18 problems

Beilinson–Hodge conjecture. The fundamental class map $\mathrm{cl}^{p,m}_{\pi}$ is surjective. This predicts that every Hodge class in $\mathrm{Hg}(H^1(\mathcal{U},\mathbf{H}(p)))$…

Griffiths–Green conjecture. For every $\alpha\in Hg^p(X)$, the associated admissible normal function $\nu_\alpha$ is singular on $\bar{S}=|L^{\otimes d}|$ for some $d>0$. This conj…

Green–Griffiths conjecture. The zero locus $Z(\nu)$ is an algebraic subvariety of $X$.

Let $S$ be a quasi-projective complex variety, let $f\colon X\to S$ be a smooth family of projective varieties of relative dimension $d$, and let $N$ be a relatively ample line bun…

Let $\Delta$ satisfy the condition referred to as, let $F$ be integrally tempered, let $\bm{\mathcal{M}}$ be the relevant moduli space, let $\bm{\nu}(\bm{\mathcal{M}})$ be the imag…

Vanishing conjecture. This set is zero unless the highest weight is

Let $F\subseteq\mathbb C$ be a subfield, and let $\nu$ be an algebraically $F$-motivated (respectively, $F$-motivated) normal function on a complex algebraic manifold, meaning that…

Let $\mathsf B$ be a complex algebraic manifold, let $\mathcal H$ be a variation of pure negative weight integral Hodge structures over $\mathsf B$, and let…

Let $X$ be a smooth complex projective variety, let $L\to X$ be a very ample line bundle, and let $\zeta$ be a non-torsion, primitive Hodge class of type $(n,n)$ on $X$. For the as…

Let an algebraic cycle on the mirror quintic determine a truncated normal function, and let its A-model expansion be expressed in the mirror-map coordinate $q$. The D-logarithm is…

Let $f:T\to\overline{\mathcal M}_{g,1}$ be a morphism from a projective curve whose image is not contained in $\Delta_0$. Let $\mathcal K$ be the normal function section of…

Let $f:T\to\overline{\mathcal M}_g$ be a morphism from a projective curve whose image is not contained in $\Delta_0$, and let $J_T$ denote the associated jumping divisor. Weak jump…

Let $\mathbb F\subset\mathbb C$ be finitely generated over $\overline{\mathbb Q}$, let $S/\mathbb F$ be a smooth quasi-projective variety of dimension $D$, and let…

Let $X$ be a smooth complex projective variety of dimension $2m$, let $L$ be a very ample line bundle, and let $\zeta$ be a primitive integral Hodge class of type $(m,m)$. Let…

Let $X$ be a smooth projective variety of dimension $2m$, let $L$ be a very ample line bundle on $X$, and let $H_{\mathrm{prim}}^{m,m}(X,\mathbb{Z}(m))$ denote the primitive integr…

Let $S$ be a smooth complex algebraic variety, let $\mathcal H$ be a variation of pure Hodge structure on $S$, and let $\nu$ be an admissible normal function, with zero locus…

GG conjecture. For every non-torsion primitive Hodge class $\zeta$, there is an integer $k$ and a resolution $f:S\to P=|\mathcal L^k|$ of the discriminant locus such that, for any…

Singularity conjecture. For every non-torsion primitive Hodge class $\zeta$, there is an integer $k$ such that $\zeta$ is singular on $|\mathcal L^k|$.