47 problems

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

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…

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

Let $X$ be a smooth projective variety, and let $\mathrm{D}^b(X)$ denote its bounded derived category of coherent sheaves. Consider semiorthogonal decompositions … of…

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 $\mathcal{Q} \to S$ be a quadric fibration that is not too singular. Let $\mathcal{C}l_0$ denote its even Clifford algebra and $\mathcal{Z}(\mathcal{C}l_0)$ its center. Let…

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…

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

Atomic semi-orthogonal decomposition conjecture. The atomic decomposition reflects a semi-orthogonal decomposition of $\mathrm{Perf}(X)$ with one component for each atom, and this…

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…

Nonexistence conjecture. If $d\leq p_a-1$, then the derived category

Let $X$ be a smooth projective variety over $\mathbb{C}$, let $H^*_{alg}(X)\subset H^*(X)$ be the subspace of algebraic classes, and let $\mathcal M^{alg}$ be the formal scheme obt…

NMMP quasi-convergence conjecture. The paths $Z_\psi(t)$ lift to paths in $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ that are quasi-convergent.

A quasi-phantom category is a nontrivial admissible full triangulated subcategory of the bounded derived category of coherent sheaves on a smooth projective variety with finite Gro…

Let $X$ be a smooth variety of dimension $d\geq 2$, and let $X^{[3]}$ denote its Hilbert scheme of zero-dimensional subschemes of length $3$. Write $\operatorname{D^b_{S_3}}(X^3)$…

Let $V$ be a vector space of dimension $2n$, let $X_2=\mathbb{P}_{\operatorname{Gr}(2,V)}(\Lambda^2\widetilde{Q}^{\vee})$, and let…

Let $V$ be a vector space of dimension $2n$, let $X_1$ be the source's variety, and let $\pi\colon X_1\to\mathbb{P}(V)$ be the projection. Let…

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 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…

The Quot formula conjecture. For every $i\in[0,\min\{d,\delta\}]$ and every $\alpha\in B_{i,\delta-i}$, there is a strong relative Fourier–Mukai transform

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…