76 problems

Let $C$ be a smooth projective curve of genus $g\geq 1$, and let $r$ and $d$ be coprime integers. Consider the moduli space of stable vector bundles on $C$ of rank $r$ and degree…

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

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 Calabi–Yau threefold with $pi_1(X)=0$. For $c\in H^4(X,\mathbf{Z})$, call $c$ a DRY class if there exist an ample class $H\in H^2(X,\mathbf{R})$ and an integer…

Shifted-binomial log-concavity conjecture. The shifted polynomial

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

Let $M/\mathbf{C}$ be a smooth projective variety, and let $E$ be a vector bundle on $M$ equipped with a flat connection. For each $i$, write $c_i(E)$ for its Chow-theoretic $i$-th…

Let $[(X,h)]\in {\mathscr U}^{il}_e(2n)$, where ${\mathscr U}^{il}_e(2n)$ is the open subset of the moduli space of polarized hyperkähler varieties of type $K3^{[n]}$ for which the…

Generalized lower-bound conjecture. One should have

Polynomiality conjecture. The quantity $c_1.c_2$ should be a polynomial in $p$ of a certain type, with its value known for small $p$.

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…

Existence conjecture. There exists a stable reflexive sheaf $V$, with respect to some ample class, such that

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,…

Let ${\cal M}_{\Lambda}(n,d)$ be the moduli space considered in the paper, with rank $n$ and genus $g$, and let $c_r$ denote its $r$th Chern class. Earl–Kirwan's conjecture. For…

Forest-basis conjecture. The dimension of $A_n$ as a vector space equals the total number of forests on $n$ marked vertices, and $A_n$ has a natural monomial basis whose elements a…

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…

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…

Hunt's Chern-ratio conjecture. There are no smooth threefolds whose Chern ratios lie above the curve

Keski-Vakkuri–Wen conductance conjecture. The conductance of the system is independent of the genus and equal to

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

Extension conjecture. The displayed inequality still holds when $\alpha_1,\ldots,\alpha_{n-2}$ are merely psef.

Let $M_{d,\chi}$ be the moduli space of one-dimensional sheaves on $\mathbb P^2$, with perverse filtration $P_\bullet H^*(M_{d,\chi})$ and tautological Chern filtration…