58 problems

Let $X$ be a smooth projective variety defined over $\overline{\mathbb{Q}}$, and let $\mathrm{H}^{n}_{\mathrm{dRB}}(X,\mathbb{Q})$ be its de Rham–Betti structure. Write…

Let $h:\mathbb{R}^n\to\mathbb{R}$ be a real polynomial, and let $d_h=d+dh\wedge\cdot$ act on the de Rham complex of tempered currents on $\mathbb{R}^n$. For sufficiently large…

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$. Write $\langle\mathrm{H}^1_{\mathrm{dRB}}(A,\mathbb{Q})\rangle^{\otimes}$ for the tensor category generated by its first…

Epsilon-class conjecture. If $\nabla$ is an integrable, admissible, absolute connection, then

Determinant de Rham formula.

Higher-rank Gauß–Manin determinant formula. Assuming these conditions, one has

Let $X$ be a smooth projective variety over a characteristic-zero field $k$ with simply connected complex-topological realization. Define its de Rham shape using the very presentab…

Let $k$ be a field of characteristic zero, let $X$ be a variety over $k$, and let $(X,\Pi_1^\mathrm{dR})$ denote the de Rham fundamental groupoid site. A site has the stated proper…

Let $\overline{\operatorname{Sh}}$ be the Siegel threefold or a unitary Shimura variety, and let $\mathfrak{m}\subseteq\mathbb{T}$ be generic and non-Eisenstein. Let…

Let $(G,X)$ be a Shimura datum of Hodge type. Let $\mathbb{T}$ be the spherical Hecke algebra and let $\mathfrak{m}\subseteq\mathbb{T}$ be a maximal ideal. For an algebraic represe…

Let $X$ be a smooth projective variety defined over $\overline{\mathbb{Q}}$, and let $0\leq j\leq\dim(X)$ be an integer. Consider the de Rham-Betti structure … If…

Grothendieck's de Rham-Betti cycle conjecture. Every de Rham-Betti class in $V_{\mathrm{B}}$ comes from a $\mathbb{Q}$-coefficient algebraic cycle on $X$ for $0\leq p\leq \dim(X)$.…

Let $X/\mathcal{O}_K$ be the fixed scheme, let $\mathfrak{S}$ be the Breuil–Kisin ring, and let $E$ be its Eisenstein polynomial. For the Breuil–Kisin cohomology module … write…

Main conjecture. For all $i\geqslant 0$,

Let $k$ be a field of characteristic $p>0$, and let the mixed Hodge module de Rham and Higgs subcomplexes $K^*_{\mathrm{dR}}$ and $K^*_{\mathrm{Hig}}$ be as in the preceding constr…

Trajectory-space invariance conjecture. The $v$-basic de Rham cohomology

Basic harmonic homology conjecture. For every degree $k$,

Openness conjecture. The set of contact forms $\beta$ on $X$ for which

Explicit splitting conjecture. For $\ell\in\{2,\dots,n-2\}$, the morphism of complexes

Čech–de Rham conjecture. For every $q\in\mathbb{N}$,

Let $k$ be a number field, let $Y$ be a smooth projective $k$-variety, and fix an embedding $k\hookrightarrow\mathbf{C}$. For a codimension-$n$ de Rham–Betti cycle, take a pair…

Let $K$ be a number field with algebraic closure $\bar K$ in $\mathbb{C}$, let $X$ be a smooth projective $K$-variety, and let $H^*_{dR}(X)$ and $H^*_B(X)$ be its de Rham and Betti…

Let $M$ be an André motive over $K$, or a homological motive over $K$. Write $\Omega_M^{\mathrm{dRB}}$ for the torsor of de Rham–Betti periods, $\Omega_M^{\mathrm{And}}$ for the to…

Let $M$ be an André motive over $K$, and let $Z_M$ be the Zariski closure of its comparison isomorphism, $\Omega_M$ the smallest $K$-subtorsor generated by that comparison,…

Degeneration and LMH conjecture. (1) This spectral sequence degenerates at $E_1$, and all $R^qf_*(\omega^p_{X/S})$ are locally free of finite rank as $\cO_S$-modules; hence…