71 problems

Completeness conjecture. The invariant $y_3^{(4)}$ should satisfy a polynomial relation involving the $x$ variables.

Mumford's conjecture. The manifold $X$ is rationally connected if and only if

Let $A_n$ be the free associative algebra and let $\varphi(d,D):A_n\to\Omega(d,D)^{even}_*$ be the algebra map sending each generator $x_i$ to $x_i$. For $m_1,m_2,m_3,m_4\in A_n$,…

Let $X$ be a smooth projective scheme defined over a field of characteristic zero. After spreading out $X$, for a prime $p$, write $X_{p=0}$ for its reduction modulo $p$, and let…

Higher direct-image vanishing conjecture. One has

Let $M$ be a compact manifold and let several closed $1$-forms on $M$ be given. Their common level is the intersection of the corresponding level sets at a generic choice of levels…

Let $A_n$ be the free associative algebra and let $\Omega(d,D)^{even}_*$ be the even differential-form algebra with two anticommuting differentials $d$ and $D$, equipped with the p…

Let $F/k$ be the field extension in the source, let $\mathfrak m$ be the object and $\mathcal C$ the category appearing in the source, let $\mathcal A$ be the category of admissibl…

Let $F/k$ be the field extension in the source, let $G$ be its field automorphism group, and let $\mathcal I_G$ be the category of homotopy-invariant smooth $G$-representations. Le…

Let $M$ be a closed manifold with zero Euler characteristic, and let $alpha$ be a non-closed differential $1$-form on $M$. Non-vanishing perturbation conjecture. There exists a smo…

Let $A$ be a projective manifold, and let ${\mathcal L} \subset \bigwedge^r \Omega^j_A$ be a line bundle. A curve $C$ on $A$ is movable if its deformations fill up $A$. The positiv…

Let $\mathrm{Vect}(\mathbb R^n)$ be the Lie algebra of vector fields on $\mathbb R^n$, let $\Omega(\mathbb R^n)$ denote the space of differential forms, and let…

Rational connectedness criterion. $X$ is rationally connected if and only if

Fundamental class smoothness conjecture. The morphism $c_Y^k$ is surjective in codimension $k$ if and only if $Y$ is smooth in codimension $k$.

Dimension-reduction conjecture. Every such integrable $1$-form is, up to multiplication by a non-vanishing function, the pull-back of an integrable $1$-form in dimension $3$ in the…

Let $({\cal M},\Omega)$ be an arbitrary multisymplectic manifold, let $D^n_m{\cal M}$ denote the decomposable $n$-vectors at $m$, and let ${\cal O}_m\subset D^n_m{\cal M}$ be the s…

Proper-form characterization conjecture. For $1\leq p\leq n$, the set of $p$-forms proper on ${\cal O}_m$ coincides with the standard copolarisation. The claim proposes an intrinsi…

Finite-dimensionality conjecture. For nonsingular foliations $\omega=0$, these exotic cohomology spaces are finite-dimensional.

Let $(M^n,g)$ be a compact Riemannian manifold whose sectional curvature $K_g$ satisfies … Assume that $M$ is star-shaped with respect to $x_0\in M$. Denote by $\lambda_{k,p}$ the…

Rational integrability conjecture. The function $f$ is rationally integrable with respect to $x_1,\ldots,x_m$ if and only if $n\neq m$. This extends the discussion of rational inte…

Logarithmic tensor–differential isomorphism conjecture. This map is an isomorphism. The map is motivated by the method used for logarithmic motives and additive-multiplicative stru…

Let $\mu$ be the measure and $V_k(\mu,x)$ the $k$-dimensional decomposability space associated with it, and let $\omega$ be a $k$-form satisfying … for every $\tau\in V_k(\mu,x)$ a…

Let $T$ be a tetrahedron, and let $C_{{\rm PF},T,\Gamma,\ell,k,p}$ denote the Poincaré–Friedrichs constant for the relevant form degree, boundary face set, and Lebesgue exponent…

A zigzag multiplicity is the multiplicity of a zigzag summand in the bicomplex of differential forms of a compact complex manifold, and a Chern number is a characteristic number ob…

Campana's birational stability conjecture. If $K_X$ is pseudo-effective, then