12 problems

Bayer–Macrì–Toda Bogomolov–Gieseker inequality. For any $(b,w)\in U$ and $\nu_{b,w}$-semistable $E\in\mathcal D(X)$, one has

Let $X$ be the smooth projective threefold under consideration, let $H$ be its ample divisor, and let $u{b,w}$ and $mathcal A(b)$ be the tilt slope and heart defined above, with…

Let $X$ be a smooth projective threefold that is the complete intersection of a quadratic and a quartic hypersurface in $\mathbb{P}^5$, and let $H$ be an ample class. For…

Let Setup be fixed, let $v \colon P \to \mathbb{R}_{>0}$ be a weight, and let $\mathcal{E}$ be a reflexive sheaf of rank $r$ that is $v$-stable. Bogomolov–Gieseker conjecture. The…

Let $(X,H)$ be a polarized smooth projective $3$-fold. For $b\bitin\mathbb{R}$ and $a>0$, let $\sigma_{a,b}=(\mathcal{A}^b,Z_{a,b})$ be the tilt-stability condition, and let…

Let $(X,H)$ be a smooth complex projective polarized variety. Let $\gamma=\gamma_{X,H}$ be a class as in the source's definition of $\gamma$, and let…

The generalized Bogomolov–Gieseker conjecture. Every polarized threefold satisfies this inequality for all $(\alpha,\beta)\in U$ and $\gamma=e^{-B}$. This was the original conjectu…

Let $v_n$ be the class under consideration, let $U$ be the stability-condition domain, and let $\ell_{v,n}$ be the line defined in the surrounding discussion. Weak Bogomolov–Giesek…

Let $X$ be a smooth projective threefold with polarization $H$, let $U$ be the domain of pairs $(b,w)$ defining the weak stability conditions $u{b,w}$, and let $u{b,w}$-semistabl…

Let $X$ be the smooth projective threefold under consideration, let $H$ be its ample divisor, and let $\beta$ and $m$ be the numerical data defining $v_n$. Set … Let the strong Bog…

Bayer–Macrì–Toda conjecture. The quantity

Strong Bogomolov–Gieseker conjecture. One has