18 problems

Algebraic homomorphism conjecture. The homomorphism property holds for all classes $d6fc,d6fd\in\mathsf{CH}^*(\mathcal{A}_g)$ defined over $d53e$.

Strong homomorphism conjecture. The map $d52d^*$ is a homomorphism of $d54a$-algebras.

Let $g$ be an integer with $g\geq 8$. Let ${\mathcal H}_g$ be the hyperelliptic locus in ${\mathcal M}_g$, let $j_h:{\mathcal H}_g\to {\mathcal A}_g$ be the restriction of the Tore…

Zeta-extension conjecture. The mixed Hodge structure on $H^{2g}_c(\mathcal{A}_g)$ has a subquotient isomorphic to such an extension whose class is given by a nonzero rational multi…

Tensor-algebra injectivity conjecture. This induced map is injective. The later sections provide various kinds of evidence for this conjecture, while the equivalent formulation in…

Let $F$ be a totally real number field of degree $e$, let $m$ be a positive integer, and set $g=me$. Let $\overline{\mathcal A}_F$ be the compactification of the Hilbert modular cy…

For an arithmetic moduli space of abelian varieties, let the bundle of modular forms denote the relevant metrized Hodge bundle, and let its arithmetic degree be the corresponding a…

Let $\overline{\mathcal{A}}_g$ be the moduli space of principally polarized abelian varieties, let $\overline{\mathcal{X}}_g$ be the universal abelian variety, and let…

Nef-cone conjecture. The cone of nef codimension-two classes is

For any integer $g\ge 3$, consider the perfect-cone compactification ${{\mathcal A}_{g}^{\operatorname{Perf}}}$ and the five surfaces obtained by embedding the surfaces $S_A$,…

Let $\overline{{\mathcal A}_{3}}$ be the toroidal compactification of the moduli space of complex principally polarized abelian threefolds. Let $S_A$, $S_F$, and $S_D$ be the three…

Let … lambda{2i+1}$and the classes of boundary strata, and no polynomial in the$Dm$, is identically zero for$g$ sufficiently large. This predicts algebraic independence of the i…

Let ${{\mathcal A}_{g,\operatorname{smooth}}^{\operatorname{Perf}}}$ denote the smooth locus of the perfect cone toroidal compactification, and let…

Let $\overline{\mathcal{A}}_5$ be a compactification of the moduli space of principally polarized abelian fivefolds, and let $s'(\overline{\mathcal{A}}_5)$ denote its moving slope.…

$\Gamma_{00}$ conjecture. If

Theta-multiplicity conjecture.

Debarre's codimension conjecture. For any $1\leq k\leq g-3$,

Vanishing conjecture. The intersection numbers $a_N^{(g)}$ for any $N$ vanish unless $G-N=k(k+1)/2$ for some $k\le g$. The claim extends the vanishing pattern established in the pa…