27 problems

Let $C$ be a smooth projective curve, let $G$ be a reductive group with Langlands dual group $G^{\vee}$, and let $w$ denote the relevant stratum or weight. Write…

Let $R$ be a ring and let $\mathcal{D}$ be an $R$-linear, smooth and proper category. A continuous path $\sigma_\bullet:[0,+\infty)\rightarrow\operatorname{Stab}(\mathcal{D}/R)$ of…

Let $X=Q_1\cap Q_2\subset\mathbb{P}^{n+2}$ be a smooth complete intersection of two quadrics in an odd-dimensional projective space. Let $C$ be the hyperelliptic curve associated w…

The Kuznetsov decomposition conjecture. For a Fano complete intersection $X\subset\mathbf{P}^n$, there exist $\tau\in\mathrm{H}^\bullet(X)$ and a sector $\mathscr{S}$ such that the…

Flapan–Macrì–O'Grady–Saccà conjecture. The Fano eightfold $F$ admits the semiorthogonal decomposition

Let $\mathcal{Y}$ be a smooth stack, let $\mathcal{S}\subset\mathcal{Y}$ be the relevant closed stratum, and let…

Let $G$ be a reductive group, let $^{L}G$ be its Langlands dual, and let $(\chi,w)\in\pi_1(G)\times Z(G)^{\vee}$. Consider the quasi-BPS semiorthogonal decompositions of the cohere…

Indecomposability conjecture. If the canonical bundle $K_X$ is nef and effective, then $X$ admits no non-trivial semiorthogonal decompositions.

Let $G$ be a finite group acting effectively on a smooth quasiprojective variety $X$. For $g\in G$, let $X^g$ denote the invariant subvariety and let $C(g)$ be the centralizer of…

Acyclicity criterion. The surface $X$ is not semiorthogonally indecomposable if and only if $\mathcal{O}_X$ is acyclic.

Let $A\subset N=\mathbb Z^d$ and $Q=\operatorname{conv}(A)$, and let $F$ be an edge of the secondary polytope. Let $Z_F$ be the associated space, let $n_{\Gamma,F}$ be the multipli…

Let $X$ be a smooth projective curve, let $\mathrm{M}(r,L)$ be the moduli space of stable rank-$r$ vector bundles on $X$ with fixed determinant $L$, and let $\operatorname{Jac}(X)$…

Let $X$ be a smooth projective curve of genus $g\geq 2$, let $L\in\operatorname{Pic}^{d}(X)$ with $(2,d)=1$ and $0<d<2$, and let $\mathrm{M}(2,L)$ be the moduli space of stable ran…

Let $X$ be a complex Fano variety such that the condition … bm{kappa}X(QS(X)) cap ({mathbb{A}}^1 setminus {0}) = {z1,dots,zN} subset {mathbb{A}}^1 setminus {0} … holds. Semiorthogo…

Let $S$ be a smooth projective surface over $\mathbb{C}$, let $X=\operatorname{Tot}_S(\omega_S)$, and let $P_n^t(X,\beta)$ denote the moduli space of $\mu_t^{\dagger}$-semistable D…

Let $C$ be a smooth projective curve of genus $g\geq 2$, let ${\mathcal L}$ be the fixed determinant, and let ${\moduli_C(2,\mathcal L)}$ be the moduli space of rank-two bundles wi…

Let $C$ be a smooth projective curve of genus $g$, let ${\mathcal L}$ be the fixed determinant, and let ${\moduli_C(2,\mathcal L)}$ be the moduli space of rank-two bundles with det…

Moduli-space conjecture. There exists an étale algebraic space $\operatorname{\mathsf{SOD}}_{\mathcal{D}/U}$ over $U$ with a functorial bijection

Let $Y\subset\operatorname{Gr}(3,V_{10})$ be a very general hyperplane section, and let $\mathsf{A}$ be the K3 subcategory of $\operatorname{D}^b(Y)$ obtained as the semiorthogonal…

Cheltsov's conjecture. The only nodal Fano threefolds of Picard rank one with Kawamata type decompositions are the quadric, $V_5$, and $V_{22}$.

Let $X$ be a smooth variety over an algebraically closed field of characteristic zero, and let a finite group $G$ act effectively on $X$. For $\lambda\in G/{\sim}$, let $X_\lambda$…

Let $X$ be a smooth projective variety with a semiorthogonal decomposition … where $\mathcal{A}$ is a Calabi–Yau category of dimension $n=\dim X$. Boundary Calabi–Yau component con…

Derived-category conjecture for Kuechle fourfolds.

Derived-category conjecture for $X_{10}$.

Let $C$ be a monotone curve in a $k\times l$ rectangle as described in the construction, with intersection points $Q_i$ and associated integers $a_i,b_i,c_i,d_i$. Define blocks…