Study Lie algebras with complex structures, focusing on degenerations and deformations.
problem Understanding the space of Lie algebras with complex structures and their transformations.
method Identifying invariants that remain consistent under degenerations and applying to four-dimensional case.
result Found invariants that help in understanding the behavior of Lie algebras under complex structures.
The paper discovers new ways Riemann surfaces can degenerate.
problem Understanding degeneration of infinite-type Riemann surfaces.
method Constructing a concrete example to prove degeneration phenomena.
result Existence of degenerations in Bers boundary of infinite-type surfaces.
In this paper we study the degeneration of convex real projective structures on bordered surfaces.
An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space PN(C). By means of the foca…
Develops new Poisson structures for moduli spaces.
problem Creating Poisson structures for moduli spaces.
method Introduces quasi Poisson and quasi Hamiltonian structures, novel momentum mappings.
result Bijective correspondence between quasi Poisson and quasi Hamiltonian structures.
Introduces a new geometric structure for statistical manifolds with degenerate metrics.
problem Degenerate metrics in statistical manifolds affect geometric structures and applications.
method Introduces quasi-Codazzi structure for degenerate metrics and coherent tangent bundles.
result Generalizes geometric structures and relations for statistical models with degenerate metrics.
We provide a complete list of two- and three-component Poisson structures of hydrodynamic type with degenerate metric, and study their homogeneous deformations. In the non-degenerate case any such deformation is trivial, that is, can be obtained via Miura transformation. We demonstrate that in the degenerate case this …
Extends arguments to limit structure in Calabi-Yau degenerations.
problem Understanding Gromov-Hausdorff limits in degenerating Calabi-Yau manifolds.
method Reduces conjecture to partial second-order estimate.
result Extends arguments to new settings.
Paper studies invariant distributions of bi-Hamiltonian structures.
problem Integrability of invariant distributions in bi-Hamiltonian structures.
method Description and investigation of invariant distributions.
result All invariant distributions of non-degenerate bi-Hamiltonian structures are described.
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…
We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.
We present new rectification theorems of degenerate quasi-conformal structures that give a meaning to quotients of Riemann surfaces with empty interior "fundamental domains". These techniques are used to define the unique renormalization of polynomials with Cantor set Julia sets.
The abstract discusses transformations on statistical and semi-Weyl manifolds with torsion.
problem Invariance and structure preservation under conformal-projective transformations.
method Proof of invariance and preservation of structures under conformal-projective transformations.
result Semi-Weyl and statistical structures with torsion are invariant under conformal-projective transformations.
The paper classifies degenerate almost complex surfaces in a nearly Kähler space.
problem Classifying degenerate almost complex surfaces in nearly Kähler spaces.
method Investigates two distinct cases based on the preservation of the tangent bundle under the almost product structure.
result Complete and explicit classification of degenerate almost complex surfaces in nearly Kähler spaces.
The paper studies connections in superintegrable systems, revealing geometric insights.
problem Understanding non- and semi-degenerate superintegrable systems.
method Analyzes two torsion-free connections associated with superintegrable systems.
result Semi-degenerate secondary structure tensor is the Ricci curvature of a natural torsion-free connection.
Study of Calabi-Yau manifold degenerations near complex structure limits.
problem Understanding polarized degenerations of Calabi-Yau manifolds.
method Improvement of metric convergence results on generic regions.
result Metric convergence for collapsing Ricci-flat Kähler metrics on generic regions.
The paper adapts metrics to anti-de Sitter structures, characterizing their degeneracies.
problem Characterizing degeneracies of metrics on anti-de Sitter structures.
method Adapting Hitchin component metrics to anti-de Sitter structures.
result Characterized degeneracies of the pressure metric and showed the Loftin metric is nowhere degenerate.
Defines log Floer cohomology for symplectic surfaces with a degenerate part.
problem Extending Floer cohomology to degenerate symplectic structures.
method Definition of log Floer cohomology for oriented log symplectic surfaces.
result Log Floer cohomology is invariant under isotopies and isomorphic to log de Rham cohomology for a single Lagrangian.
Let M be a hyperkaehler manifold, and η a closed, positive (1,1)-form which is degenerate everywhere on M. We associate to η a family of complex structures on M, called a degenerate twistor family, and parametrized by a complex line. When η is a pullback of a Kaehler form under a Lagrangian fibration L, a…
Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle
problem Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle
method Classification using rigidity of structures
result Complete classification in the homogeneous setting
For 3-dimensional hyperbolic cone structures with cone angles θ, local rigidity is known for 0≤θ≤2π, but global rigidity is known only for 0≤θ≤π. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most π do not degenerate in defo…
A quadratic line complex is a three-parameter family of lines in projective space P^3 specified by a single quadratic relation in the Plucker coordinates. Fixing a point p in P^3 and taking all lines of the complex passing through p we obtain a quadratic cone with vertex at p. This family of cones supplies P^3 with a c…
The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.
problem Degenerate complex structures in transport twistor spaces.
method Holomorphic blow-down structure maps to resolve degeneracy and gain insight into complex geometry.
result Global and local β-maps for various metrics, proving a Newlander-Nirenberg theorem for degenerate complex structures.
In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space degenerating into the transversal union of two smooth Fano hypersurfaces in a generic …
This paper proves unique determination of certain non-quasi-Fuchsian manifolds by their end structure and bending lamination.
problem Unique determination of non-quasi-Fuchsian manifolds by their end structure and bending lamination.
method Analysis of the end structure (parabolic locus, ending laminations, conformal structures) and bending lamination.
result Non-quasi-Fuchsian manifolds are uniquely determined by their end structure and bending lamination.
We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and para-quaternion sectional curvature respect…
In this paper the notion of an M-th order invariant bilinear differential pairing is introduced and a formal definition is given. If the manifold has an AHS structure, then various first order pairings are constructed. This yields a classification of all first order invariant bilinear differential pairings on homogeneo…
We study the class K2+K4 of homogeneous pseudo-Kähler structures in the strongly degenerate case. The local form and the holonomy of a pseudo-Kähler manifold admitting such a structure is obtained, leading to a possible complex generalization of homogeneous plane waves. The same question is …
The paper identifies all flat CR Lie groups and their structures.
problem Identifying flat CR Lie groups and their structures.
method Analyzing Lie algebras and structures of Lie groups.
result Only specific Lie algebras are Cartan flat non-degenerate CR Lie algebras.
Superintegrable systems on surfaces are classified geometrically.
problem Classifying superintegrable systems on conformal surfaces.
method Geometric structures on conformal surfaces, conformal covariant structural equations.
result Explicit set of algebraic equations defining superintegrable systems on all constant curvature surfaces.
We study the local differential geometry of varieties Xn⊂CPn+a with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined v…
In this note, we extend the notion of a Monge hypersurface from its roots in semi-Euclidean space to more general spaces. For the degenerate case, the geometry of these structures is studied using the Bejancu-Duggal method of screen distributions.
We exhibit examples of projective varieties with degenerate Gauss mappings and determine numerical invariants of such varieties. Our examples provide counter-examples to an asserted structure theorem of Griffiths and Harris (Ann. Sci. ENS 1979).
We continue the study of blow-ups in generalized complex geometry with the blow-up theory for generalized Kähler manifolds. The natural candidates for submanifolds to be blown-up are those which are generalized Poisson for one of the two generalized complex structures and can be blown up in a generalized complex manner…
Study on special Lie groups with Lorentzian metrics.
problem Characterize structure of 2-step nilpotent Lorentzian naturally reductive Lie groups. method Develop framework for naturally reductive Lie groups, extend to Lorentzian context, analyze degenerate and non-degenerate cases.
result Complete structural description of naturally reductive 2-step Lorentzian nilpotent Lie groups. Stability of Morse index for harmonic maps on degenerating surfaces analyzed.
problem Analyzing stability of Morse index for harmonic maps on degenerating Riemann surfaces.
method Analysis of second variation of energy, identification of conditions for upper semicontinuity, explicit contribution of geodesics.
result Sharper control of spectrum of Jacobi operator, explicit contribution of geodesic segments to Morse index.
New symplectic forms derived from Lagrangian fibrations on symplectic manifolds.
problem Deriving new symplectic forms from existing ones.
method Proving existence of degenerate twistorial deformations.
result Existence of degenerate twistorial deformations preserving complex structures.
A Kaehler-Nijenhuis manifold is a Kaehler manifold M, with metric g, complex structure J and Kaehler form F, endowed with a Nijenhuis tensor field A that is compatible with the Poisson stucture defined by F in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if either AJ=JA or AJ=-JA, M is …
Superintegrable systems on curved manifolds found to have Hessian structures.
problem Characterizing superintegrable systems on curved manifolds.
method Identifying and computing Hessian coordinates for superintegrable systems.
result Examples of superintegrable systems in 2D and 3D have natural Hessian coordinates.
Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on S×R by the cotangent bundle of the Teichmüller space of S, we study how some geometric quantities, such as the Lorentzian Hausdorff dimension of the limit set, the width of the convex core and the Hölder exponen…
The authors study in detail new types of varieties with degenerate Gauss maps: varieties with multiple foci and their particular case, the so-called twisted cones. They prove an existence theorem for twisted cones and describe their structure.
Study of degenerate contrast functions on Lie groupoids and their geometric structures.
problem Understanding geometric structures on Lie groupoids with degenerate metrics.
method Using Lie groupoids and algebroids, analyze contrast functions and degenerate two-forms.
result Reduction of degenerate two-forms to pseudometric structures under regular conditions.
There exist non-degenerate 3-form dωI, ωI(X,Y)=g(IX,Y), for each leftinvariant almost Hermitian structure (g,I), where g is Killing-Cartan metric on the M=S3×S3=SU(2)×SU(2). Known \cite{H1}, that arbitrary non-degenerate 3-form on the 6-dimensional manifold, with some additional properties def…
We study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a C∞ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result \textit{à la Kodaira-Spencer} for the dimension o…
Study explores unstable 3-forms on Calabi-Yau 3-folds.
problem Understanding degenerations of Calabi-Yau 3-folds via 3-forms.
method Investigates geometries of 3-forms on symplectic 6-manifolds.
result Unstable 3-forms reveal rich geometric properties related to SYZ conjecture.
We discuss contact invariant structures on the space of solutions of a third-order ordinary differential equation. Associated to any third-order differential equation modulo contact transformations, Chern introduced a degenerate conformal Lorentzian metric on the space of 2-jets of functions of one variable. When the W…
Study on deformations of (p,q)-forms and spectral sequence degenerations.
problem Understanding deformations of (p,q)-forms under complex structure changes. method Analyzing Frölicher spectral sequence conditions for (p,q)-form deformations. result Unobstructed deformations of (p,q)-forms under specific spectral sequence conditions. Any singular level of a completely integrable system (c.i.s.) with non-degenerate singularities has a singular affine structure. We shall show how to construct a simple c.i.s. around the level, having the above affine structure. The cotangent budle of the desingularised level is used to perform the construction, and th…