arXiv research

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

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

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

Study compact symplectic solvmanifolds' hard Lefschetz property.

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.

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

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

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…

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

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…

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 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…

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.

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…

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

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.

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 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.

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…

Study properties of balanced hyperbolic compact complex manifolds.

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…

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

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

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

Proves cohomology theorems for tropical varieties.

New operators in Khovanov-Rozansky homology exhibit symmetry.

A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is eas…

We study properties concerning decomposition in cohomology by means of generalized-complex structures. This notion includes the $\mathcal{C}^\infty$-pure-and-fullness introduced by Li and Zhang in the complex case and the Hard Lefschetz Condition in the symplectic case. Explicit examples on the moduli space of the Iwas…

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of $sl(2,\mathbb{C})$, generalizing the well known structure on the harmonic f…

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…

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 …

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…

Abstract study of HKT manifolds, proving Hodge theory and formality properties.

Study on twisted Dolbeault cohomology in Kähler foliations.

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

We study the Morse-Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In…

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…

We discuss how quantitative cohomological informations could provide qualitative properties on complex and symplectic manifolds. In particular we focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since they represent useful tools in studying non Kähler geometry. We give an overview on the comparis…

We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang, in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by dis…

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