arXiv research

The Hard Lefschetz Theorem extends to certain Kähler Lie Algebroids with ellipticity.

Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.

This short review is the result of a minicourse at the Sapienza University of Rome the author gave about the proof of the $g$-theorem. We review the hard Lefschetz theorem for simplicial spheres, as well as the theory at its core: perturbations of maps, biased Poincaré pairings and a cobordism argument that relates the…

We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliatio…

The paper proves a Hard Lefschetz Theorem for push-forwards of polarized twistor modules.

We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral geometry. Then we formulate a conjectural analogue of this result for odd valuations.

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…

We provide a simpler proof of the hard Lefschetz Theorem for face rings of PL spheres: While the algebraic theory remains the same, we replace the geometric constructions by Pachner's Theorem. This simplifies the reasoning for an important special case of the main result of the first author in arxiv:1812.10454, and alr…

We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) mani…

Study properties of balanced hyperbolic compact complex manifolds.

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new…

Symplectic structures simplified for compact manifolds.

We find a family of five dimensional completely solvable compact manifolds that constitute the first examples of $K$-contact manifolds which satisfy the Hard Lefschetz Theorem and have a model of Tievsky type just as Sasakian manifolds but do not admit any Sasakian structure.

The paper proves a generalized Lefschetz duality for a specific type of manifold.

The paper defines and proves a new property for symplectic manifolds.

Extends Donaldson's techniques to symplectic orbifolds, proving existence of sections and computing cohomology.

Study compact symplectic solvmanifolds' hard Lefschetz property.

Using the Hard Lefschetz Theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions five and seven, respectively

For a simply connected solvable Lie group G with a cocompact discrete subgroup Γ, we consider the space of differential forms on the solvmanifold G/Γ with values in certain flat bundle so that this space has a structure of a differential graded algebra(DGA). We construct Sullivan's minimal model of this DGA. This resul…

Proves cohomology theorems for tropical varieties.

The study constructs symplectic solvmanifolds satisfying the hard-Lefschetz condition.

New symplectic structures found on complex manifolds without Kähler structures.

Paper shows examples of almost Kähler manifolds satisfying Hard Lefschetz but not Betti-Hodge equality.

We introduce a Lefschetz filtration for integer cohomology and explore its applications.

The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometrical, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on holomorphic functions in one or more variables and the definition and first examples of …

For a Lie group $G=\R^{n}\ltimes_φ\R^{m}$ with the semi-simple action $φ:\R^{n}\to {\rm Aut}(\R^{m})$, we show that if $Γ$ is a finite extension of a lattice of $G$ then $K(Γ, 1)$ is formal. Moreover we show that a compact symplectic aspherical manifold with the fundamental group $Γ$ satisfies the hard Lefschetz proper…

Constructing compact non-Kähler manifolds with and without the Hard Lefschetz Condition

New findings show some symplectic solvmanifolds fail hard-Lefschetz condition.

Extends Hard Lefschetz Property to isometric flows and shows equivalence.

Study defines and proves Hard Lefschetz Property for S^3-actions.

The paper explores Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds.

We study (i) asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic bundles and polarized wild pure twistor $D$-modules. As an application, we show the hard Lefschetz theorem for algebraic semis…

For a symplectic manifold $(M,ω)$, not necessarily hard Lefschetz, we prove a version of the Merkulov $dδ$--lemma. We also study the $dδ$--lemma and related cohomologies for compact symplectic solvmanifolds.

Study on twisted Dolbeault cohomology in Kähler foliations.

The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard…

It is known that the hard Lefschetz action, together with Kähler identities for Kähler (resp. hyperkähler) manifolds, determines a $\mathfrak{su}(1,1)_{sup}$ (resp. $\mathfrak{sp}(1,1)_{sup}$) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also gener…

The paper develops $L^2$-Hodge theory on almost Kähler manifolds and proves the Hopf conjecture.

Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among o…

Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data…

In this paper we obtain theorems of Barth-Lefschetz type in Sasakian geometry. As corollaries, this implis connectedness principle and Frankel's type theorem.

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 …

Study cohomologies of complex manifolds with symplectic forms and their stability.

T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irr…

The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integ…

The study shows that symplectic Lefschetz fibrations can have infinitely many sections.

Using an approach based on the heat kernel we prove an Atiyah-Bott-Lefschetz theorem for the $L^2-$Lefschetz numbers associated to an elliptic complex of cone differential operators over a compact manifold with conical singularities. We then apply our results to the case of the de Rham complex.

We construct real polarizable Hodge structures on the reduced leafwise cohomology of Kähler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's Kählerian analogue of the Weil conjectures carries over as well. Generalizing a construction of …

New space for valuations in non-Archimedean setting with duality properties.