37 problems

Let $N$ be a closed manifold, let $g$ be a Riemannian metric on $N$, and let $D_1T^*N=\{(q,p)\mid |p|_g\leq 1\}$. Let $\mathfrak L(T^*N)$ denote the relevant class of closed exact…

Let $M$ be a smooth algebraic variety, set $X=T^*M$, and let … be the natural map, assumed to be projective and birational. Demailly–Campana–Peternell conjecture. Then $M=G/P$, whe…

Consider the cotangent bundle $T^*(\mathbb{R}^n/\mathbb{Z}^n)$ with its standard symplectic structure. A homogeneous Hamiltonian torus action is an action compatible with the cotan…

Let $(T^*\mathbb T^2,\omega_{\rm can})$ be the cotangent bundle of the two-torus, and let $\mathcal S$ denote the class of systolically convex domains defined in the source. A symp…

Let $G=\operatorname{GL}_n$, let $P$ be a parabolic subgroup containing the Borel subgroup $B$, and let $\mathfrak{u}_P$ be the nilpotent radical of the Lie algebra of $P$. Conside…

Equiconstitution conjecture. The tilting generator $\mathcal{Z}$ is equiconstituted with any Toda–Uehara tilting generator.

Let $X$ be a smooth projective variety, and let $1 \leq q \leq \dim X$. A vector bundle $\mathcal{E}$ on $X$ is pseudoeffective when the tautological line bundle…

Let $(M,g)$) be a closed Riemannian manifold. Define … and let $0_M$ denote the image of the zero section of $T^*M$. For a compactly supported Hamiltonian diffeomorphism…

Let $Q$ be a closed oriented manifold that is not a $K(\pi,1)$. The cotangent-bundle conjecture. Then … and the almost existence property for closed contractible characteristics ho…

Let $N$ be a manifold, let $T^*N$ carry the cotangent-bundle structure, and let … DrT^N={(q,p)in T^N:|p|leq r}. … is bounded on the space of exact Lagrangians contained in…

Let $L$ be a closed smooth manifold and let $T^*L$ denote its cotangent bundle. For every compactly supported Hamiltonian diffeomorphism $\phi$ of $T^*L$, consider the intersection…

Generation conjecture. The category $\mathcal{W}_\kappa(T^*Q)$ is split-generated by any $\kappa$-tuple of disjoint cotangent fibers.

Cyclic coordinate-function conjecture. If $T^*Q$ admits a cyclic coordinate function, then $Q$ is diffeomorphic to $S^1\times\Sigma_g$, where $g\geq0$, or a spherical space form. T…

Ganatra-Pomerleano's conjecture. If $T^*Q$ admits a coordinate function, then $Q$ is diffeomorphic to $S^1\times\Sigma_g$ for some $g\geq0$. The conjecture is presented as motivate…

Let $X$ be a projective manifold such that the cotangent bundle $\Omega_X$ is pseudoeffective. Assume that $X$ admits an abelian fibration $f:X\to C$ onto a curve $C$, and let $F$…

Let $X$ be a projective manifold, let $q$ satisfy $1 e q$ and $q e \dim X$ only if excluded?

Let $X$ be a smooth complex projective manifold. If $K_X$ is pseudoeffective, write $\nu_1(X):=\nu(X,\Omega_X^1)$. If $X$ is uniruled, let $r:X\to R_X$ be its maximal rationally co…

Let $X$ be a compact Kähler manifold that is special, and let $p>0$. For every rank-one coherent subsheaf $L\subset\Omega_X^p$, the numerical cotangent bound conjecture. One has ……

Let $X$ be a compact connected Kähler manifold and let $p>0$. Write … The cotangent-dimension equality conjecture. One has … This extends the canonical-bundle equality to all exter…

Generalized abundance conjecture. For every $X$ and every integer $p>0$,

Let $2n$ be an even natural number. Let $(X,H)$ be a polarised hyperkähler manifold, where $X$ has dimension $2n$ and $H$ is an ample Cartier divisor. Write … for the projectivised…

Let $X$ and $Y$ be smooth affine varieties over $\mathbb{C}$, and let $T^*(X)$ and $T^*(Y)$ denote their cotangent bundles. The cotangent-bundle isomorphism conjecture. If … then ……

Let $L$ be a closed manifold and let $T^*L$ denote its cotangent bundle. Let $K\subset T^*L$ be an exact Lagrangian submanifold. Assume that some Weinstein neighbourhood of $K$ con…

Characteristic-cycle push-forward conjecture. We have

Let … and let $\vec{\operatorname{gr}}(\mathcal{T})$ be the associated graded object of the Hodge-module kernel $\mathcal{T}$. Dense-open line-bundle conjecture. There exists a den…