69 problems

Let $N$ be a closed manifold, let $T^*N$ be its cotangent bundle, and let $\mathcal{O}_N$ denote the zero section. Write $\mathscr{L}(T^*N)$ for the space of closed exact Lagrangia…

Debarre's conjecture. The cotangent bundle $\Omega_X^1$ is ample.

Arnol'd's nearby Lagrangian conjecture. The Hamiltonian orbit of the zero section $L$ should be the set of all exact Lagrangians $L'$. This conjecture concerns the classification o…

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 $M$ be a closed connected manifold, and let $\mathcal{L}(M)$ be the space of closed connected exact Lagrangian submanifolds of $T^*M$. Strong nearby Lagrangian conjecture. The…

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

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 $N$ be a compact $C^infty$-manifold, let $\mathscr{X}=T^*N$, and let $I$ be an open interval containing $0$. For a Hamiltonian symplectic isotopy…

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…

Let $M$ be a smooth manifold and let $T^*M$ denote its cotangent bundle with the canonical symplectic structure. A Lagrangian submanifold $L\subset T^*M$ is exact if the canonical…

Let $X$ be the intersection in ${\bf P}^n$ of $c$ general hypersurfaces of sufficiently high degrees, and let $m$ be an integer. For $r\gg 0$, consider the cohomology groups…

Let $X$ be the intersection in ${\bf P}^n$ of at least $n/2$ general hypersurfaces of sufficiently high degrees. Cotangent ampleness conjecture. The cotangent bundle $\Omega_X$ is…

Let ${\mathcal R}=\bigoplus_{d=0}^{\infty}{\mathcal R}^d$ be the graded algebra in the paper, let ${\mathcal D}=\bigoplus_{d=0}^{\infty}{\mathcal F}^d$ be the splitting of the orde…

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…

Cotangent-bundle hyperkähler conjecture. Every total space $T^*M$ admits a hyperkähler structure.

Let $G$ be the reductive group in the construction above, with Weyl group $W$, universal centralizer base $\mathfrak{c}$, and Whittaker cotangent bundle … Let $\overline{T^*(G/U)}$…

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…

Irrational ellipticity conjecture. All closed Reeb orbits in such a pseudo-rotation are irrationally elliptic.

Multiplicity conjecture. The number of closed Reeb orbits is at least $n$ for star-shaped hypersurfaces in $\mathbb{R}^{2n}$, and at least

Capacity finiteness conjecture. This capacity is finite for every manifold $Q$ that is not a $K(\pi,1)$.

Contractible almost-existence conjecture. There exists $\varepsilon>0$ such that almost all energy levels of $H$ in $(c-\varepsilon,c+\varepsilon)$ carry a contractible periodic or…

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…

Tseu's equiconstitution conjecture. For all $n$, the tilting generator $\mathcal{T}_n$ is equiconstituted to $\mathcal{Z}$ up to tensor product with a line bundle.

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

Let $C\subset \mathbb{R}^n$ be a convex body with smooth boundary. For $q\in C$, define $L_q=\operatorname{graph}({\rm d}\langle q,-\rangle)\subset T^*S^{n-1}$, and let…