The boundary Calabi–Yau component conjecture for semiorthogonal decompositions

Let $X$ be a smooth projective variety with a semiorthogonal decomposition

$$\operatorname{D}^b(X)=\langle \mathcal{A},\mathcal{B}\rangle$$

where $\mathcal{A}$ is a Calabi–Yau category of dimension $n=\dim X$. Boundary Calabi–Yau component conjecture. Then $X$ is a blowup of a Calabi–Yau variety $Y$ of dimension $n$, and $\mathcal{A}\cong \operatorname{D}^b(Y)$. The preceding argument shows that any such Calabi–Yau component has dimension at most that of $X$; the conjecture describes the restrictive boundary case of equality and remains unresolved in the stated generality.