The paper proves a generalized Lefschetz duality for a specific type of manifold.
problem Proving the hard Lefschetz duality for a new class of manifolds.
method Generalizing Kähler identities to prove the duality for locally conformally almost Kähler manifolds.
result The hard Lefschetz duality is established for locally conformally almost Kähler manifolds.
Jones polynomials compute weighted sums of Lefschetz numbers.
problem Computing Lefschetz numbers for braids.
method Colored Jones polynomials of braid closures.
result Jones polynomials compute abelianized Lefschetz numbers.
The paper connects Chern-Gauss-Bonnet and Lefschetz Duality using a novel approach.
problem Exploring the relationship between Chern-Gauss-Bonnet and Lefschetz Duality.
method Using the mapping cone for relative deRham cohomology and Thom isomorphism.
result Shows the Chern-Gauss-Bonnet Theorem as a manifestation of Lefschetz Duality.
The paper explores geometric and topological properties of almost Kähler manifolds using harmonic theory.
problem Understanding the geometric and topological aspects of almost Kähler manifolds.
method Deduction of geometric and topological consequences from extended Kähler identities for compact almost Kähler manifolds.
result Generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition for d-harmonic forms. Extends Hard Lefschetz Property to isometric flows and shows equivalence.
problem Studying the Hard Lefschetz Property in isometric flows.
method Extending the Hard Lefschetz Property to isometric flows and proving equivalence.
result Equivalence of transverse and non-transverse Hard Lefschetz Properties in isometric flows.
Heegaard Floer homology duality applied to 4D mapping tori.
problem Computing Heegaard Floer invariants of 4D mapping tori.
method Applied Heegaard Floer homology and duality theory.
result Computed invariants of 4D mapping tori using Lefschetz numbers.
Extends Poincaré-Lefschetz duality to pairs of ∞-categories.
problem Generalizing Poincaré-Lefschetz duality to ∞-categories.
method Introduces Poincaré duality pairs of ∞-categories and uses them to study various diagrams of spaces.
result Unified treatment of Wall's Poincaré ads and iterated Poincaré cobordisms.
Proves new fixed point formulae for complex manifolds with boundary.
problem Fixed points on complex manifolds with boundary conditions.
method Logarithmic Lefschetz fixed point formulae, normal rescaling, relative duality.
result Resonant boundary terms record normal contact and tangential multiplicity.
We survey work by the author and Ralf Meyer on equivariant KK-theory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action of a group or groupoid called the Lefschetz map. The Lefschetz map associates an …
Harmonic forms on complex manifolds satisfy special symmetries.
problem Understanding symmetries in harmonic forms on complex manifolds.
method Representation of sl(2,C) to generalize Kähler manifold properties. result Harmonic forms on Hermitian manifolds exhibit hard Lefschetz duality.
A theory of differential characters is developed for manifolds with boundary. This is done from both the Cheeger-Simons and the deRham-Federer viewpoints. The central result of the paper is the formulation and proof of a Lefschetz-Pontrjagin Duality Theorem, which asserts that the pairing: Ch^k(X,dX) x Ch^{n-k-1}(X) --…
Study on twisted Dolbeault cohomology in Kähler foliations.
problem Exploring cohomology in transverse Kähler foliations.
method Analysis of twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem.
result Proved Kodaira-Serre type duality for twisted basic Dolbeault cohomology.
Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.
problem Proving properties of convex valuations analogous to Kähler manifolds.
method Elliptic operator theory and perturbation theory applied to unbounded operators on a Hilbert space.
result Establishes hard Lefschetz theorem and Hodge-Riemann relations for convex bodies.
New space for valuations in non-Archimedean setting with duality properties.
problem Developing a non-Archimedean analogue of classical valuation spaces.
method Construction of a new space of valuations with similar structures to classical spaces.
result The new space satisfies Poincaré duality and hard Lefschetz theorem.
Study shows Lefschetz fibrations on Milnor fibers of certain singularities.
problem Understanding Lefschetz fibrations on Milnor fibers of specific singularities.
method Analyzes Milnor fibers of cusp and simple elliptic singularities to construct Lefschetz fibrations.
result Milnor fibers of cusp and simple elliptic singularities admit genus-one Lefschetz fibrations.
Differential refinement of homology satisfies Poincaré duality.
problem Constructing a refined homology theory that satisfies Poincaré duality.
method Defining differential cap product and constructing differential refinement of singular homology.
result Essential uniqueness of differential homology and cap product proven.
The main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …
We prove that the complement of any affine 2-arrangement in R^d is minimal, that is, it is homotopy equivalent to a cell complex with as many i-cells as its i-th rational Betti number. For the proof, we provide a Lefschetz-type hyperplane theorem for complements of 2-arrangements, and introduce Alexander duality for co…
The paper develops L2-Hodge theory on almost Kähler manifolds and proves the Hopf conjecture.
problem Proving the Hopf conjecture for almost Kähler manifolds.
method Developed L2-Hodge theory identities and applied them to prove vanishing theorems and refine estimates. result Proved the Hopf conjecture for compact almost Kähler manifolds with negative sectional curvature.
This thesis explores Hamiltonian systems and Kähler structures on complex coadjoint orbits.
problem Investigating Hamiltonian systems and Kähler structures on complex coadjoint orbits.
method Analyzes (pseudo)-holomorphic Hamiltonian systems, Lefschetz and almost toric fibrations, and introduces pseudo-holomorphic Hamiltonian systems.
result Complex coadjoint orbits exhibit both Hyperkähler and holomorphic Kähler structures, suggesting Kähler duality.
Study knot spaces and Atiyah duality in spectral categories.
problem Understanding the space of embeddings of a circle into higher-dimensional manifolds.
method Develop a cosimplicial model using Atiyah duality and prove a comodule version.
result Compute knot spaces in low degrees and establish isomorphisms on fundamental groups.
Study cohomologies on manifolds with locally conformally symplectic structures.
problem Understanding cohomologies on manifolds with locally conformally symplectic structures.
method Introduced lcs cohomologies, studied elliptic Hodge theory, dualities, and Hard Lefschetz Condition.
result Oeljeklaus-Toma manifolds with precisely one complex place and under an arithmetic condition satisfy the Mostow property.
Extended Moser's theorem to manifolds with corners.
problem Diffeomorphism group action on smooth densities.
method Based on Banyaga's 1974 paper, with cohomological interpretation.
result Moser's theorem proven for simplices.
In this note we introduce certain invariants of real Lefschetz fibrations. We call these invariants {\em real Lefschetz chains}. We prove that if the fiber genus is greater than 1, then the real Lefschetz chains are complete invariants of real Lefschetz fibrations with only real critical values. If however the fiber ge…
Researchers find multiple supporting minimal Lefschetz pencils for a complex fibration.
problem Finding multiple supporting minimal Lefschetz pencils for a specific Lefschetz fibration.
method Analyzing maximal disjoint (−1)-sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration. result The Matsumoto-Cadavid-Korkmaz Lefschetz fibration has more than one supporting minimal Lefschetz pencils.
Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.
problem Understanding achiral Lefschetz fibrations from non-orientable ones.
method Composition of standard orientation double covering map and non-orientable Lefschetz fibration.
result Specification of a monodromy factorization for the composition.
In this paper we construct six-dimensional compact non-Kähler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known counter examples to the question whether the strong Lefschetz property descends to th…
We construct universal Lefschetz fibrations, defined in analogy with classical universal bundles. We also introduce the cobordism groups of Lefschetz fibrations, and we see how these groups are quotients of the singular bordism groups via the universal Lefschetz fibrations.
The study examines K-polystability on Fano 4-folds with specific Lefschetz defects.
problem Investigating K-polystability on Fano 4-folds with Lefschetz defect at least 2.
method Examining 19 families of Fano 4-folds with Lefschetz defect 3 and 175 families with Lefschetz defect 2, proving K-polystability and instability.
result Exactly 5 out of 19 families of Fano 4-folds with Lefschetz defect 3 are K-polystable, and 5 out of 175 Casagrande-Druel Fano 4-folds with Lefschetz defect 2 are K-polystable.
The paper bounds the first Betti number and discusses properties of Lefschetz fibrations.
problem Understanding the topology of Lefschetz fibrations.
method Analyzes upper bounds for the first Betti number, examines monodromy transitivity, and discusses potential fibrations.
result Upper bounds for the first Betti number and insights into Lefschetz fibration properties.
Constructs Lefschetz fibrations with slopes near 2.
problem Finding Lefschetz fibrations with slopes close to 2.
method Constructs minimal, simply connected genus-g Lefschetz fibrations over the sphere.
result The infimum and supremum of slopes are not achievable.
We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above …
Study 1-Lefschetz contact solvmanifolds, proving their characterization.
problem Characterize 1-Lefschetz contact solvmanifolds.
method Prove the 1-Lefschetz condition on Lie algebras via 1-dimensional central extensions and explicit cohomology relations.
result Unimodular symplectic Lie algebras are 1-Lefschetz if and only if their contactizations are.
The harmonic cohomology of a Donaldson symplectic submanifold and of an Auroux symplectic submanifold are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the s-Lefschetz propery. In particular, we consider the symplectic blow-ups of the comp…
The paper embeds 4-manifolds into CP^2 x CP^1 using Lefschetz fibrations.
problem Embedding 4-manifolds into CP^2 x CP^1.
method Proving Lefschetz fibration embeddings of 4-manifolds into CP^2 x CP^1.
result Every closed, connected, orientable 4-manifold admits a smooth Lefschetz fibration embedding into CP^2 x CP^1.
The paper extends trisection construction for Lefschetz fibrations with (−n)-sections.
problem Constructing trisections for 4-manifolds with specific Lefschetz fibrations.
method Explicit construction of trisections using monodromies of Lefschetz fibrations.
result Trisections for various 4-manifolds constructed from Lefschetz fibrations.
Proves existence of regular Lagrangians via Weinstein Lefschetz fibrations.
problem Existence of regular Lagrangians.
method Weinstein Lefschetz fibrations with a hypothesis.
result Existence of regular Lagrangians can be characterized by Lefschetz fibrations.
The study constructs and analyzes new symplectic 4-manifolds from hyperelliptic Lefschetz fibrations.
problem Understanding and constructing new symplectic 4-manifolds.
method Explicitly constructing Lefschetz pencils on hyperelliptic Lefschetz fibrations.
result Infinite families of symplectic 4-manifolds are diffeomorphic to fiber sums of standard hyperelliptic Lefschetz fibrations.
New proof of indecomposable minimal genus-2 Lefschetz fibration.
problem Existence of indecomposable minimal genus-2 Lefschetz fibration.
method Proved existence by counterexample to the converse of Usher's theorem.
result Existence of an indecomposable minimal genus-2 Lefschetz fibration.
In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identify…
A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and…
Study calculates Poisson cohomology of broken Lefschetz fibrations.
problem Computing Poisson cohomology of broken Lefschetz fibrations.
method Calculates cohomology at fold and Lefschetz singularities, adapting techniques for Sklyanin algebra.
result Compact formulas for Poisson coboundary operator in 4 dimensions.
Characterizes Lefschetz fibrations and multisections via monodromy factorizations.
problem Classifying Lefschetz fibrations and multisections.
method Using monodromy factorizations and Hurwitz equivalence.
result Two Lefschetz fibrations with multisections are isomorphic if and only if their monodromy factorizations are related by a finite collection of modifications.
In this paper we study the behaviour of the Lefschetz property under the blow-up construction. We show that it is possible to reduce the dimension of the kernel of the Lefschetz map if we blow up along a suitable submanifold satisfying the Lefschetz property. We use that, together with results about Massey products, to…
This paper provides a new proof of the Lefschetz fixed point formula using groupoids.
problem The Lefschetz fixed point formula for elliptic complexes.
method Defines a relative tangent groupoid and pseudodifferential calculi to prove the formula.
result A new proof of the Lefschetz fixed point formula using groupoids.
Simplicial spheres have a weak Lefschetz property in characteristic 2.
problem Proving the weak Lefschetz property for simplicial spheres.
method Using bistellar moves to show the property is preserved.
result The weak Lefschetz property is preserved by bistellar moves for PL-spheres.
Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for hyperelliptic Lefschetz fibrations, and show that any hyperelliptic Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations.
We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz …