2026/07/28 by Alastair Craw, Ryo Yamagishi
Mathematics · #math.AG #math.RT
For a finite subgroup G⊂ SL(3,ℂ), the Cautis--Logvinenko conjecture states that for each nontrivial irreducible representation ρ of G, the image of the sheaf O0⊗ ρ under the derived equivalence of Bridgeland--King--Reid is a pure sheaf on the G-Hilbert scheme. We prove this when the McKay quiver of G contains no loops; this includes many dihedral and trihedral subgroups of SL(3,ℂ), as well as six of the eight sporadic finite subgroups. 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.