AI 中文总结
研究Kuznetsov-Shinder关于锥奇点的范畴吸收，推广其对节点奇点的结果。给出特定法诺簇上锥的范畴吸收中倾斜对象自同态代数的描述，得到代数奇点范畴与锥奇点范畴的三角等价及负K理论的消失结果，附录给出加权射影空间倾斜对象描述。
AI 中文摘要
我们研究了Kuznetsov-Shinder关于某些锥奇点的范畴吸收，推广了他们关于节点奇点的结果。特别地，对于某些允许几何例外序列的法诺簇上的锥的范畴吸收，我们给出了倾斜对象的自同态代数的明确描述。在最简单的“分裂情形”下，这些代数是Keller意义下某些卡拉比-丘完备化的截断。许多法诺簇上的反典范射影锥会出现分裂情形。一般来说，这些代数是分裂情形下的变形。由此，我们得到有限维代数的奇点范畴与某些锥奇点的奇点范畴之间的三角等价，这也在负K理论中产生了消失结果。在与Yujiro Kawamata的联合附录中，我们给出了加权射影空间\(\mathbb{P}(1^d, m)\)的倾斜对象的明确描述。
英文摘要
We study Kuznetsov-Shinder's categorical absorption for certain cone singularities, generalizing their results for nodal singularities. In particular, we give explicit descriptions of the endomorphism algebras of tilting objects for categorical absorptions of cones over certain Fano varieties admitting a geometric exceptional sequence, in the sense of Bridgeland and Stern. In the simplest ``split case'', these algebras are truncations of certain Calabi-Yau completions in the sense of Keller. The split case occurs for anticanonical projective cones over many Fano varieties (like projective spaces, smooth quadrics, del Pezzo surfaces of degree greater than $4$, smooth del Pezzo threefolds of degree five, and finite products of these varieties). In general, the algebras are deformations of the split case. As a consequence, we obtain triangle equivalances between singularity categories of finite dimensional algebras and singularity categories of certain cone singularities, which also yields vanishing results in negative $\mathsf{K}$-theory. In a joint appendix with Yujiro Kawamata, we give an explicit description of tilting objects for weighted projective spaces $\mathbb{P}(1^d, m)$.
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