Study shows how volume forms on degenerating varieties converge to a non-Archimedean measure.
problem Convergence of volume forms on degenerating log-Calabi-Yau varieties.
method Extending a result of Boucksom and Jonsson, the study uses a hybrid space filled with Berkovich analytification.
result Measures induced by meromorphic volume forms on fibers converge to a measure on the Berkovich analytification as the puncture is approached.
Study on Kähler-Einstein metrics on log Calabi-Yau families, proving local triviality and discrediting a metric conjecture.
problem Understanding Kähler-Einstein metrics on log Calabi-Yau families.
method Investigation of relative Ricci-flat Kähler metrics, focusing on Kodaira dimension zero.
result Disproof of a folklore conjecture about the semipositivity of relative flat metrics.
We present a new method to solve certain ∂ˉ-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a ∂ˉ-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness…
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…
Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.
problem K-stability of Calabi-Yau fibrations over curves.
method Adiabatic uniform K-stability and log-twisted K-stability of base curves.
result Uniform K-stability of Calabi-Yau fibrations if and only if base curves are K-stable.
Study geometric properties of log Calabi-Yau manifolds, focusing on Fano manifolds with smooth or two proportional components.
problem Geometric properties of log Calabi-Yau manifolds in two specific cases.
method Analysis of various geometric properties, focusing on Bochner principle, local triviality, polystability, and compactifiability of universal cover.
result The universal cover of X∖D is a Calabi-Yau manifold of infinite topological type when D has two components. Log minimality proven for weak K-moduli compactifications of Calabi-Yau varieties.
problem Constructing weak K-moduli compactifications of Calabi-Yau varieties.
method Revisiting classical problem, proving log minimality of normalizations under conditions.
result Log minimality of weak K-moduli compactifications under certain conditions.
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.
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.
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.
New polystability theory connects Calabi-Yau varieties to gravitational instantons.
problem Understanding the structure of Calabi-Yau manifolds and their metrics.
method Introducing a new concept of poly-stability and relating it to gravitational instantons.
result Polystability is equivalent to the existence of certain gravitational instantons.
Researchers create special fibrations on Calabi-Yau hypersurfaces.
problem Understanding special Lagrangian fibrations on Calabi-Yau hypersurfaces.
method Produced special Lagrangian T^n-fibrations on Calabi-Yau hypersurfaces in the Fermat family.
result Demonstrated fibrations on generic regions of Calabi-Yau hypersurfaces in the large complex structure limit.
Study solves Monge-Ampère equation for complete Calabi-Yau metrics.
problem Existence of complete Calabi-Yau metrics on log Calabi-Yau pairs.
method Free boundary Monge-Ampère equation, Legendre duality, existence proof in smooth and Hölder spaces.
result Existence and strict convexity of solutions in smooth and Hölder spaces.
The paper classifies certain singular projective varieties with specific properties.
problem Classifying projective klt pairs with nef anti-log canonical divisors.
method Establishes a structure theorem using locally trivial rationally connected fibrations.
result Projective klt pairs can be decomposed into rationally connected and Calabi-Yau varieties.
Solves non-Archimedean Calabi-Yau equation on complex log pairs.
problem Non-Archimedean Monge-Ampère equation on Berkovich analytification.
method Solves complex Monge-Ampère equation, then adapts to non-Archimedean setting.
result Non-Archimedean analog of Ricci-flat metric potentials on complex affine varieties.
Constructs Calabi-Yau metrics on 3-folds with properties similar to Taub-NUT and Ooguri-Vafa.
problem Creating metrics on Calabi-Yau 3-folds with specific properties.
method Constructs families of Calabi-Yau metrics on $\C^3$ and Calabi-Yau 3-folds with properties similar to Taub-NUT and Ooguri-Vafa.
result Demonstrates construction of Calabi-Yau metrics with properties similar to Taub-NUT and Ooguri-Vafa.
Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.
problem Finding metrics for toric Kähler cones with conical singularities.
method Parametrized family of Calabi-Yau cone metrics with conical singularities.
result Any toric Calabi-Yau cone metric with conical singularities belongs to this optimal family.
Researchers prove birational invariance of BCOV invariant using motivic integration.
problem Proving the birational invariance of BCOV invariant for Calabi-Yau manifolds and varieties.
method Motivic integration theory applied to Calabi-Yau varieties with Kawamata log terminal singularities.
result Birational Calabi-Yau manifolds have the same BCOV invariant.
We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi-Yau manifolds. For example we prove that given any real-analytic one parameter family of Riemannian metrics gt on a 3-dimensional manifold Y with volume form independent of t and with a real-analytic family of nowhere vanishin…
In this paper, by applying Greene-Shapere-Vafa-Yau semi-flat metric, we give a new proof of closed formula of Weil-Petersson metric on moduli space of Calabi-Yau varieties.
New one-parameter families of SU(2)2-invariant instantons found on Calabi-Yau 3-folds.
problem Behavior of Calabi-Yau instantons and monopoles with SU(2)2-symmetry. method Gauge theory on asymptotically conical Calabi-Yau 3-folds with SU(2)2 co-homogeneity one action. result New one-parameter families of invariant instantons found.
This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certai…
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.
We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of L2, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invarian…
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…
We prove that the mirror map is trivial for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author. In other words, the natural coordinate in a canonical Calabi-Yau family is a canonical coordinate in the sense of Hodge theory. This implies that the higher weight periods direct…
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.
Given a polarized family of varieties over the unit disc, smooth except over the origin and with smooth fibers Calabi-Yau, we show that the origin lies at finite Weil-Petersson distance if and only if after a finite base change the family is birational to one with central fiber a Calabi-Yau variety with at worst canoni…
New metrics on C^3 defy uniqueness, differing even at infinity.
problem Non-uniqueness of Calabi-Yau metrics with maximal volume growth.
method Constructed a family of inequivalent metrics on C^3.
result First example of non-uniqueness in Calabi-Yau metrics asymptotic to a fixed cone.
Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they …
Study on Calabi-Yau metrics collapsing and complex structure degenerations.
problem Understanding the behavior of Calabi-Yau metrics on degenerating families of manifolds.
method Gluing and singular perturbation techniques, construction of Kähler metrics with torus symmetry.
result Explicit and precise relationships between metric collapsing and complex structure degenerations established in all dimensions.
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. Method finds approximate Ricci-flat metrics on Calabi-Yau manifolds.
problem Finding analytic Kähler potentials for Calabi-Yau manifolds.
method Numerically calculating Ricci-flat Kähler potentials via machine learning and fitting to Donaldson's Ansatz.
result Simple analytic expressions for approximately Ricci-flat Kähler potentials are found, including explicit dependence on complex structure parameter.
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…
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…
The study finds infinitely many special Lagrangian submanifolds in certain complex manifolds.
problem Existence of special Lagrangian submanifolds in log Calabi-Yau manifolds.
method Analyzes the existence of special Lagrangian submanifolds in log Calabi-Yau manifolds using Ricci-flat Kähler metrics.
result Proves existence of infinitely many disjoint special Lagrangian submanifolds in certain log Calabi-Yau manifolds.
We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.
We give a simple proof of the local version of a result of R. Bryant, stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We refine the theorem proving that a certain class of one-parameter families of metrics on a 3-torus can be…
Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.
problem Understanding the moduli spaces of quasimaps and Calabi-Yau fibrations.
method Constructing a projective K-moduli space of quasimaps and investigating relationships with Calabi-Yau fibrations.
result Entire quasi-projectivity and ampleness of the CM line bundle on the normalization of the K-moduli space of Calabi-Yau fibrations.
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…
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.
Estimates for complex Monge-Ampère equations lead to insights on moduli spaces and singular metrics.
problem Uniform estimates for complex Monge-Ampère equations on Kähler manifolds.
method Refined techniques to control degenerate equations and analyze families of singular Kähler-Einstein metrics.
result Uniform integrability properties and insights into moduli spaces of stable varieties.
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 note gives a simple formula for the unique asymptotically conical Calabi-Yau metrics on the canonical bundle of a flag variety known to exist by the work of R. Goto and others. This is done by generalizing the well known Calabi Ansatz to general Kähler classes. We give some examples of explicit families, in partic…
New linking numbers link complex cycles to Calabi-Yau 3-folds.
problem Understanding complex analytic cycles and their Massey products.
method Relating ABC Massey products to holomorphic linking numbers.
result Constructed a family of Calabi-Yau 3-folds with non-vanishing Massey products.
Constructs unique bases for CY varieties over valued fields.
problem Finding unique bases for CY varieties over valued fields.
method Uses techniques from higher rank degenerations in K-stability.
result Induces canonical functions on skeletons and agrees with tropicalizations of theta functions.
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.
Elliptic surfaces have unique Lefschetz pencils and Calabi-Yau diffeomorphisms.
problem Understanding symplectic structures on elliptic surfaces.
method Analyzing R. Inanc Baykur's family of symplectic manifolds.
result Elliptic surfaces admit Lefschetz pencils and diffeomorphic to K3 surfaces.