-Yau locally conformally Kähler manifolds are Vaisman.
Estimates symplectic Calabi-Yau equation using Cheng-Yau method.
problem Estimating solutions to the Calabi-Yau equation on symplectic 4-manifolds.
method Applies a Cheng-Yau type estimate in the symplectic setting.
result Proves an a priori estimate for the symplectic Calabi-Yau equation.
Introduces new limit spaces for degenerating Calabi-Yau families.
problem Understanding degenerating Calabi-Yau families and their limit structures.
method Introduces galaxy spaces as dense subspace of infinite open Calabi-Yau varieties.
result Galaxy spaces are projective limits of toroidal compactifications.
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.
We generalize Calabi-Yau 3-folds from the special Lagrangian perspective. More precisely, we study SU(3)-structures which admit as "nice" a local special Lagrangian geometry as the flat C3 or a Calabi-Yau structure does. The underlying almost complex structure may not be integrable. Such SU(3)-structures ar…
We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some applications are included, such as construction of new examples of Calabi--Yau 3-folds …
Study of large mass limits of G2 and Calabi-Yau monopoles on specific manifolds.
problem Understanding the behavior of monopoles in the large mass limit on G2 and Calabi-Yau manifolds.
method Developed a structure theory for the limit of SU(2) G2-monopoles and Calabi-Yau monopoles, extracting singular abelian G2-monopoles with Dirac singularities. result Proved an energy identity for monopole bubbles in the large mass limit.
We develop some consequences of the connection between Calabi-Yau structures and torsion-free G2 structures on compact and asymptotically cylindrical six- and seven-dimensional manifolds. Firstly, we improve the known proof that matching asymptotically cylindrical Calabi-Yau threefolds can be glued. Secondly, we giv…
New Calabi-Yau metrics found on complex symmetric spaces.
problem Finding Calabi-Yau metrics on complex symmetric spaces.
method Complete Calabi-Yau metrics with prescribed horospherical singular tangent cone.
result First examples of Calabi-Yau smoothings of singular tangent cones.
New examples of Calabi-Yau metrics on cones with irregular smooth links.
problem Finding new Calabi-Yau metrics on cones with irregular smooth links.
method Explicit computation of Reeb field and Minkowski decompositions of toric Calabi-Yau cones.
result Examples of complete Calabi-Yau metrics on cones with irregular smooth links.
Machine learning uncovers hidden patterns in Calabi-Yau hypersurfaces.
problem Identifying and clustering Calabi-Yau hypersurfaces from weighted-P4s.
method Supervised and unsupervised machine learning techniques.
result High accuracy in predicting topological parameters and identifying hypersurfaces.
We use the gluing method to give a refined description of the collapsing Calabi-Yau metrics on Calabi-Yau 3-folds admitting a Lefschetz K3 fibration.