14 problems

Let $(M,\omega)$ be a positive monotone symplectic manifold of dimension six admitting a Hamiltonian circle action. A Fano manifold is a compact complex manifold whose anticanonica…

Assume the setting of the Lie monoid action conjecture in Floer theory, with the actions Hamiltonian. Let $\mathcal{W}(T^*G)$ and $\mathcal{W}(T^*H)$ denote the wrapped Fukaya cate…

Unimodality conjecture. If $(M,\omega)$ is a closed symplectically rational $8$-manifold admitting a Hamiltonian $S^1$-action, then the even Betti numbers of $M$ are unimodal.

Fine–Panov conjecture. $M$ is diffeomorphic to a smooth Fano threefold.

Let $X$ be a Liouville manifold equipped with a Hamiltonian $S^1$ action such that the fixed locus $X^{S^1}$ has codimension four and there are no finite non-trivial stabiliser gro…

Let $(M,\omega)$ be a six-dimensional closed monotone symplectic manifold equipped with an effective Hamiltonian circle action. A Kähler realization conjecture asserts that…

Let $(M,\omega)$ be a $2n$-dimensional closed symplectic manifold equipped with a Hamiltonian $S^1$-action whose fixed points are all isolated. Write $b_j$ for the $j$-th Betti num…

Let $G$ act Hamiltonianly on a monotone symplectic manifold $X$. The action gives an $E_2$-algebra map … Lekili and Evans' construction gives a map from the endomorphisms of the co…

Let $(M,\omega)$ be a six dimensional closed monotone symplectic manifold, meaning that $\langle c_1(TM),[\Sigma]\rangle>0$ for every symplectic surface $\Sigma$, equipped with an…

Kählerness conjecture. Then $(M,\omega)$ is $S^1$-equivariantly symplectomorphic to some Kähler manifold $(X,\omega_X,J)$ with some holomorphic Hamiltonian $S^1$-action.

Equivariant symplectic Fano conjecture. $M$ is diffeomorphic to a complex projective Fano $3$-fold.

Let $(Z,\omega)$ be a six-dimensional symplectic Fano manifold with a Hamiltonian $S^1$-action. An integrable $S^1$-invariant complex structure compatible with $\omega$ is an integ…

Let $(X^6,\omega)$ be a closed symplectic six-manifold equipped with a symplectic $S^1$ action, and suppose that its fixed-point set contains at least one isolated fixed point. Ham…

Let $K$ be a connected compact Lie group, let $M$ be a multiplicity-free compact Hamiltonian $K$-manifold with moment map $\mu:M\rightarrow\mathfrak k$, and fix a maximal torus…