5 problems

Serre functor conjecture. The functor $S$ is right exact, and its left derived functor ${\mathcal L}S$ is a Serre functor. This is a proposed description of the Serre functor for c…

Perverse sheaves on flag varieties form a bounded derived category with a Serre functor. Beilinson–Bezrukavnikov–Mirkovic conjecture. The Serre functor of the bounded derived categ…

Let $\mathcal{A}$ be an Ext-finite abelian category with Serre duality, and let $(\mathcal{T},\mathcal{F})$ be a torsion pair on $\mathcal{A}$. Denote by $U_{\mathcal{T}}$ the aisl…

Let $M \subset \mathbb{P}^n$ be a smooth complete intersection of type $(d_1,d_2,\dots,d_{k-1})$, let $X \subset M$ be a smooth hypersurface section of degree $d_k$, and assume…

Parity-shift Serre functor conjecture. Under the assumption $\Pi V\cong V$, the functor $\Pi$ is the Serre functor on