AI 中文总结
研究考蒂斯 - 洛维年科猜想，当有限子群\(G\subset \operatorname{SL}(3,\mathbb{C})\)的麦凯箭图无圈时证明该猜想，涵盖多种子群，明确计算支撑维数为一层，得出定义盖尔对偶矩阵符号相干及可读出纯层相关信息的结论。
AI 中文摘要
对于有限子群\(G\subset \operatorname{SL}(3,\mathbb{C})\)，考蒂斯 - 洛维年科猜想指出，对于\(G\)的每个非平凡不可约表示\(\rho\)，在布里奇兰德 - 金 - 里德导出等价下，层\(\mathcal{O}_0\otimes \rho\)的像为\(G\)-希尔伯特概型上的纯层。当\(G\)的麦凯箭图无圈时我们证明了此猜想，涵盖许多\(\operatorname{SL}(3,\mathbb{C})\)的二面体和三面体子群以及八个散在有限子群中的六个。我们还在支撑维数为一时明确计算了相关层。主要结果表明定义线性化映射的盖尔对偶的矩阵是符号相干的，可直接从矩阵读出纯层的支撑和上同调度数。
英文摘要
For a finite subgroup $G\subset \operatorname{SL}(3,\mathbb{C})$, the Cautis--Logvinenko conjecture states that for each nontrivial irreducible representation $ρ$ of $G$, the image of the sheaf $\mathcal{O}_0\otimes ρ$ under the derived equivalence of Bridgeland--King--Reid is a pure sheaf on the $G$-Hilbert scheme. We prove a strong form of this conjecture in complete generality, and in doing so, we compute the relevant sheaf explicitly whenever its support is of dimension one. Our main result implies that a matrix defining the Gale dual of the linearisation map is sign-coherent, thereby allowing us to read off the support and cohomological degree of the pure sheaves directly from the matrix.
Commentsv2: 22 pages, we prove the conjecture in complete generality, without having to assume that there are no loops at certain vertices in the McKay quiver. Coauthor added
