For a finite subgroup $G\subset \operatorname{SL}(3,\mathbb{C})$, the Cautis--Logvinenko conjecture states that for each nontrivial irreducible representation $\rho$ of $G$, the image of the sheaf $\mathcal{O}_0\otimes \rho$ 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.
Let $\mathscr{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$, and let $\mathfrak{X}$ be a proper formal scheme over $\mathscr{O}_K$ with semistable reduction. Answering a question of Fontaine and Jannsen, a classical theorem of Hyodo and Kato gives a rational identification between the log...
Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. We study the smooth mod $p$ representations of $\mathrm{GL}_2(K)$ appearing in a tower of mod $p$ Hecke eigenspaces of the cohomology of Shimura curves, under mild genericity assumptions but notably no multiplicity one assumption at tame...
We show that the Tate module of the $\ell$-adic unramified cohomology $H^3_{\nr}(X, {\Q_\ell}/{\Z_\ell})$ of a smooth projective surface $X$ over a local or strictly local field $k$ of residue characteristic prime to $\ell$ vanishes under suitable conditions. We apply this to answer some questions concerning the left a...
We show that for a finite group $G$, the $G$-category $\underline{\mathrm{Sp}}_{G,\mathbb{Q}}$ of rational genuine $G$-spectra is parametrized semiadditive, in the sense of Cnossen--Lenz--Linskens, for morphisms of genuine $G$-spaces whose fixed point maps have $\pi$-finite fibers. In particular, this means that the ca...
We prove that two finite-dimensional linear representations of the braid group are isomorphic. One representation comes from the monodromy of the boundary Casimir connection, and the other comes from the $\iota$quantum Weyl group. Both representations are defined for any split symmetric pair $\mathfrak{k}\subset \mathf...
We investigate how Heegner points detect the finiteness of the $p$-primary part of Tate-Shafarevich groups of semi-stable elliptic curves over the rationals of \emph{arbitrary} rank with any good reduction prime $p \geq 5$ with irreducible mod $p$ representation and partially recover \c{C}iperiani--Wiles'theorem on sol...
Chan-Ho Kim· 0 citations
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