Related papers: A decomposition theorem for Lefschetz modules

The Alesker product turns the space of smooth translation-invariant valuations on convex bodies into a commutative associative unital algebra, satisfying Poincar\'e duality and the hard Lefschetz theorem. In this article, a version of the…

We give a geometric proof of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. The method rests on new…

Recently Tewari and van Willigenburg constructed modules of the 0-Hecke algebra that are mapped to the quasisymmetric Schur functions by the quasisymmetric characteristic and decomposed them into a direct sum of certain submodules. We show…

We develop a Helmholtz-like theorem for differential forms in Euclidean space $E_{n}$ using a uniqueness theorem similar to the one for vector fields. We then apply it to Riemannian manifolds, $R_{n}$, which, by virtue of the…

We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove…

Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact K\"ahler manifold or on the…

We discuss a 'Lefschetz filtration' of $\Lambda^*(\mathbb Z^{2g})$ and prove its subquotients are isomorphic as $\text{Sp}(2g)$-modules to primitive subspaces $P^k(\mathbb Z^{2g})$. This gives a sort of integral version of the Lefschetz…

We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations for the Chow ring, recovering…

The Lefschetz algebra $L^*(X)$ of a smooth complex projective variety $X$ is the subalgebra of the cohomology algebra of $X$ generated by divisor classes. We construct smooth complex projective varieties whose Lefschetz algebras do not…

Given a bounded subanalytic submanifold of $\mathbb{R}^n$, possibly admitting singularities within its closure, we study the cohomology of $L^p$ differential forms having an $L^p$ exterior differential (in the sense of currents) and…

Compact K\"{a}hler manifolds satisfy several nice Hodge-theoretic properties such as the Hodge symmetry, the Hard Lefschetz property and the Hodge-Riemann bilinear relations, etc. In this note, we investigate when such nice properties hold…

There are significant differences between Helmholtz and Hodge's decomposition theorems, but both share a common flavor. This paper is a first step to bring them together. We here produce Helmholtz theorems for differential 1-forms and…

We investigate the Weak Lefschetz Properties for modules whose minimal free resolutions are given by generalized Kosuzl complexes in dimension three through a careful study of their Betti numbers and the symmetry and unimodality of their…

Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base non-archimedean field $K$ is of residue…

We study Soergel modules for arbitrary Coxeter groups. For infinite Coxeter groups, we show that the homomorphisms between Soergel modules are in general more than those coming from morphisms of Soergel bimodules. This result provides a…

The Hard Lefschetz theorem for intersection cohomology of nonrational polytopes was recently proved by K. Karu [Ka]. This theorem implies the conjecture of R. Stanley on the unimodularity of the generalized $h$-vector. In this paper we…

In this paper, we study compatible Leibniz algebras. We characterize compatible Leibniz algebras in terms of Maurer-Cartan elements of a suitable differential graded Lie algebra. We define a cohomology theory of compatible Leibniz algebras…

Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W := \wedge (V \oplus…

As a corollary of nonabelian Hodge theory, Simpson proved a strong Lefschetz theorem for complex polarized variations of Hodge structure. We show an arithmetic analog. Our primary technique is $p$-adic nonabelian Hodge theory. Conditional…

Taking a compact K\"{a}hler manifold as playground, we explore the powerfulness of Hodge index theorem. A main object is the Lorentzian classes on a compact K\"{a}hler manifold, behind which the characterization via Lorentzian polynomials…