32 problems

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 $A$ 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…

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 $i\geqslant 0$,

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 $M$ 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…

Let $f=f_1\cdots f_d\in R=\mathbb{C}[x_1,\dots,x_n]$ define a reduced hyperplane arrangement $\mathscr{A}$ in $X=\mathbb{C}^n$, and let $U=X\setminus\mathscr{A}$. Write…

Let $\mathcal{X}/S$ be a smooth projective scheme, where $S=\operatorname{Spec}(k[[t]])$ for a field $k$ of characteristic zero. Set $S_n=\operatorname{Spec}(k[t]/t^{n+1})$ and…

Let $\mathcal O_K$ be the ring of integers of a finite extension of $\mathbb Q_p$, let $k$ be its residue field, and let $\pi\colon X\to Y$ be a conical resolution over…

Periodic de Rham conjecture.

Weak periodic de Rham conjecture. The natural inclusion functor

Extension conjecture. The canonical parabolic extension of an irreducible periodic de Rham bundle over $U$ is again periodic.