40 problems

Bayer–Macrì–Toda conjecture. The heart obtained by the double-tilt construction on $\operatorname{Coh}(X)$, together with the central charge $Z^X_{a,b}$, defines a Bridgeland stabi…

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 $X$ 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…

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…

Folklore product conjecture. The form $\Omega_{\mathbf{a}^{(i)}}$ induces a stability condition on the Fukaya–Seidel-type category mirror to $\operatorname{D^b}(X_i)$, and stable o…

Folklore conjecture. For generic coefficients $\mathbf{a}$, the Fukaya–Seidel-type category associated with $W_{\mathbf{a}}$ admits a Bridgeland stability condition whose central c…

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…

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.

Generic hyperplane-section stability conjecture. Theorem holds generically for every GM fourfold and cubic fourfold.

dHYM–Bridgeland stability conjecture. The existence of a dHYM instanton $\nabla$ on ${\bf E}$ should be equivalent to Bridgeland stability of ${\bf E}$. This conjecture proposes an…

Let $X$ be a singular Calabi–Yau threefold, let $\rho:\widehat{X}\to X$ be a non-Kähler small resolution, let $\alpha\in\operatorname{Br}(\widehat{X})$, and let $\widehat{X}'$ be a…

Let $X$ be a singular Calabi–Yau threefold, let $\rho:\widehat{X}\to X$ be a non-Kähler small resolution, and let $\alpha\in\operatorname{Br}(\widehat{X})$. Let $K(X)$ denote the K…

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…

Let $X$ be the variety under consideration, let $L\in\operatorname{Pic}(X)$ be a line bundle, and set $[\psi]=c_1(L)$. Conjectural dHYM–Bridgeland correspondence. The existence of…

Consider the wall-and-chamber structure for asymptotic $\lambda_{\alpha,\beta,s}$-stability along a vertical line $\{\beta=\overline{\beta}\}$, with $s>0$. A wall is understood as…

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…

Vanishing-entry conjecture. Every element of the destabilization set of $U$ arises by imposing vanishing conditions on entries of the map $M$ of the minimal free resolution of $U$.

The minimal-resolution conjecture. Under these hypotheses, every sheaf destabilized at the second-last wall in this way has the asserted Gaeta resolution and block-matrix structure…

Let $X$ be a smooth projective variety over … for any Bridgeland stability condition on $D^b(X)$. This conjecture extends Toda's result for K3 surfaces to arbitrary smooth projecti…