论文标题
Fano品种的补充和核心性
Complements and coregularity of Fano varieties
论文作者
论文摘要
我们研究了核心性，log calabi-yau对的索引与Fano品种的补充之间的关系。我们表明，log calabi-yau对$（x，b）coregularity $ 1 $的索引最多为$120λ^2 $，其中$λ$是$ k_x+b $的Weil索引。由于Filipazzi，Mauri和Moraga，这延伸了最近的结果。我们证明，一系列绝对核心的fano $ 0 $可以承认$ 1 $结合或$ 2 $结合。对于具有绝对核心的Fano品种$ 1 $，我们表明他们承认与$ n $的$ n $最多6 $ 6.使用先前的结果，我们证明了绝对核心的klt奇异性$ 0 $承认要么承认$ 1 $ complement complement或$ 2美元。此外，绝对核心$ 1 $的KLT奇异性最多可以用$ n $ 6 $ n $ 6。这将$ a，d，e $ type klt klt klt表面奇异性的经典分类扩展到任意维度。在Coregularity $ 2 $的情况下，证明了类似的结果。在证明过程中，我们证明了一个新型的规范捆绑套件，用于具有有限的相对核心性的对。在核心性至少$ 3 $的情况下，我们在索引猜想的假设和b-pressentations的界限下建立了类似的陈述。
We study the relation between the coregularity, the index of log Calabi-Yau pairs, and the complements of Fano varieties. We show that the index of a log Calabi-Yau pair $(X,B)$ of coregularity $1$ is at most $120λ^2$, where $λ$ is the Weil index of $K_X+B$. This extends a recent result due to Filipazzi, Mauri, and Moraga. We prove that a Fano variety of absolute coregularity $0$ admits either a $1$-complement or a $2$-complement. In the case of Fano varieties of absolute coregularity $1$, we show that they admit an $N$-complement with $N$ at most 6. Applying the previous results, we prove that a klt singularity of absolute coregularity $0$ admits either a $1$-complement or $2$-complement. Furthermore, a klt singularity of absolute coregularity $1$ admits an $N$-complement with $N$ at most 6. This extends the classic classification of $A,D,E$-type klt surface singularities to arbitrary dimensions. Similar results are proved in the case of coregularity $2$. In the course of the proof, we prove a novel canonical bundle formula for pairs with bounded relative coregularity. In the case of coregularity at least $3$, we establish analogous statements under the assumption of the index conjecture and the boundedness of B-representations.
