21 problems

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

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 $a\in\mathbb R^E_{>0}$. 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

Hard Lefschetz conjecture. For every $J\subseteq[n]$, the quotient ring $A_J$ satisfies the Hard Lefschetz property.

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 $0\leq r\leq n$.

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…

Let $X$ be a smooth projective variety of dimension $n$, and let $L$ be an ample line bundle. Write $A^j_{AJ}(X)_{\Q}$ for the subgroup of Abel–Jacobi trivial codimension-$j$ cycle…

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 $\mathcal{M}_\B$ be the relevant character variety, let $\dim=\dim(\mathcal{M}_\B)$, and let $\alpha\in W_4H^2(\mathcal{M}_\B)$, where $W_\bullet$ is the weight filtration. Cur…

Hard Lefschetz conjecture for Chow groups. This map is injective. The conjecture is a Chow-theoretic analogue of the hard Lefschetz theorem on cohomology; surjectivity generally fa…

Let $w\in{\widehat{\mathcal W}}$ and suppose that $\operatorname{char} K>N(w)$. For every $x\le w$, let … be the indicated map between the costalk and stalk of the reduced Braden–M…