14 problems

Let $X$ be a smooth projective variety and let $D^b\operatorname{Coh}(X)$ denote its derived category of coherent sheaves. A semi-orthogonal decomposition is a decomposition of thi…

Nef-and-effective canonical bundle conjecture. If $\omega_X$ is nef and effective, then ${\mathbf{D}}^b(X)$ has no non-trivial semi-orthogonal decompositions.

Let $X$ be a minimal surface of Kodaira dimension $1$ with an elliptic fibration … Suppose that $P_g(X)=0$ and $q(X)=1$. Elliptic-fibration decomposability conjecture. Then $D(X)$…

Let $X$ be a smooth projective variety, let $K_X$ be nef and effective, and define the irregularity by $q(X)=h^1(\mathcal{O}_X)$. Refined indecomposability conjecture. If…

Let $X$ be a smooth projective variety. A variety is derived-indecomposable if $D(X)$ has no nontrivial semi-orthogonal decompositions. Kawatani's converse conjecture. If $D(X)$ ha…

Let $X$ be a smooth projective variety. Assume that the canonical bundle $K_X$ is nef and effective. A minimality conjecture for indecomposable derived categories asserts that…

Let $Z$ be the toric variety associated to a wall-crossing, and suppose its derived category has a semi-orthogonal decomposition … where each $D^b(Z_i)$ occurs some number of times…

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Let $\mathrm{M}_K$ be the abelian group freely generated by isomorphism classes of smooth proper…

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. For a field $L$, let $K_0(\mathrm{GT}_L)$ and $K_0(\mathrm{sGT}_L)$ denote the Grothendieck grou…

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. For a field $L$, let $K_0(\mathrm{sGT}_L)$ be the Grothendieck group generated by bounded derive…

Let $X$ be a smooth projective variety. Let $K_X$ be its canonical bundle and let $h^0(K_X)$ denote the dimension of its global sections. A line bundle is nef if it has nonnegative…

Let $C$ be a projective smooth curve of genus $g\geq 2$, and let $C_d$ denote its $d$-th symmetric product. Let $D(C_d)$ be its derived category, and call a semi-orthogonal decompo…

Ultimate-dimension finiteness conjecture. If the ultimate dimensions of $\mathcal A$ and $\mathcal B$ are finite, then the ultimate dimension of $\mathcal T$ is finite. The source…

Let $X$ be a smooth algebraic variety, and let … be a semi-orthogonal decomposition of $\operatorname{D}^{\operatorname{b}}(\operatorname{coh} X)$. A gap conjecture for semi-orthog…