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 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.
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 …
Paper shows examples of almost Kähler manifolds satisfying Hard Lefschetz but not Betti-Hodge equality.
problem Understanding the Hard Lefschetz condition in almost Kähler manifolds.
method Examples and counterexamples of compact almost Kähler manifolds.
result The Hard Lefschetz condition does not imply the equality between Betti and Hodge numbers in almost Kähler manifolds.
The study constructs symplectic solvmanifolds satisfying the hard-Lefschetz condition.
problem Developing an analogue of Hodge theory for symplectic manifolds.
method Analyzing specific Lie algebras and their associated Lie groups, exploiting connections with Kneser graphs.
result Examples of almost-Kähler solvmanifolds satisfying the hard-Lefschetz condition are constructed.
Symplectic structures simplified for compact manifolds.
problem Locally conformally symplectic structures on compact manifolds.
method Symplectic analogue of Vaisman's theorem.
result Locally conformally symplectic structures become globally symplectic.
New symplectic structures found on complex manifolds without Kähler structures.
problem Finding symplectic structures on complex manifolds without Kähler structures.
method Constructing explicit lattices and cohomological computations.
result Compact complex manifolds with symplectic structures satisfying the Hard Lefschetz Condition.
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…
New findings show some symplectic solvmanifolds fail hard-Lefschetz condition.
problem Characterizing symplectic solvmanifolds that do not satisfy the hard-Lefschetz condition.
method Detailed analysis of Lie algebra cohomology groups and construction of lattices.
result Symplectic solvmanifolds with non-semisimple actions fail the hard-Lefschetz condition at degree 1 or 2.
The paper explores Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds.
problem Analyzing harmonic forms and Hodge decomposition on almost Kähler manifolds.
method Using Hodge decomposition and the Hard Lefschetz Condition to study almost Kähler manifolds.
result The spaces of harmonic forms have the Hodge decomposition and the Hard Lefschetz Condition is satisfied.
In [8] the authors introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of new boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with…
For any two continuous maps f,g between two solvmanifolds of same dimension satisfying the Mostow condition, we give a technique of computation of the Lefschetz coincidence number of f,g. This result is an extension of the result of Ha, Lee and Penninckx for completely solvable case.
The Hard Lefschetz Theorem extends to certain Kähler Lie Algebroids with ellipticity.
problem Extending the Hard Lefschetz Theorem to Kähler Lie Algebroids.
method Analyzing a specific class of Kähler Lie Algebroids with ellipticity requirements.
result A class of Kähler Lie Algebroids satisfy the Hard Lefschetz Theorem with ellipticity.
Constructing compact non-Kähler manifolds with and without the Hard Lefschetz Condition
problem Symplectic non-Kähler manifolds
method One-parameter family of symplectic forms on orbifold
result Symplectic manifolds with HLC and non-HLC structures
The paper derives a formula for Lefschetz number of a geometric endomorphism.
problem Calculating the Lefschetz number for a singular foliation.
method Adapting the Atiyah-Bott theorem to a geometric endomorphism of a complex of LT-parallel sections. result A formula for the Lefschetz number of a geometric endomorphism.
Proves unique symplectic Lefschetz fibration from Morse functions.
problem Mapping Morse functions to symplectic Lefschetz fibrations.
method Homotopically unique complex-valued symplectic Lefschetz fibration on cotangent bundles.
result Existence and uniqueness of symplectic Lefschetz fibrations.
The paper explores conditions for sections in Lefschetz fibrations and bundles over 2-complexes.
problem Conditions for sections in Lefschetz fibrations and bundles over 2-complexes.
method Algebraic criteria and topological conditions.
result Complete algebraic criterion for sections in Lefschetz fibrations over the disk.
We construct two types of non-holomorphic Lefschetz fibrations over S2 with (−1)-sections ---hence, they are fiber sum indecomposable--- by giving the corresponding positive relators. One type of the two does not satisfy the slope inequality (a necessary condition for a fibration to be holomorphic) and has a simpl…
Develops analytic methods for Lefschetz and Morse theories on stratified pseudomanifolds.
problem Analytic framework for Lefschetz and Morse theories on stratified pseudomanifolds.
method Heat kernel and Witten deformation based techniques for global and local Lefschetz numbers and Morse polynomials.
result Formulas for Lefschetz numbers and Morse polynomials as supertraces over cohomology groups of Hilbert complexes.
The paper proves rigidity of certain Dirac operators using theta functions.
problem Rigidity of twisted Dirac operators on specific bundles.
method Lefschetz formula, Atiyah-Bott localization, theta function properties.
result Lefschetz numbers are constant under certain conditions, proving operator rigidity.
In the literature, there are two different versions of Hard Lefschetz theorems for a compact Sasakian manifold. The first version, due to Kacimi-Alaoui, asserts that the basic cohomology of a compact Sasakian manifold satisfies the transverse Lefschetz property. The second version, established far more recently by Capp…
The abstract explores analogues of Hodge theory in Lie algebroids.
problem Exploring analogues of Hodge theory in Lie algebroids.
method Establishing equivalence of conditions and applying algebraic theory to geometric setting.
result Equivalence of analogues of Hodge theory conditions in Lie algebroids.
The paper studies extPin±-structures on non-oriented 4-manifolds via Lefschetz fibrations.
problem Understanding extPin±-structures on non-oriented 4-manifolds and Lefschetz fibrations. method Extending work on orientable settings, the paper uses Lefschetz fibrations to analyze extPin±-structures on non-orientable 4-manifolds and vector bundles. result The paper provides existence results of extPin+ and extPin−-structures on closed non-orientable 4-manifolds and Lefschetz fibrations over the 2-sphere. 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 …
Study defines and proves Hard Lefschetz Property for S^3-actions.
problem Formulating Hard Lefschetz Property for S^3-actions.
method Defined and proved two versions of HLP for S^3-actions.
result Agreement of two versions of HLP for 3-Sasakian manifolds.
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…
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.
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.
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.
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.
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 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.
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…
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.
We give a maximal set of disjoint (−1)-sections of the well-known Lefschetz fibration constructed by Matsumoto, Cadavid and Korkmaz. In fact, we obtain several such sets for a fixed genus, which implies that the Matsumoto-Cadavid-Korkmaz Lefschetz fibration has more than one supporting minimal Lefschetz pencils. We a…
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…
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.
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.
A condition for the existence of false gauge field copies in terms of the Lefschetz number of a certain differential operator is presented.
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 …
Proves existence of Lefschetz fibrations with arbitrary slopes.
problem Finding Lefschetz fibrations with specific slopes.
method Proves existence for rational numbers in (2,8) with large genus.
result Arbitrary slopes (r in (2,8)) are possible for genus-g Lefschetz fibrations.
The study shows that symplectic Lefschetz fibrations can have infinitely many sections.
problem The finiteness of sections in Lefschetz fibrations.
method General criterion and examples for symplectic Lefschetz fibrations with infinitely many sections.
result Symplectic Lefschetz fibrations can have infinitely many homologically distinct sections.
Spin Lefschetz fibrations can represent any group and lattice point.
problem Representing groups and lattice points using Lefschetz fibrations.
method Proving any finitely presented group and admissible lattice point can be realized as a spin Lefschetz fibration.
result Spin Lefschetz fibrations can represent any group and lattice point.
Authors construct symplectic Lefschetz pencils on complex projective plane.
problem Construct symplectic Lefschetz pencils on complex projective plane.
method Differential topological construction, analogous to holomorphic pencils.
result Explicit monodromy factorization and topological construction for d=4.