Examples of non-Kähler Hamiltonian circle manifolds with the strong Lefschetz property

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 counterexamples to the question whether the strong Lefschetz property descends to the symplectic quotient. We also give examples of Hamiltonian strong Lefschetz circle manifolds which have a non-Lefschetz fixed point submanifold. In addition, we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a strong Lefschetz manifold. We then use it to show the existence of Lefschetz four-manifolds with non-Lefschetz finite covering spaces.     

Hyperplane sections and the subtlety of the Lefschetz properties

The weak and strong Lefschetz properties are two basic properties that Artinian algebras may have. Both Lefschetz properties may vary under small perturbations or changes of the characteristic. We study these subtleties by proposing a systematic way of deforming a monomial ideal failing the weak Lefschetz property to an ideal with the same Hilbert function and the weak Lefschetz property. In particular, we lift a family of Artinian monomial ideals to finite level sets of points in projective space with the property that a general hyperplane section has the weak Lefschetz property in almost all characteristics, whereas a special hyperplane section does not have this property in any characteristic.     

On decomposing monomial algebras with the Lefschetz properties

We introduce a general technique for decomposing monomial algebras which we use to study the Lefschetz properties. In particular, we prove that Gorenstein codimension three algebras arising from numerical semigroups have the strong Lefschetz property, and we give partial results on monomial almost complete intersections. We also study the reverse of the decomposition process – a gluing operation – which gives a way to construct monomial algebras with the Lefschetz properties.     

Joint torsion of several commuting operators

We introduce the notion of joint torsion for several commuting operators satisfying a Fredholm condition. This new secondary invariant takes values in the group of invertibles of a field. It is constructed by comparing determinants associated with different filtrations of a Koszul complex. Our notion of joint torsion generalize the Carey–Pincus joint torsion of a pair of commuting Fredholm operators. As an example, under more restrictive invertibility assumptions, we show that the joint torsion recovers the multiplicative Lefschetz numbers. Furthermore, in the case of Toeplitz operators over the polydisc we provide a link between the joint torsion and the Cauchy integral formula. We will also consider the algebraic properties of the joint torsion. They include a cocycle property, a triviality property and a multiplicativity property. The proof of these results relies on a quite general comparison theorem for vertical and horizontal torsion isomorphisms associated with certain diagrams of chain complexes.     

The strong Lefschetz property for complete intersections defined by products of linear forms

We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules.

     

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.     

Ordinary Curves,Webs and the Ubiquity of the Weak Lefschetz Property

In this short note we establish a close relationship between two a priori unrelated problems: (1) the existence of ordinary curves in ? ⁿ and, (2) the existence of artinian graded algebras satisfying the Weak Lefschetz Property.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Hodge structures on cohomology algebras and geometry

We study restrictions on cohomology algebras of compact Kähler manifolds, imposed by the presence of a polarized Hodge structure on cohomology groups, compatible with the cup-product, but not depending on the h ^p,q numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the formality condition, the Lefschetz property and having commutative or trivial π ₁, but not having the cohomology algebra of a compact Kaehler manifold. We also prove a stability theorem for these restrictions : if a compact Kähler manifold is homeomorphic to a product X × Y, with one summand satisfying b ₁ = 0, then the cohomology algebra of each summand carries a polarized Hodge structure.     

Lefschetz decomposition for de Rham cohomology on weakly Lefschetz symplectic manifolds

For a compact symplectic manifold which is s-Lefschetz which is weaker than the hard Lefschetz property, we prove that the Lefschetz decomposition for de Rham cohomology also holds.