Proves a vanishing property for symplectic manifold cohomology.
problem Generalizing complex geometry results to symplectic geometry.
method Based on Tseng and Zhou's vanishing property under symplectic flatness.
result Establishes necessity of symplectic flatness for certain results.
Averages vanish in ℓ1 cohomology of Heisenberg groups.
problem Vanishing of averages in ℓ1 cohomology. method Proof of vanishing averages for abelian and Heisenberg groups.
result Vanishing of ℓ1 cohomology for Heisenberg groups. Uniform criterion for vanishing products in bounded cohomology.
problem Vanishing of cup products and Massey products in bounded cohomology.
method Uniform vanishing criterion for products in bounded cohomology.
result Reproved and extended previous vanishing results.
Study shows bounded cohomology vanishes for higher dimensional sphere diffeomorphisms.
problem Vanishing of bounded cohomology for higher dimensional sphere diffeomorphism groups.
method Proved vanishing of bounded cohomology with real coefficients for n≥4 and 1≤r≤∞. result Vanishing of bounded cohomology for higher dimensional spheres.
We show vanishing theorems of L2-cohomology groups of Kodaira-Nakano type on complete Hessian manifolds. We obtain further vanishing theorems of L2-cohomology groups L2Hp,q(Ω) on a regular convex cone Ω with the Cheng-Yau metric for p>q.
Study confirms optimal bounds for group cohomology of Lie groups.
problem Optimal bounds for group cohomology of Lie groups.
method Combining complementary vanishings with spectral sequences and quasi-isometry invariance.
result Non-vanishing of group Lp-cohomology for large p and equal degree to rank. We prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.
The paper shows Morse-Novikov cohomology vanishes for certain curved manifolds.
problem Analyzing cohomology groups of curved manifolds.
method Examining Morse-Novikov cohomology groups for manifolds with specific curvature properties.
result Morse-Novikov cohomology groups vanish for almost nonnegatively curved manifolds with nonzero first de Rham cohomology.
The article studies cohomology on complex manifolds and proves vanishing theorems.
problem Investigating topological properties of complex manifolds.
method Using Dolbeault-Morse-Novikov cohomology and integral inequalities.
result The Hirzebruch χ_y-genus vanishes on certain complex manifolds.
The abstract discusses p-harmonic forms and their geometric properties, proving new theorems about Lp-cohomology.
problem The abstract tackles the geometric properties of p-harmonic forms and their role in Lp-cohomology.
method The approach involves using p-harmonic and p-coclosed forms to reprove vanishing theorems and provide injectivity theorems.
result The main finding is the reproof of vanishing theorems and the provision of injectivity theorems for Lp-cohomology.
The first cohomology of Poisson algebras is described and conditions for its vanishing are established.
problem Understanding the first cohomology of Poisson algebras.
method Description and mapping of first cohomology to intrinsic cohomologies of Poisson submanifolds, formulation of vanishing conditions.
result Necessary and sufficient conditions for the vanishing of the first cohomology of infinitesimal Poisson algebras are derived.
We study some relation between some geometrically defined classes of diffeomorphisms between manifolds and the Lq,p-cohomology of these manifolds. Some applications to vanishing and non vanishing results in Lq,p-cohomology are given.
In this paper we find sufficient conditions for the vanishing of the Morse-Novikov cohomology on Riemannian foliations. We work out a Bochner technique for twisted cohomological complexes, obtaining corresponding vanishing results. Also, we generalize for our setting vanishing results from the case of closed Riemannian…
Vanishing theorem for L2-harmonic forms on Riemannian manifolds with parallel 1-form.
problem Proving vanishing of L2-harmonic forms on Riemannian manifolds with a parallel 1-form. method Using L2 Morse-Novikov cohomology and a vanishing theorem. result The L2-harmonic forms on the manifold are identically zero. We find a presentation of symplectic Steinberg modules and show vanishing cohomology for certain groups.
problem Cohomology vanishing for specific groups and modules.
method Presented a symplectic Steinberg module and used it to prove cohomology vanishing.
result Cohomology of Sp2n(Z) vanishes in a specific degree for n≥2. Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.
problem Computing Čech cohomology groups of Morse boundaries.
method Analyzing cusped hyperbolic n-manifolds and relatively hyperbolic groups.
result Reduced Čech cohomology vanishes in dimensions ≤ n-3 and does not vanish in dimension n-2.
New displacement technique vanishes bounded cohomology in all degrees.
problem Vanishing of bounded cohomology in all positive degrees and dual separable coefficients.
method Introducing the property of commuting cyclic conjugates as a new displacement technique.
result Vanishes bounded cohomology in all positive degrees and all dual separable coefficients.
We study the simplicial {\ell} q,p cohomology of Carnot groups G. We show vanishing and non-vanishing results depending of the range of the (p, q) gap with respect to the weight gaps in the Lie algebra cohomology of G.
Study cup products on CAT(0) cube complexes, proving quasimorphisms' vanishing results.
problem Understanding cup products in higher bounded cohomology groups.
method Extending vanishing results from Brooks quasimorphisms to median quasimorphisms on CAT(0) cube complexes.
result Vanishing results for cup products of median quasimorphisms on CAT(0) cube complexes.
Vanishing result for cohomology leads to extension theorem for pluriharmonic functions.
problem Extension of pluriharmonic functions on complex manifolds.
method Vanishing result for Bott-Chern cohomology combined with Ehrenpreis technique.
result Hartogs extension theorem for pluriharmonic functions on cohomologically (n−1)-complete manifolds. Study vanishing theorems for CR manifolds using contact forms and Laplacian formulas.
problem Vanishing theorems for Kohn-Rossi cohomology of spherical CR manifolds.
method Used a canonical contact form and Weitzenböck-type formulae for the Kohn Laplacian.
result Results are optimal in some cases and prove vanishing theorems.
New proofs and refined theorems on bounded cohomology.
problem Properties of bounded cohomology and comparison map.
method Homotopy-theoretic properties and generalizations.
result New proofs and refined versions of vanishing and covering theorems.
We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic c…
The paper proves vanishing cohomology groups for free boundary hypersurfaces.
problem Proving vanishing cohomology groups for free boundary hypersurfaces.
method Using a universal constant and traceless second fundamental form condition.
result The pth cohomology group of a compact free boundary submanifold vanishes. Modified Θ-invariant for non-vanishing cohomology groups.
problem Defining Θ-invariant for non-vanishing cohomology groups. method A slightly modified version of the Θ-invariant. result Can define Θ-invariant even when cohomology group is not vanishing. New cohomology vanishing result for twisted cylinders.
problem Cohomology of twisted cylinders.
method Sobolev--Poincaré inequality methods.
result Vanishing result for Lq,p-cohomology. Study proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.
problem Hodge symmetry on complex manifolds with line bundles.
method Analyzes Dolbeault cohomology of Oeljeklaus-Toma manifolds with holomorphic line bundles.
result Proves Hodge symmetry and vanishing/non-vanishing of Dolbeault cohomology.
Local vanishing theorems for complex spaces with smooth boundaries.
problem Vanishing of cohomology groups for complex spaces with smooth boundaries.
method Local vanishing theorem for Dolbeault cohomology groups.
result Vanishing of L2 and L2,loc Dolbeault cohomology groups for q>0. The paper proves vanishing bounded cohomology for various groups.
problem Vanishing bounded cohomology for specific groups.
method Elementary algebraic criterion called the commuting cyclic conjugates condition.
result Many groups of interest have vanishing bounded cohomology.
Study groups acting on trees with specific local actions, proving cohomology vanishing or infinite.
problem Understanding bounded cohomology of groups with prescribed local actions.
method Proving vanishing or infinite bounded cohomology based on the 2-transitivity of F′. result Vanishing or infinite bounded cohomology depending on F′'s 2-transitivity. We prove that the first reduced cohomology with values in a mixing Lp-representation, p larger than 1, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group h…
The dualizing module of GL_n(O) varies, affecting cohomology vanishing and nonvanishing.
problem Understanding the dualizing module of GL_n(O) and its impact on cohomology.
method Analyzing the Steinberg module and a variant for GL_n(O), proving vanishing and nonvanishing theorems.
result The dualizing module of GL_n(O) is not always the Steinberg module, but a variant that accounts for orientation.
The paper studies topological properties of Ricci shrinkers using weighted L2 cohomology.
problem Proving topological results for smooth gradient Ricci shrinkers.
method Weighted L2 cohomology and extensions to mean curvature flow self-shrinkers. result Establishes upper bounds for Betti numbers, vanishing theorem for cohomology, and dichotomy for ends.
Study shows non-abelian free groups' 4th cohomology is non-zero.
problem Proving non-vanishing of fourth bounded cohomology of free groups.
method Splitting argument applied to n-manifolds with codimension 2 submanifolds. result Non-zero fourth bounded cohomology of non-abelian free groups.
This study shows that certain cohomology groups of symplectic manifolds are always even-dimensional.
problem Understanding the cohomology structure of symplectic manifolds.
method Constructing and deforming a skew-adjoint operator to prove the vanishing property.
result The even dimensionality of even-degree cohomology groups in (4n+2)-dimensional symplectic manifolds.
Direct proof of cohomology isomorphism for pullback Lie algebroids.
problem Cohomology of pullback Lie algebroids under surjective submersions.
method Differential-geometric proof, leafwise cohomology vanishing, exhaustion-and-patching argument.
result Cohomology isomorphism and injectivity properties for pullback Lie algebroids.
Vanishing results for reduced Lp,q-cohomology are established in the case of twisted products, which are a~generalization of warped products. Only the case q≤p is considered. This is an extension of some results by Gol'dshtein, Kuz'minov and Shvedov about the Lp-cohomology of warped cylinders. One of t…
The paper shows conditions under which certain cohomology groups vanish for Hermitian manifolds.
problem Conditions for vanishing certain cohomology groups on Hermitian manifolds.
method Analyzes the cohomology groups of Hermitian manifolds with specific conditions on metrics and forms.
result If the Aeppli class of ωn−p vanishes, then HBCp,0(M)=0. Study convexity of Mabuchi functional in big cohomology classes.
problem Convexity of Mabuchi functional in big cohomology classes.
method Defined an invariant related to transcendental Fujita approximations and established convexity under vanishing of this invariant.
result Established almost convexity along weak geodesics in big cohomology classes.
Formula for Dolbeault cohomology groups on complex manifolds with vector bundles.
problem Calculating Dolbeault cohomology groups on complex manifolds with vector bundles.
method Sheaf-theoretic approach to derive a blow-up formula.
result Uniform presentation of vanishing theorems and blow-up invariance of holomorphic invariants.
We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with ddc-harmonic Kähler form and positive (1,1)-part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such…
Researchers compute Hochschild cohomology of Grassmannians.
problem Computing Hochschild cohomology of Grassmannians.
method Explicit description of Gerstenhaber algebra structure, vanishing of higher cohomology.
result Decomposition of Hochschild cohomology concentrated in global sections for certain Grassmannians.
It is proved that the continuous bounded cohomology of SL_2(k) vanishes in all positive degrees whenever k is a non-Archimedean local field. This holds more generally for boundary-transitive groups of tree automorphisms and implies low degree vanishing for SL_2 over S-integers.
When $X=Γ\backslash \H^n$ is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of L2 harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.
The study shows conditions for Kähler manifolds to have rational cohomology.
problem Conditions for Kähler manifolds to have rational cohomology.
method Analyzes the Kähler curvature operator and its eigenvalues.
result Compact Kähler manifolds have rational cohomology under certain conditions.
The Euler characteristic of transitive Lie algebroids vanishes unless they are tangent bundles.
problem Calculating the Euler characteristic of transitive Lie algebroids.
method Atiyah-Singer index theorem and tensor products of elliptic complexes.
result The Euler characteristic of a transitive Lie algebroid vanishes unless it is the tangent bundle.
Extends Gromov's theorem with amenable covers.
problem Gromov's Vanishing Theorem limitations.
method Gromov's multicomplex theory, relative bounded cohomology.
result Relative version of Gromov's Vanishing Theorem.
The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. …