43 problems

For $n=2$, write the Chern class in the Schur basis as … and expand each coefficient in the binomial basis: … A polynomial is binomially positive when all its binomial-basis coeffi…

Positivity conjecture. For all ${\underline\alpha}$, $c_{\text{SM}}({\mathbb{S}}({\underline\alpha})^\circ)\in A_*({\mathbb{S}}({\underline\alpha}))$ is represented by an effective…

Let $G$ be a finite group, or more generally a linear algebraic group, acting on a variety $X$, and let $X\to Y:=X/G$ be the quotient. A $G$-variety is a variety with a $G$-action,…

Shifted-binomial log-concavity conjecture. The shifted polynomial

Strong binomiality conjecture. For each fixed $k$ and each partition $\lambda\vdash k$, the coefficient $f_\lambda(n,d)$ is a polynomial in finitely many binomial coefficients of t…

Separate polynomiality conjecture. For each fixed $k$ and each partition $\lambda\vdash k$, the coefficient $f_\lambda(n,d)$ is polynomial in $d$ for fixed $n$, and polynomial in…

Chern-class vanishing conjecture. All Chern classes of $\mathfrak{E}$ vanish.

Bruzzo and Graña Otero conjecture. The Higgs bundle $\mathfrak{E}$ is semistable with respect to some polarization $H$ and $\Delta(E)=0$.

Strong conjecture. The class $c_k\big(\Pol^d(\mathbb{C}^n)\big)$ is of the form $P(B_1,\dots,B_s)$, where $P$ is a polynomial. This strengthens the weak conjecture by proposing an…

Let $\Pol^d(\mathbb{C}^n)$ denote the relevant polynomial-space bundle, and let $c_k\big(\Pol^d(\mathbb{C}^n)\big)$ be its $k$th Chern class. Weak conjecture. For any fixed $k$ and…

Let $X$ be a projective hyper-Kähler manifold, and let $C^*$ be the subalgebra of $CH^*(X)$ generated by divisors and Chern classes. Voisin's conjecture. The cycle class map is inj…

Tropical Chern-class adjunction conjecture. For every $k\in\mathbb{Z}_{\geq 0}$,

Let $R$ be a smooth affine algebra of Krull dimension $d\geq 3$ over an algebraically closed field. Let $P$ be a projective $R$-module of rank $d-1$. Murthy's splitting conjecture.…

Let $X$ be a compact tropical manifold of dimension $n$. The Todd class of $X$ is obtained from the formal power series in its Chern classes by substituting…

Let $X$ be a smooth complex variety, let $F$ be the bundle appearing in the tautological integral of Theorem, and write that integral as a polynomial in the Chern classes of $F$ an…

Extension conjecture. The vanishing assertion $c_s(chat{E})\cap T=0$ for $s>\operatorname{rank}E$ and the positivity assertion that $c_s(chat{E})$ is a closed positive $(s,s)$-curr…

Assume that the base field $k$ has characteristic zero. Let $f:\tilde X\to X$ be the morphism under consideration, and let $\mathcal F$ be a rank $r$ vector bundle on $\tilde X$. W…

Alternating-sign conjecture. For every $u,v,w\in W$,

Let $X$ be a compact Kähler manifold, let $\mathcal{E}$ be a torsion-free coherent sheaf of rank $r$, and let $\alpha_1,\ldots,\alpha_{n-2}$ be Kähler classes on $X$. A sheaf is ge…

Generalized Fujiki positivity conjecture.

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $k$ of characteristic zero, let $H$ be an ample divisor, and let $(E,\theta)$ be a slope…

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $k$, let $H$ be an ample divisor on $X$, and let $(E,\theta)$ be a slope $H$-stable Higgs…

Let $X$ be a smooth complex quasi-projective variety with base point $x$, and let … be Lie irreducible with quasi-unipotent monodromy at infinity. Let $(V_\rho,\nabla_\rho)$ be the…

Let $X$ be a smooth complex quasi-projective variety and let $\mathbb V$ be a local system of geometric origin on $X$. Let $(V,\nabla)$ be the associated algebraic flat vector bund…

Let $X$ be a smooth complex quasi-projective variety with base point $x$, and let … be a properly rigid representation with quasi-unipotent monodromy at infinity. Let…