Kähler structures on products of 2-spheres have identical Chern classes.
problem Classifying Kähler structures on products of 2-spheres.
method Using complex Bott manifolds and iterated P1-bundle constructions, we classify Kähler structures up to biholomorphism via Bott diagrams. result Kähler structures on products of 2-spheres have identical Chern classes.
The paper constructs non-trivial cocycles for long embeddings with more than one loop.
problem Constructing non-trivial cocycles for long embeddings with more than one loop.
method Integral over configuration spaces associated with Bott-Cattaneo-Rossi graphs with more than one loop.
result Explicit construction of a non-trivial family of trivial long embeddings for odd dimensions.
New definition of knot invariants for higher-dimensional knots.
problem Defining invariants for long knots in high-dimensional spaces.
method Flexible definition as counts of diagrams, extending to asymptotic homology.
result Generalized invariants for long knots in more general spaces.
We describe Taylor towers for spaces of knots arising from Goodwillie-Weiss calculus of the embedding functor and extend the configuration space integrals of Bott and Taubes from spaces of knots to the stages of the towers. We show that certain combinations of integrals, indexed by trivalent diagrams, yield cohomology …
We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel an…
Study cohomology of braid spaces using rational formality and configuration space integrals.
problem Cohomology of braid spaces in dimensions 4 and above.
method Rational formality of configuration spaces, bar construction, diagram complexes, quasi-isomorphisms, configuration space integrals.
result Quasi-isomorphism between diagram complex and de Rham cochains on braid space, compatible with Bott-Taubes integrals.
Proposes a new Hodge conjecture in Bott-Chern cohomology.
problem Hodge conjecture in Bott-Chern cohomology.
method Characterization of real holomorphic chains, atomic section theory, refined Bott-Chern classes.
result Proof of a new Hodge conjecture in Bott-Chern cohomology.
Discrete Morse-Bott theory on CW complexes generalizes Forman's theory.
problem No specific problem stated; focuses on theory development.
method Derived a discrete Morse-Bott theory on CW complexes.
result Discrete Morse-Bott theory is a generalization of Forman's theory.
Formula derived for Bott-Chern classes in complex blow-ups.
problem Calculating Bott-Chern classes in blow-ups of complex manifolds.
method Proved blow-up formula for Bott-Chern classes, established Riemann-Roch without denominators.
result Formula for Bott-Chern classes in blow-ups.
Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.
problem Characterize Bott-Chern cohomology of Vaisman manifolds.
method Explicit description via basic cohomology, infer relationships between cohomology groups, show invariants are unbounded, cohomological characterization of formality.
result Bott-Chern and Dolbeault numbers determine each other for Vaisman manifolds, and cohomological invariants are unbounded.
A notion of geometric formality in the context of Bott-Chern and Aeppli cohomologies on a complex manifold is discussed. In particular, by using Aeppli-Bott-Chern-Massey triple products, it is proved that geometric Aeppli-Bott-Chern formality is not stable under small deformations of the complex structure.
Paper computes Stiefel-Whitney classes on real Bott manifolds.
problem Computing Stiefel-Whitney classes on real Bott manifolds.
method Analyzes real Bott manifolds with holonomy group Z2k and diagonal type. result Extends results to compute even and odd Stiefel-Whitney classes.
New findings on complex manifold properties under deformations.
problem Properties of Dolbeault and Bott-Chern formalities are not preserved under holomorphic deformations.
method Construction of a complex manifold to demonstrate non-preservation of properties.
result Existence of a manifold satisfying ∂∂-lemma but with non-vanishing Aeppli-Bott-Chern-Massey product. In studying the Bott-Chern and Aeppli cohomologies for q-complete manifolds, we introduce the class of cohomologically Bott-Chern q-complete manifolds.
The paper generalizes the Bott-Virasoro group and derives new Euler equations.
problem Understanding the generalized Bott-Virasoro group and its dynamics.
method Generalizing the Bott-Virasoro group using connection cochain and deriving Euler equations.
result New Euler equations derived from the generalized Bott-Virasoro group.
In the present paper, we define Morse-Bott functions on manifolds with boundary which are generalizations of Morse functions and show Morse-Bott inequalities for these manifolds.
Study of deformed Bott-Chern cohomology on complex manifolds.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that these assumptions are insufficient with an explicit counterexample…
Study on Kähler manifolds proves weak decompositions and relates harmonic forms.
problem Analyzing harmonic forms on Kähler manifolds.
method Proves weak W1,2 Bott-Chern and Dolbeault decompositions. result Strict relation between W1,2 Bott-Chern harmonic forms and the W1,2 Bott-Chern decomposition. The study describes Lefschetz-Bott fibrations on symplectic manifolds and their applications.
problem Understanding Lefschetz-Bott fibrations on symplectic manifolds.
method Explicit construction and analysis of Lefschetz-Bott fibrations over line bundles.
result Construction of strong symplectic fillings of symplectic manifolds.
Spin-structures on real Bott manifolds with Kähler structure are characterized.
problem Existence of spin-structures on real Bott manifolds with Kähler structure.
method Ishida characterization and techniques from \cite{PS16} using characteristic classes.
result Necessary and sufficient condition for the existence of spin-structures on M. 3D foliation study finds Poisson structure with specific singularities.
problem Characterizing Poisson structures on Bott-Morse foliations in 3D.
method Analyzes Bott-Morse foliations, computes Poisson bivectors and symplectic forms.
result Linear, singular Poisson structure of rank 2 with Bott-Morse singularities found.
Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.
problem Well-definedness of Morse-Bott-Smale chain complex.
method Unified five degeneracy relations into a single condition.
result Quasi-isomorphic to Morse-Smale-Witten chain complex, alternative proof of Morse Homology Theorem.
Condition found for spinc structures on a specific type of manifold.
problem Existence of spinc structures on real Bott manifolds.
method Provided a necessary and sufficient condition.
result Condition for spinc structures on real Bott manifolds established.
Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.
problem Proving inequalities for four-manifolds of Seiberg-Witten simple type.
method Uses virtual Morse-Bott indices and Hirzebruch-Riemann-Roch Theorem.
result Proves positivity of virtual Morse-Bott indices, leading to inequalities.
The Alexander polynomial is linked to Bott-Cattaneo-Rossi invariants via Chern-Simons theory.
problem Expressing Alexander polynomial of long knots in terms of invariants.
method Using a previously established formula relating Bott-Cattaneo-Rossi invariants to the Alexander polynomial and Chern-Simons theory.
result Relating Bott-Cattaneo-Rossi invariants to the Alexander polynomial and Chern-Simons theory.
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on Lorentzian Lie groups.
method Computed Bott connections and their curvature; classified Ricci solitons.
result Classification of Ricci solitons on three-dimensional Lorentzian Lie groups.
Paper detects non-trivial cycles in embedding spaces using graph integrals.
problem Detecting non-trivial cycles in embedding spaces.
method Construct cycles from chord diagrams, use modified configuration space integrals, and pair arguments.
result Non-trivial cycles in embedding spaces are detected.
Research resolves sign conventions in Floer theory for Morse-Bott case.
problem Sign conventions in filtered A∞-operations for Lagrangian Floer theory. method Defined filtered A∞-operations and verified formulae using de Rham model. result Resolved sign issues in Bott-Morse setting.
Study bigraded formality and Aeppli-Bott-Chern-Massey products on complex manifolds.
problem Formality and higher Aeppli-Bott-Chern-Massey products on complex manifolds.
method Introduce and study bigraded formality and Aeppli-Bott-Chern-Massey products, showing non-trivial pullbacks on blow-ups.
result Aeppli-Bott-Chern-Massey products on complex manifolds pull back non-trivially to blow-ups under certain conditions.
Proof that critical knots of Morse-Bott functions are graph knots.
problem Characterizing critical knots in Morse-Bott functions.
method Inductive proof on the number of index-1 critical knots.
result Critical knots of Morse-Bott functions are graph knots.
We study Kähler geometry of Bott manifolds and find extremal metrics.
problem Understanding Kähler geometry on Bott manifolds.
method Simple induction and generalized Calabi construction.
result Any stage n Bott manifold admits an extremal Kähler metric.
Proves a Baum--Bott formula for foliations by curves with logarithmic terms.
problem Analyzing singularities and smoothness in foliations by curves.
method Logarithmic Baum--Bott residues for foliated triples (X,F,D), relating to Poincaré's Problem and GSV indices. result Logarithmic Baum--Bott residues generalize Aleksandrov logarithmic index for vector fields on hypersurfaces.
The paper extends Bott-Chern Laplacian definition and explores its properties on almost Hermitian manifolds.
problem Exploring the properties of Bott-Chern Laplacian on almost Hermitian manifolds.
method Extending the definition of Bott-Chern Laplacian, proving ellipticity, and analyzing kernels on different types of manifolds.
result The dimensions of Bott-Chern and Dolbeault harmonic forms differ on almost complex 4-manifolds with specific metrics.
Spin-structures on real Bott manifolds with Kähler structures are characterized.
problem Existence of Spin-structures on real Bott manifolds with Kähler structures.
method Ishida characterization and Popko-Szczepański cohomological rigidity technique.
result Necessary and sufficient condition for the existence of Spin-structures on M.
This paper extends Bott-Chern cohomology to coherent sheaves using superconnections.
problem Extending Bott-Chern cohomology to coherent sheaves.
method Using superconnections and the theory of superconnections in the sense of Quillen.
result Characteristic classes are independent of Hermitian metrics and depend on the homotopy class of the coherent module.
A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a …
The paper calculates cohomology for complex 3-folds using new cohomology types.
problem Calculating cohomology for compact complex 3-folds.
method Defined new cohomology types Kp,q and used them to calculate cohomology. result Complete Bott-Chern-Aeppli cohomology for various complex 3-folds.
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
The paper proves a blow-up formula for Bott-Chern cohomology and shows the bimeromorphic invariance of non-Kählerness degrees on threefolds.
problem The bimeromorphic invariance of the ∂∂ˉ-Lemma on threefolds. method Proves a blow-up formula for Bott-Chern cohomology and applies it to show bimeromorphic invariance of non-Kählerness degrees.
result The non-Kählerness degrees are bimeromorphic invariants on compact complex threefolds, implying the bimeromorphic invariance of the ∂∂ˉ-Lemma. Study L2 Hilbert complexes on complex manifolds.
problem Analyse L2 Hilbert complexes on complex manifolds. method Define and study L2 Aeppli-Bott-Chern Hilbert complex; examine properties on various manifolds; use self-adjoint extensions of differential operators. result Kernels of operators on compact Hermitian manifolds are isomorphic to Aeppli or Bott-Chern cohomology.
Study new invariants in complex geometry using Bott-Chern hypercohomology.
problem Understanding geometry through Bott-Chern hypercohomology and bimeromorphic invariants.
method Construct new invariants involving sheaf cohomology, establish blow-up formula and canonical morphism.
result Compute invariants for specific complex threefolds like Iwasawa manifolds and quintic threefolds.
Study primitive decompositions for harmonic forms on almost Kähler manifolds.
problem Decomposing harmonic forms on almost Kähler manifolds.
method Proved primitive decompositions for Bott-Chern and Aeppli harmonic forms in specific bidegrees.
result Optimal bidegrees for primitive decompositions of harmonic forms.
The paper develops a theory of Bott-Chern cohomology and proves a residue theorem.
problem Localization of Bott-Chern classes and Hermitian residues.
method Developed a unified theory of Cech-Bott-Chern cohomology, defined cup product and integration, and introduced local and global duality homomorphisms.
result Proved a residue theorem for vector bundles with compatible Hermitian connections.
Smooth SE structures on Sasaki-joins and Bott orbifolds constructed.
problem Constructing smooth Sasaki-Einstein structures on Sasaki-joins.
method Categorical relationship between Sasaki-joins and Bott orbifolds, explicit construction.
result Infinitely many smooth SE structures up to dimension eleven, conjecture for all odd dimensions.
Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.
problem Establishing equivalence of Morse-Bott volume forms.
method Adapting Moser's trick to Morse-Bott volume forms.
result Two Morse-Bott volume forms with the same zero set are diffeomorphic if and only if they have equal total volumes.
Only products of projective lines have vanishing Futaki invariants for all Kähler classes.
problem Finding Bott manifolds with vanishing Futaki invariants for all Kähler classes.
method Proving isomorphism to products of projective lines.
result Only products of projective lines satisfy the condition.
Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.
problem Finding homology 3-spheres with non-Morse-Bott Chern-Simons functions.
method Constructing specific surgeries on torus knots.
result Examples of homology 3-spheres with non-Morse-Bott Chern-Simons functions.