Remarks on approximate decompositions of the diagonal

Some remarks on modified diagonals

We prove a number of basic vanishing results for modified diagonal classes. We also obtain some sharp results for modified diagonals of curves and abelian varieties, and we prove a conjecture of O'Grady about modified diagonals on double covers.

关于代数堆的对角维数的评论

Remarks on diagonal dimension for algebraic stacks

本文研究代数堆有界导出范畴的Rouquier维数，提出通过研究映射的对角维数来获得上界，并为光滑映射提供明确上界，恢复Elagin-Lunts-Schnürer的结果，并证明具有轻微奇点的变种的对角维数不超过其Krull维数的两倍。

Further remarks on derived categories of algebraic stacks

Let $X$ be an algebraic stack with quasi-affine diagonal of finite type over a field $k$ of characteristic $0$. We extend the well-known equivalence $\mathsf{D}^+(\mathsf{QCoh}(X)) \simeq \mathsf{D}_{\mathrm{qc}}^+(X)$ to unbounded derived categories. We also prove that if $X$ is smooth over $k$, then $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated. We accomplish the former using the descendable algebras of Mathew. We also establish related results in positive and mixed characteristics.

Remarks on the Grothendieck-Teichmüller group and A. Beilinson's gluing

In this note, we study A. Beilinson's gluing for perverse sheaves in the case of the diagonal arrangement and its relation to the Grothendieck-Teichmüller group. We also explain a relation to the Etingof-Kazhdan quantisation