Algebraically trivial automorphisms of irreducible holomorphic symplectic manifolds
Stevell Muller
TL;DR
This work extends Brandhorst–Cattaneo’s lattice framework to classify algebraically trivial nonsymplectic automorphisms of projective IHS manifolds across known deformation types, up to deformation and birational conjugacy. The authors reduce the geometric problem to classifying finite-order isometries of even unimodular lattices with cyclotomic minimal polynomials $\Phi_1\Phi_m$ (and $\Phi_1\Phi_2\Phi_m$ in the nontrivial discriminant case) via the cyclotomic trace correspondence with hermitian lattices, equivariant primitive extensions, and local $p$-adic data. They provide both necessary and sufficient conditions for the existence of such isometries, classify conjugacy classes, and map these to monodromy actions to yield concrete classifications for all known IHS deformation types, including explicit examples. A central outcome is a finiteness/resultative classification: up to deformation and birational conjugacy, the action on cohomology is determined by the deformation type, the order $m$, and discriminant data, with a small number of exceptional cases where multiple classes arise. The results give a concrete lattice-theoretic pipeline to realize or rule out nonsymplectic, algebraically trivial automorphisms on IHS manifolds and provide explicit instances across OG6, OG10, K3$^{[n]}$, and Kum$_n$ types.
Abstract
We extend the lattice-theoretic approach of Brandhorst--Cattaneo to classify algebraically trivial actions on the known IHS manifolds, up to deformation and birational conjugacy. In particular, we classify even order algebraically trivial nonsymplectic automorphisms, with or without trivial discriminant action. In the case of nontrivial discriminant actions, we show that such automorphisms exist only for finitely many known deformation types and orders.