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.
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.
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…
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.
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.
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.
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 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.
Extends Kodaira Spencer to higher-dimensional Calabi-Yau manifolds.
problem No specific problem stated; extension of functional.
method Extension to Calabi-Yau manifolds of arbitrary dimension.
result Extension of the Kodaira Spencer functional.
Constructs non-Kähler Calabi-Yau manifolds with large Betti numbers.
problem Finding non-Kähler Calabi-Yau manifolds with high Betti numbers.
method Smoothing normal crossing varieties to create K3 fibrations over smooth projective varieties.
result Examples of non-Kähler Calabi-Yau manifolds with arbitrarily large 2nd Betti numbers.
In this paper, we study the convergence of Calabi-Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi-Yau manifolds.
We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles…
We show that Calabi-Yau manifolds are emergent from the commutative limit of six-dimensional noncommutative Hermitian U(1) instantons. Therefore we argue that the noncommutative Hermitian U(1) instantons correspond to quantized Calabi-Yau manifolds.
Compactifies Calabi-Yau to weak Fano manifolds.
problem Compactifying Calabi-Yau manifolds to weak Fano manifolds.
method Generalized Tian-Yau construction and asymptotically Calabi metrics.
result Calabi-Yau structure arises from compactification.
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.
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…
Simplified argument for second order estimate in quaternionic Calabi-Yau problem.
problem Second order estimates for quaternionic Calabi-Yau problem on hyperkähler manifolds.
method Simplified argument to derive the estimate.
result Simplified derivation of second order estimate.
In this paper, the relationship between the existence of special lagrangian submanifolds and the collapsing of Calabi-Yau manifolds is studied. First, special lagrangian fibrations are constructed on some regions of bounded curvature and sufficiently collapsed in Ricci-flat Calabi-Yau manifolds. Then, in the opposite d…
In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path c…
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. Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.
problem Solving the Calabi-Yau equation on symplectic manifolds.
method Global deformation of almost complex structures compatible with symplectic form, constructing measurable Lipschitz Kahler metric.
result Existence theorem for solutions to the one-form type Calabi-Yau equation on closed symplectic manifolds.
This paper classifies Calabi-Yau manifolds near cones.
problem Classifying Calabi-Yau manifolds near cones.
method Complete classification of smooth complete Calabi-Yau manifolds asymptotic to a given cone.
result Classification of all smooth complete Calabi-Yau manifolds near a given cone.
Superminimal surfaces in certain Einstein manifolds have a Calabi-Yau property.
problem Characterizing superminimal surfaces in specific Einstein manifolds.
method Utilizing twistor spaces and properties of holomorphic Legendrian curves.
result Superminimal surfaces in self-dual or anti-self-dual Einstein four-manifolds can be uniformly approximated by complete superminimal surfaces.
Study explores Laplacian coflow versions on Calabi-Yau 7-manifolds.
problem Analyzing G2-structures on Calabi-Yau 7-manifolds. method Reduced Ansätze for the Laplacian coflow on various Calabi-Yau 7-manifolds.
result Obtained a modified Kähler-Ricci flow.
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.
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 …
Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.
problem Preserving symplectic forms under deformations on Calabi-Yau manifolds.
method Dynamical stability of symplectic curvature flow.
result Any small symplectic deformation of a Kähler form remains Kähler on a compact Calabi-Yau manifold.
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…
We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous démontrons que les seules variétés projectives de Calabi--Yau qui possèdent des géométrie holomorphes de Cartan sont les variétés abéliennes.)
Holomorphic splitting theorem for Calabi-Yau manifolds with specific properties.
problem Constructing a complete Calabi-Yau metric on a manifold with a specific divisor.
method Solved Monge-Ampère equation on generalized ALG manifolds, used solution to prove holomorphic splitting theorem.
result Proved biholomorphic equivalence of a Calabi-Yau manifold to a product space.
We consider a generalization of Calabi-Yau structures in the context of α-Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional nilmanifolds. Finally we generalize to codimension r.
This is a short expository note about Calabi-Yau manifolds and degenerations of their Ricci-flat metrics.
We study orientability issues of moduli spaces from gauge theories on Calabi-Yau manifolds. Our results generalize and strengthen those for Donaldson-Thomas theory on Calabi-Yau manifolds of dimensions 3 and 4. We also prove a corresponding result in the relative situation which is relevant to the gluing problem in DT …
New octonionic Kähler metrics solve an octonionic Calabi-Yau theorem.
problem Finding metrics on 16D manifolds.
method Introduced octonionic Kähler metrics and solved an octonionic Monge-Ampère equation.
result Solved an octonionic Calabi-Yau theorem.
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.
Proves a conjecture for Calabi-Yau manifolds.
problem Maximal degeneration of Calabi-Yau manifolds.
method Valuative independence condition for section ring.
result Metric SYZ conjecture proven.
Two Calabi-Yau theorems for Kähler manifold degenerations.
problem Understanding degenerations of compact Kähler manifolds.
method Direct proof for big test configurations and broader class of degenerations in non-Archimedean Kähler geometry.
result Established a connection between big cohomology classes and their volumes.
Proves Calabi-Yau theorem for certain nonnegative curvature manifolds.
problem Proving a Calabi-Yau type theorem for specific manifolds.
method Existence result for bounded regions with weakly mean-concave boundary.
result Proves contractibility of certain manifolds with positive scalar curvature.
Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.
problem Analyzing the behavior of G2-structures under Laplacian and Hitchin flows.
method Investigation of Laplacian and Hitchin flows on contact Calabi-Yau 7-manifolds.
result Ancient solutions of the Laplacian flow with finite time Type I singularity and immortal solutions of the Laplacian coflow with infinite time Type IIb singularity.
Study contact instantons on Sasakian 5-manifolds with Calabi-Yau structures.
problem Anti-self-dual contact instantons on Sasakian 5-manifolds with transverse Calabi-Yau structures.
method Singularity data of leaf spaces and computation of moduli spaces.
result Explicit computation of complex dimensions of moduli spaces for 95 orbifold K3 surfaces.
This paper shows hypercommutative algebras on Calabi-Yau manifolds are formal.
problem Formality of hypercommutative algebras on Calabi-Yau manifolds.
method Using purity of mixed Hodge structures.
result The canonical hypercommutative algebra on compact Calabi-Yau manifolds is formal.
Study finite group actions on symplectic Calabi-Yau 4-manifolds with non-zero first Betti number.
problem Determine fixed-point set structure of symplectic Calabi-Yau 4-manifolds with certain group actions.
method Analysis of symplectic surfaces and disjoint embeddings in rational 4-manifolds.
result Symplectic Calabi-Yau 4-manifolds are shown to be T2-bundles under specific group actions. 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.
Study existence of harmonic 1-forms on Calabi-Yau manifolds.
problem Tackles existence of harmonic 1-forms on Calabi-Yau manifolds.
method Uses neural networks to approximate metrics and harmonic 1-forms.
result Suggests existence of harmonic 1-forms on some Calabi-Yau manifolds.
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.
Neural and numerical methods approximate G2-structures on Calabi-Yau manifolds.
problem Approximating G2-structures on Calabi-Yau manifolds.
method Three stages: Ricci-flat metric computation, numerical approximations, and neural architecture training.
result Validated neural architecture for learning G2-structures and their metrics.
We prove several formulas related to Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory of Kähler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds and also a construction of global canonical family of ho…
This paper is a sequel to arXiv:1012.2940. We further investigate the Gromov-Hausdorff convergence of Ricci-flat Kähler metrics under degenerations of Calabi-Yau manifolds. We extend Theorem 1.1 in arXiv:1012.2940 by removing the condition on existence of crepant resolutions for Calabi-Yau varieties.