26 problems

Let $X$ be a Calabi–Yau threefold with only terminal conifold singularities and no Kähler small resolution. Let $\mathcal{S}$ be the set of all small resolutions $\widehat{X}$ of…

Let $\check{\mathcal{X}}$ be the mirror geometry, let $L$ be a Lagrangian in $\check{\mathcal{X}}$, and let $\mathcal{E}$ be a holomorphic vector bundle on $\mathcal{X}$. A Bridgel…

Let $S$ be an Enriques realizable K3 surface, meaning a K3 surface arising in the Enriques-related moduli setting considered in the source. Bridgeland conjecture. $S$ is Bridgeland…

Let $X$ be a smooth complex projective surface, let $B,\omega\bigin\operatorname{NS}(X)\otimes\mathbb{R}$ with $\omega$ ample, and let…

Joyce's conjecture. There exists a Bridgeland stability condition $\sigma=(Z,\mathcal{P})$ on $\operatorname{DFuk}(M)$ such that the central charge is induced by

Let $(X,H)$ be a smooth polarized variety over $\mathbb{C}$ and let $d\geq0$. Let $\mathfrak B_n^{>d}$ and $\pm\mathfrak B_n^{>d}$ denote the reduced-central-charge spaces satisfyi…

Let $(X,H)$ be a smooth polarized variety over $\mathbb{C}$ and let $d\geq0$. Let $\mathfrak U_n^{>d}$ be the subspace of central charges defined by the interlacing conditions and…

Let $(X,H)$ be a smooth polarized variety over $\mathbb{C}$, and let $\Lambda_H$ be the $H$-polarized lattice. Let $\mathfrak B_n$ and $\pm\mathfrak B_n$ be the spaces of reduced c…

Let $(X,H)$ be an $n$-dimensional irreducible smooth polarized variety over $\mathbb{C}$. Let $\Lambda_H$ be the $H$-polarized lattice, and let $\mathfrak U_n$ be the specified ope…

Let $X$ be a smooth projective surface with a fixed Kähler class $[\omega_0]$ and a line bundle $L$. The stability condition $\sigma_{[\omega_0]}$ is the stability condition suppor…

Syzygy behavior conjecture. A sheaf $U$ is $F$-admissible, where $F$ is a Bridgeland destabilizing object that yields the wall $W$ when $U$ enters the stable base locus at the wall…

Generic projected-push-forward conjecture. For every $[E]\in U$, the object $\operatorname{pr}_X(j_*E)$ is $\sigma_X$-semistable.

Let $X$ be the compact Kähler manifold under consideration, and let $L$ be a holomorphic line bundle on $X$. A metric solving the deformed Hermitian–Yang–Mills equation is a metric…

Characterization conjecture. Every element of the destabilization set of $U$ comes from vanishing conditions on the map $M$. Moreover, knowing the destabilization set of $U$ is equ…

Fukaya-category stability–equation conjecture. There is a class of stability conditions on the Fukaya category such that for each stability condition in this class, there is an ass…

Let $X$ be a Calabi–Yau manifold, let $L\rightarrow X$ be a holomorphic line bundle, and let $L\hookrightarrow X$ be a Lagrangian submanifold. The objects $L$ are viewed respective…

Let $(X,\omega)$ be a Kähler $3$-fold and let $L\to X$ be a holomorphic line bundle. Assume that $L$ admits a solution of the deformed Hermitian–Yang–Mills equation with lifted ang…

Let $X$ be a Calabi–Yau manifold with mirror $\check{X}$, let $L$ be a Lagrangian in $\check{X}$, and let $E$ be a holomorphic vector bundle on $X$. Write…

Let $X$ be a projective Calabi–Yau $n$-fold, let $\mathcal{M}_{\mathrm{Kah}}(X)$ be its stringy Kähler moduli space, and let $\mathcal{D}_X=\mathrm{D^bCoh}(X)$. Let…

Fourier–Mukai compatibility conjecture. One has

Fourier–Mukai compatibility conjecture. For $1\leq k\leq g-1$, there are suitable complexified ample classes $\Omega_k$ and $\Omega'_k$ such that

Let $S$ be a Del Pezzo surface, and consider the program of constructing suitable hearts of Bridgeland stability conditions from exceptional collections and applying it to moduli s…

Let $X$ be an abelian threefold, let $B\in\operatorname{NS}_{\mathbb{R}}(X)$, and let $\omega\in\operatorname{NS}_{\mathbb{R}}(X)$ be an ample class. With $\mathcal{B}_{\omega,B}$…

Let $I_Z$ be the ideal sheaf of a point $Z\in\mathbb{P}^{2[n]}$. A Bridgeland wall is a wall $W_x$ in the family of walls $W_{I_Z,(r',c',d')}$ along which some ideal sheaf $I_Z$ is…

Bridgeland's conjecture. Given a stability condition $\sigma$ on a K3 surface and a numerical class $v$, there exists a coarse moduli space $M_{\sigma}(v)$ of $\sigma$-semistable c…