34 problems

Brylinski's conjecture. Every $k$-de Rham cohomology class contains a symplectic harmonic representative.

Let $M$ be a closed symplectic manifold admitting a Hamiltonian torus action with isolated fixed points. The Hard Lefschetz conjecture. The manifold $M$ satisfies the Hard Lefschet…

Let $V$ be an $n$-dimensional Euclidean vector space and let $G$ be a compact group acting transitively on the unit sphere of $V$. Write $Val_i^G(V)$ for the degree-$i$ component o…

Hard Lefschetz conjecture for Lawson homology. The map $h^k\bullet$ is injective for $k\geq1$. This conjecture is presented as a Lawson-homology analogue of hard Lefschetz; the sup…

Let $M$ be a simply connected compact symplectic manifold. The Hard Lefschetz condition is the standard Lefschetz isomorphism condition associated with the symplectic form, and a c…

Let $V$ be an $n$-dimensional Euclidean vector space, and let $V_1\in Val_1^\infty(V)$ denote the first intrinsic volume. For each integer $i$ with $0\leq i<n/2$, consider multipli…

Choose a generic line and the associated graded algebra ${\mathbb R}[T]$, with $T$ of degree $2$, and let ${\mathscr B}(x)^{\delta y}_T$ be the corresponding module. The Hard Lefsc…

Choose a generic line in $V^\ast$ and let $\rho:S\to{\mathbb R}[T]$ be the corresponding graded surjection, with $T$ of degree $2$. For an $S$-module $M$, set…

Hard Lefschetz conjecture. For all $k$, the Lefschetz operator induces an isomorphism

Bressler–Lunts hard Lefschetz conjecture. For every $i<d/2$, this map is a bijection. This conjecture proposes the hard Lefschetz property for intersection cohomology of arbitrary…

Beckmann's conjecture. The element $\operatorname{obs}_E(\eta)$ has the hard Lefschetz property for the Yoneda product on $\operatorname{Ext}^{\bullet}(E,E)$.

Shenfeld–van Handel's conjecture. One has

Variational filtered convergent log hard Lefschetz conjecture. For the relevant integers $i$, the cup product

Let $M=(E,\mathcal I)$ be a matroid of rank $d$, let $\A(M)$ be its associated Gorenstein ring, and let $a\in\mathbb R^E_{>0}$. Kähler-package conjecture. The ring $\A(M)$ satisfie…

Let $M=(E,\mathcal I)$ be a matroid of rank $d$, let $\A(M)$ be its associated Gorenstein ring, and let $a\in\mathbb R^E_{>0}$. All-degree Hard Lefschetz conjecture. The ring…

Let $X$ be a smooth projective variety of dimension $n$, and let $\mathfrak{L}=(L_1,\ldots,L_{n-2})$ be a collection of nef classes on $X$. Define … and let … where the span is ove…

Supercritical collection conjecture. One has

Let $\Delta$ be a homology sphere, and let $\mathbf{k}[\Delta]$ be its face ring over a field $\mathbf{k}$. The hard Lefschetz theorem concerns the existence of a suitable Artinian…

Intersection curious hard Lefschetz conjecture. There exists a class $\alpha\in H^2(M_{\mathrm{B}}(C,G))$ such that, for every $k$, cup product induces isomorphisms

Let $f\colon X\to S$ be in the log convergent setting of the paper, with relative dimension $d$, and let $P$ denote the weight filtration. Let $N$ be the monodromy parameter and as…

Let $X/S_0$ be a proper SNCL scheme with structural morphism $f\colon X\longrightarrow S_0\overset{\subset}{\longrightarrow} S$, and suppose that the relative dimension of…

Let $M$ be a compact completely solvable manifold, and let $\omega$ be a symplectic structure on $M$. The Hard Lefschetz invariance conjecture. If $\omega$ satisfies the Hard Lefsc…

Log hard Lefschetz conjecture. If $\mathcal{L}$ is ample, then $L^r$ is an isomorphism for all $0\leq r\leq n$.

Standard conjecture of Lefschetz type. For all $n\leq d$, there is an isomorphism

Let $X$ be a smooth projective variety of dimension $n$, and let $L$ be an ample line bundle. Let $N^1H^i(X,\mathbb{Q})$ denote the first coniveau piece of cohomology. Hard Lefsche…