Study on characteristic classes for foliation deformations.
problem Characterizing and understanding characteristic classes for foliation deformations.
method Introduced a differential graded algebra (DGA) to recover Bott vanishing and formulae, and discussed properties of its cohomology.
result Discovered new classes that cannot be described by existing classes like Godbillon--Vey and Fuks--Lodder--Kotschick.
Introduces new deformation classes in generalized Kähler geometry.
problem No specific problem stated; focuses on new concepts.
method Uses Courant symmetry group to introduce deformation classes.
result Generalized Kähler cone is preserved by the generalized Kähler-Ricci flow.
Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.
problem Deformation theory of Lie groupoids and algebroids.
method Defining a morphism between deformation complexes and Hochschild complexes, applying to adiabatic groupoids.
result Induced van Est map from geometric to algebraic deformation cohomology.
This paper studies a deformation retraction of Teichmüller space and its analogy with well-rounded retractions.
problem Understanding the well-rounded deformation retraction of Teichmüller space.
method Examining the mapping class group-equivariant deformation retraction of Teichmüller space onto a CW complex and comparing it to well-rounded retractions of other spaces.
result The well-rounded deformation retraction of Teichmüller space is analogous to well-rounded retractions of other spaces.
Study shows that deformed Liouville metrics on tori remain Liouville.
problem Tackles the conjecture that only Liouville metrics are integrable on tori.
method Examines deformations of non-flat Liouville metrics and proves they remain Liouville.
result For a broad class of deformations, the deformed metric remains Liouville.
Geometrically deforms L∞ algebras to Lie algebroids, revealing new invariants.
problem Classifying geometric invariants of L∞ algebras arising from vector bundles. method Define geometric deformations of curved L∞ algebras and show they correspond to Lie algebroid structures. result Geometric deformations of L∞ algebras classify new geometric invariants. Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.
problem Mapping Teichmüller space to Thurston spine.
method Equivariant deformation retraction of Teichmüller space onto a cell complex.
result Thurston spine contains points corresponding to hyperbolic surfaces with shortest geodesics forming polygons.
Generalizes Frobenius theorem to quasiconformal deformations.
problem Integrability of plane fields generated by quasiconformal deformations.
method Generalization of classical Frobenius theorem to CQ plane fields. result A.e. involutive CQ plane fields are integrable. This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into PU(n,1) and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sulliva…
New integrable deformations for topological hierarchies from Frobenius manifolds.
problem Integrable deformations of topological hierarchies from Frobenius manifolds.
method Construction of integrable deformations with polynomial tau-structures.
result Conjecture of universal object for Riemann--Hopf hierarchy.
Completes the classification of Moebius deformable hypersurfaces for dimensions 5 and above.
problem Missing examples in the classification of Moebius deformable hypersurfaces for dimensions 5 and above.
method Investigates the class of Moebius deformable hypersurfaces and completes the classification for dimensions 5 and above.
result Completes the classification of Moebius deformable hypersurfaces for dimensions 5 and above.
In this note, we show that the obstruction classes of deforming vector forms on a compact Kähler manifold is annihilated by cohomology classes.
Study shows deformations of quaternionic Kähler manifolds are locally inhomogeneous.
problem Understanding deformations of quaternionic Kähler manifolds.
method Proved one-loop deformation of quaternionic Kähler manifolds are locally inhomogeneous.
result Full isometry group of one-loop deformations has cohomogeneity one.
The paper explores deformations of quasi-Hamiltonian spaces to Hamiltonian spaces.
problem Deforming quasi-Hamiltonian spaces to Hamiltonian spaces.
method Introducing and proving examples of deformations, including Lie groups and conjugacy classes.
result Moduli space of flat G-connections deforms to T*G^r+g.
Study infinitesimal deformations of Lie algebroid pairs.
problem Infinitesimal deformations of Lie algebroid pairs.
method Investigate isomorphism classes of infinitesimal deformations of (L,A) modulo automorphisms from exponentials of derivations of L and those from the exponentials of inner derivations of L. result Find the associated governing L∞-algebras in the sense of extended deformation theory. Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
The study shows boundedness of certain fibered varieties in algebraic geometry.
problem Bounding fibered varieties in algebraic geometry.
method Analyzing Calabi-Yau varieties and their fibrations by abelian or symplectic varieties.
result There are only finitely many deformation classes of certain fibered varieties.
This research classifies singular foliations and finds a universal deformation.
problem Classifying singular foliations on (C2,0). method Topological universal deformation through fixed invariants.
result Every equisingular deformation uniquely factors through the topological universal deformation.
In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family of deformation…
Study shows compact Sasakian manifolds are locally Heisenberg up to deformation.
problem Characterizing compact Sasakian manifolds.
method Analyzing basic Chern classes and using left invariant Sasakian structures.
result Compact Sasakian manifolds are locally isomorphic to the real Heisenberg group.
Study of wild mapping class groups on complex reflection groups.
problem Understanding deformations of wild Riemann surfaces.
method Construction of configuration spaces and combinatorial fission forests.
result Sharp parameterisation of admissible deformation classes of wild Riemann surfaces.
We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study th…
Paper introduces kernel deformed exponential families for sparse continuous attention.
problem Creating efficient attention mechanisms for sparse data.
method Developed kernel deformed exponential families, theoretically and experimentally.
result Kernel deformed exponential families can attend to multiple compact regions of data.
Study complex structure deformations on Lie algebras and Dolbeault cohomology.
problem Deformations of complex structures on Lie algebras and their associated Dolbeault cohomology.
method Construct a complete deformation of complex structures similar to the Kuranishi family, showing extension isomorphism validity.
result Analytic open subset of deformations where Dolbeault cohomology can be computed by left invariant tensor fields.
Study instanton metrics via Taub-NUT deformations.
problem Understanding deformations of instanton metrics.
method Using generalized Legendre transform on bow varieties.
result Found Kähler potential on instanton moduli spaces.
Study on new metrics on para-Kähler-Norden manifolds with conformal deformation.
problem Exploring geometric and harmonic properties of new metrics.
method Conformal deformation of Berger-type metric, analysis of Levi-Civita link, study of curvature varieties, and harmonic maps.
result Detailed examination of curvature varieties and harmonic maps on the manifold.
Study on special metrics and deformations of solvmanifolds.
problem Existence and properties of Kähler metrics on solvmanifolds.
method Investigation of strong Kähler with torsion metrics and balanced metrics on deformations of specific solvmanifolds.
result Non-existence of certain metrics on specific solvmanifolds.
In the present paper, we deform isolated singularities of a certain class of polar weighted homogeneous mixed polynomials, and show that there exists a deformation which has only definite fold singularities and mixed Morse singularities.
Study curvature of direct image bundles in deformations of maps.
problem Understanding curvature in deformations of maps with fixed targets.
method Analyzing curvature of direct image bundles related to deformation data.
result Proved seminegativity for a vector bundle of relative forms.
The paper explores moduli space of heterotic system using two deformation paths.
problem Exploring the moduli space of the heterotic system.
method Considering two dual deformation paths starting from a Kähler solution, one along Bott-Chern cohomology class and the other along Aeppli cohomology class. Using the implicit function theorem to prove local existence of heterotic solutions.
result Established an initial step to construct local moduli coordinates around a Kähler solution.
In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold (C∞ or complex analytic).
In this paper, the Douglas curvature of (α,β)-metrics, a special class of Finsler metrics defined by a Riemannian metric αand a 1-form β, is studied. These metrics with vanishing Douglas curvature in dimension n\geq3 are classified by using a new class of metrical deformations called β-deformations. The result shows th…
I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness --up to equivalence-- of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products …
According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformati…
Global fixed points in low-dimensional surface group space correspond to trivial representations.
problem Understanding global fixed points in surface group deformation spaces.
method Direct analysis of the deformation space, focusing on the trivial representation.
result Global fixed points in low-dimensional surface group deformation spaces correspond to the trivial representation of the pure mapping class group.
The study explores polarized deformations of SKT Calabi-Yau manifolds using Aeppli classes.
problem Understanding polarized deformations of SKT Calabi-Yau manifolds.
method Introducing small deformations polarized by Aeppli classes and investigating their properties.
result Existence of primitive elements in Bott-Chern classes and metrics comparison.
The paper proves finiteness for stable Lagrangian fibrations with a given divisor.
problem Finiteness of deformation classes of hyperkähler Lagrangian fibrations.
method Survey and proof of finiteness for stable Lagrangian fibrations with a given discriminant divisor.
result Finiteness for stable Lagrangian fibrations with a given discriminant divisor.
We prove a generalized version of the classic deformation lemma from Morse Theory that considers functions going to −∞ at a compact set, and allowing the lower value of the deformation to be −∞. The result is valid for a class of functions satisfying a suitable growth condition.
We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class obtained by differentiating the star-product with respect to the deformation par…
Study YB operators and their deformations, finding integrable and nontrivial cases.
problem Understanding deformations of Yang-Baxter operators and their integrability.
method Relating deformations to Lie algebra deformations, analyzing cohomology groups.
result Existence of integrable YB deformations and nontrivial cases not arising from SD deformations.
We prove in this paper the quasitriviality of a class of deformations of the one component bihamiltonian structures of hydrodynamic type.
Schmutz Schaller and Thurston's approaches are dual.
problem Mapping class group-equivariant deformation retractions of Teichmüller space.
method Comparing Schmutz Schaller's and Thurston's methods.
result Schmutz Schaller and Thurston's approaches are dual.
The goal of this article is to generalise the Witten deformation to even dimensional conic manifolds and a class of functions called admissible Morse functions.
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.
In trying to generalize Bianchi's Bäcklund transformation of quadrics to Bäcklund transformations of isometric deformations of other (classes of) surfaces, we investigate basic features of the isometric deformation of surfaces via the Bäcklund transformation with isometric correspondence of leaves of a general nature (…
We present explicit universal strict deformation quantization formulae for actions of Iwasawa subgroups AN of SU(1,n). This answers a question raised by Rieffel.
In this paper we study a class of Finsler metrics defined by a Riemannian metric and an 1-form. We classify those of projectively flat in dimension n≥3 by a special class of deformations. The results show that the projective flatness of such kind of Finsler metrics always arises from that of some Riemannian metric…
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.