210 problems

For every integer $m\ge 1$, let $X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbb{P}^5$ be the Fermat fourfold. The Hodge conjecture asserts that every rational Hodge…

Let $D$ be a Mumford--Tate domain parameterizing pure, effective, weight $\mathsf{w}$, $Q$-polarized Hodge structures on a finite-dimensional rational vector space $V$. Let…

Let $B$ be the base of a period map $\Phi:B\to\Gamma\backslash D$, let $\overline{B}$ be a completion with boundary divisors $Z_i$, and let $\Lambda_\mathrm{e}$ denote the extended…

Let $V$ be a complex projective variety, let $M=V\setminus\operatorname{sing} V$ be its regular part, and equip $M$ with the metric induced by a smooth Kähler metric on projective…

Let $X$ be a smooth projective variety of dimension $g$, and let $V\subset H^m(X,\mathbb{Q})$ be a $\mathbb{Q}$-Hodge substructure of level at most $m-2p$. A nonsingular projective…

Let $(w,u_1,\ldots,u_n,\xi,T)$ be data with $w\in{\mathds Z}$, $u_i\in{\mathds C}$ pairwise distinct, $\xi\in S^1$ satisfying $\Re((u_i-u_j)/\xi)<0$ for $i<j$, and…

Let $(H,\nabla,H'_{\mathds R},P)$ be a TERP-structure. It requires no ramification when its formal decomposition can be made without ramification; it is a mixed TERP-structure when…

Let $k$ be a field of characteristic $0$, and let ${\mathcal A}$ be a smooth proper dg category over $k$. For every integer $n\geq 1$, consider the complex … where…

Let $X$ be a smooth projective variety of dimension $m$, let ${\mathbb F}$ be the coefficient field used in the source, and let $\widetilde{L}^tH^k(X;{\mathbb F})$ be the image of…

Let $X$ be a numerical $(K3)^{[2]}$ and suppose that Items (1) through (6) of Proposition (hodgeprop) hold. O'Grady's conditional theorem. Item (1) of Theorem (mainthm1) holds. The…

Let $Y$ be a Calabi–Yau variety of dimension $n$. Infinitesimal mirror symmetry conjecture. There exists a Calabi–Yau variety $Y^{\circ}$ and isomorphisms of complex vector spaces…

Quasi-adjunction eigenspace conjecture. Then $h^{n,0}_{\chi}(\bar U_G)=0$ unless the lift of $\chi$ belongs to a contributing global face of quasi-adjunction $\mathcal F$; in the l…

Dubrovin's reduced quantum cohomology conjecture. The variety $X$ has generically semisimple reduced quantum cohomology, that is, $M'$ is generically semisimple, if and only if the…

Let $S^{[3]}$ and $S^{[4]}$ be the strata appearing in the semistable degeneration, and let $[N]$ denote the cohomology class of the monodromy operator. The relevant restriction ma…

Let $X$ be smooth and proper over $\mathbb C$. Let $\operatorname{gr}^0_{F_m}CH^p(X)$ denote the degree-zero graded piece of the motivic filtration on codimension-$p$ Chow groups,…

Let $X$ be a proper smooth scheme over $\mathbb C$. For integers $i,j$, define the coniveau filtration by … The inclusion … identifies the left-hand side with a subspace of rationa…

Let $H$ be a geometric Hodge structure, and let $H^{\nabla}$ denote its horizontal subspace. Horizontal-subspace conjecture. For any geometric Hodge structure $H$, the horizontal s…

Let $X$ be a smooth Calabi–Yau manifold with an action of a finite group $H$ preserving the holomorphic volume form, let $Y=X/H$, and let $h_{st}^{p,q}(Y)$ denote its stringy Hodge…

Let $X$ be a smooth complex projective variety. Define the coniveau filtration by … where $Z\subset X$ ranges over subvarieties, and let $H^{m,i}_{\operatorname{Hdg}}(X)$ be the la…

Let $X$ be a smooth projective variety defined over a finitely generated subfield $F\subset\mathbb{C}$. Under the Artin comparison identification … let…

Let $X$ be a smooth, projective variety over $\mathbb{C}$, and write $h^{p,0}(X)$ for the Hodge number measuring global holomorphic $p$-forms. Bloch's conjecture. If … for every…

Tadpole conjecture. The tadpole contribution satisfies

Extremality conjecture.

Grothendieck's standard conjecture of Hodge type. For any $i \le d/2$, the symmetric bilinear form

Hodge realization conjecture. The functor $\iota_{X,\sigma}^{\textup{Hdg}}$ is fully faithful, with essential image stable under subquotients. This is a Hodge-type conjecture for N…