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.
Vanishing theorem for tautological classes on aspherical manifolds.
problem Understanding rational cohomology of classifying spaces for diffeomorphism groups.
method Using tautological classes and rational cohomology to show vanishing for aspherical manifolds.
result Rational tautological classes depend only on the underlying topological block bundle, leading to vanishing results.
Vanishing cup product of Brooks quasimorphisms proved.
problem Vanishing cup product of Brooks quasimorphisms.
method Vanishing of the square of a universal class for tree automorphism groups.
result Vanishing cup product of Brooks quasimorphisms.
Study virtual fundamental classes of derived manifolds, proving invariant vanishes.
problem Computing virtual fundamental classes for derived manifolds.
method Combining derived differential geometry and cosection localization.
result Stable pair invariants of hyperkähler fourfolds are zero.
Study on smooth bundles with nonpositive curvature, proving vanishing classes and rigidity.
problem Characterizing smooth bundles with nonpositive curvature fibers.
method Proving vanishing results for Miller-Morita-Mumford classes and showing topological rigidity.
result Vanishing of generalized Miller-Morita-Mumford classes and topological rigidity of vertical tangent bundles.
The Euler class vanishes under certain curvature conditions.
problem Conditions for the vanishing of the Euler class in positive scalar curvature manifolds.
method Generalization of Lichnerowicz vanishing theorem for flat vector bundles.
result The Euler class vanishes for specific flat vector bundles over manifolds with positive scalar curvature.
Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more…
The paper addresses deformations of Kähler spaces with vanishing first Chern class.
problem Deformations of Kähler spaces with specific properties.
method Analyzes locally trivial deformation spaces and uses cohomological vanishing conditions.
result Shows that under certain conditions, deformations of Kähler spaces are projective varieties.
Geometrically interprets Atiyah class vanishing for Lie pairs.
problem Atiyah class vanishing for Lie pairs and Lie algebroids.
method New characterisation of ideal systems and geometric interpretation.
result Atiyah class vanishes for Lie pairs with kernel of fibration.
Study categorizes Vaisman manifolds with vanishing first Chern class and finds canonical metrics.
problem Characterizing Vaisman manifolds with vanishing first Chern class.
method Categorization into three types based on Bott-Chern class sign, showing canonical metrics, quasi-regularity, stability, and automorphism group behavior.
result Vaisman manifolds with non-positive Bott-Chern class admit canonical metrics and are stable under deformations.
Study shows non-vanishing Stiefel-Whitney classes and absence of spin^C structures in certain hyperbolic manifolds.
problem Existence of manifolds without spin^C structures and non-vanishing higher order Stiefel-Whitney classes.
method Analysis of cusped arithmetic hyperbolic manifolds of simplest type.
result Existence of manifolds with non-vanishing Stiefel-Whitney classes and absence of spin^C structures.
The paper proves conditions for minimal compact Kähler manifolds with vanishing second Chern class.
problem Conditions for minimal compact Kähler manifolds with vanishing second Chern class.
method Study of the abundance conjecture and associated Iitaka fibrations.
result For a minimal compact Kähler manifold, the second Chern class vanishes if and only if the cotangent bundle is nef and the canonical bundle has numerical dimension 0 or 1.
Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.
problem Characterizing curves that can be vanishing cycles in degenerations of linear systems.
method Computing mapping class group-valued monodromy and identifying it with r-spin mapping class groups.
result Identifies simple closed curves as vanishing cycles and provides characterizations of discriminants and Lefschetz fibrations.
The study classifies metrics with vanishing curvature on complex manifolds.
problem Understanding metrics with vanishing curvature on complex manifolds.
method Analyzing Hermitian metrics, pluriclosed metrics, and metrics with real bisectional curvature.
result Hermitian metrics with vanishing curvature are Kähler and conformally balanced.
An odd Seiberg-Witten invariant imposes bounds on the signature of a closed, almost complex 4-manifold with vanishing first Chern class. This applies in particular to symplectic 4-manifolds of Kodaira dimension zero.
Paper shows Euler class vanishes in certain subgroup of mapping class group.
problem Vanishing of the Euler class in power subgroups of mapping class group.
method Used Dahmani's result to show triviality of Euler class.
result Euler class vanishes in power subgroup of mapping class group.
New quasi-homomorphisms on surface groups vanish on handlebody groups.
problem Understanding quasi-homomorphisms on mapping class groups and their behavior on handlebody subgroups.
method Constructing infinitely many linearly independent quasi-homomorphisms that vanish on a handlebody subgroup.
result Disproved a conjecture about bounded width in Heegaard splittings and found infinitely many quasi-invariants on Heegaard splittings.
The paper constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.
problem Constructing 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.
method Constructs infinitely many rational homology 3-spheres using Dehn surgeries and Heegaard Floer homology.
result Found rational homology 3-spheres that admit co-orientable taut foliations but none with vanishing Euler class.
Study vanishing cycles on curves in toric surfaces, identifying specific types of obstructions.
problem Identify vanishing cycles on curves in toric surfaces with specific types of obstructions.
method Investigate monodromy maps and their target space, the mapping class group, to determine vanishing cycles.
result Explicit generators for the Spin mapping class group are provided.
Study on Finsler metrics finds no P-reducible metrics with vanishing S-curvature.
problem Existence of P-reducible metrics with vanishing S-curvature.
method Investigation of generalized P-reducible metrics and proving their reduction to Berwald or C-reducible metrics.
result No concrete P-reducible (α,β)-metric with vanishing S-curvature exists. 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. The paper finds metrics with vanishing Douglas curvature in a specific class of Finsler metrics.
problem Identifying Finsler metrics with vanishing Douglas curvature.
method Defined and studied general (α,β)-metrics, solved an equation for vanishing Douglas curvature. result Explicitly constructed new non-trivial examples of general (α,β)-metrics with vanishing Douglas curvature. Paper proves non-extendability of quasimorphism and non-vanishing of Reznikov's class.
problem Non-extendability of Shelukhin's quasimorphism and non-vanishing of Reznikov's class.
method Analyzing specific symplectic manifolds like blow-ups of tori and products of surfaces.
result Non-extendability of Shelukhin's quasimorphism and non-vanishing of Reznikov's class for certain symplectic manifolds.
We construct closed symplectic manifolds for which spherical classes generate arbitrarily large subspaces in 2-homology, such that the first Chern class and cohomology class of the symplectic form both vanish on all spherical classes. We construct both Kaehler and non-Kaehler examples, and show independence of the cond…
CR Q-curvature flow solves CR manifold curvature conjecture.
problem Proving existence and convergence of CR Q-curvature flow in CR 3-manifolds.
method Deforming a contact form according to CR Q-curvature flow.
result Existence and smooth asymptotic convergence of CR Q-curvature flow.
Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.
problem Negative curvature manifolds and their volume classes.
method Integration of volume forms and isoperimetric constants.
result Vanishing of bounded volume class implies positivity of Cheeger constant and vice versa.
A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same …
The paper uses symplectic homology to study 3D Besse manifolds with vanishing first Chern class.
problem Identifying 3D Besse manifolds with vanishing first Chern class.
method Computing the first Chern class, analyzing periodic Reeb orbits, and using symplectic homology.
result Classifies 3D Besse manifolds with vanishing first Chern class.
We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki's class C with vanishing first Bott-Chern cla…
This paper solves a question about curves on toric surfaces.
problem Which curves can be vanishing cycles for degenerations?
method Reformulated as a mapping class group problem and determined the monodromy group.
result A complete answer to the question of which curves can be vanishing cycles.
Classifies connections on central extensions and finds vanishing obstruction classes.
problem Classifying connections on central extensions of Lie groups.
method Introducing a new connective structure on the lifting gerbe and proving a vanishing result for Neeb's obstruction classes.
result Admissible connections correspond to parallel trivializations of the lifting gerbe and vanishing obstruction classes.
Computes invariant for smooth h-cobordisms families, proving duality and vanishing theorems.
problem Computing invariants for smooth h-cobordisms families.
method Using Dwyer, Weiss, and Williams work, fiberwise generalized Morse function, fiberwise Poincaré--Hopf theory.
result Duality theorem for smooth structure class, vanishing theorem for Rigidity Conjecture.
New Douglas metrics found in a specific Finsler class.
problem Finding new Douglas metrics in Finsler geometry.
method Defined general (α,β)-metrics and solved PDEs for vanishing Douglas curvature. result Many new Douglas metrics constructed.
Study non-vanishing ℓ2-Betti numbers for specific groups.
problem Calculating non-vanishing ℓ2-Betti numbers for certain groups. method Using Euler characteristics, higher Kazhdan projections, and Baum-Connes assembly map.
result Non-vanishing calculations for delocalised ℓ2-Betti numbers. We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the i…
We show that a compact orientable 4-manifold M has a CR regular immersion into C3 if and only if both its first Pontryagin class and its Euler characteristic vanish, and has a CR regular embedding into C3 if and only if in addition the second Stiefel-Whitney class of M vanishes.
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.
We study the bounded fundamental class in the top dimensional bounded cohomology of negatively curved manifolds with infinite volume. We prove that the bounded fundamental class of M vanishes if M is geometrically finite. Furthermore, when M is a R-rank one locally symmetric space, we show that the bou…
Study compact Kähler manifolds with nonpositive holomorphic sectional curvature and their canonical bundles.
problem Characterizing compact Kähler manifolds with nonpositive holomorphic sectional curvature and properties of their canonical bundles.
method Analyzing properties of Hermitian metrics and canonical bundles on compact Kähler manifolds.
result Proves nefness of canonical bundle and ampleness in complex dimension two for negative holomorphic sectional curvature.
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.
The paper classifies (α,β)-metrics with vanishing Douglas curvature.
problem Classifying (α,β)-metrics with vanishing Douglas curvature.
method Using β-deformations to classify metrics with vanishing Douglas curvature.
result Conformal 1-forms of Riemannian metrics play a key role, and an effective way to construct such 1-forms is provided by β-deformations.
An (α,β)-metric is defined by a Riemannian metric α and 1-form β. In this paper, we study a known class of two-dimensional (α,β)-metrics of vanishing S-curvature. We determine the local structure of those metrics and show that those metrics are Einsteinian (equivalently, isotropic flag curvature) but generall…
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.
Characterizes two-dimensional generalized Berwald metrics with vanishing S-curvature.
problem Characterizing metrics with specific curvature properties.
method Analyzing two-dimensional generalized Berwald (α,β)-metrics with vanishing S-curvature. result Provides a generalization of Szabó rigidity theorem for (α,β)-metrics. We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern …
The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…
New invariant from Seiberg-Witten equations constrains embedded surfaces.
problem Constraints on embedded surfaces in 4-manifolds.
method Constructs an invariant using Seiberg-Witten equations on families of embedded surfaces.
result Non-vanishing invariant provides new constraints on Seiberg-Witten vanishing.
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.