Higher-rank instantons sheaves on Fano threefolds

Gaia Comaschi and Daniele Faenzi

Vol. 344 (2026), No. 2, 293–358

DOI: 10.2140/pjm.2026.344.293

Abstract

We define higher-rank instanton sheaves on smooth Fano threefolds $X$ of Picard rank one and show that their topological type depends on two integers, namely the rank $n$ (or half of it if the Fano index of $X$ is odd) and the charge $k$. With the possible exception of Fano threefolds of index one and genus at most $3$, or genus $4$ contained in a singular quadric, we determine the minimal charge $k_0$ of slope-stable $n$-instanton bundles as an integer depending only on $H_X^3$ and $n$, and we prove the existence of slope-stable $n$-instanton bundles of charge $k\ge k_0$.

Next, we study several properties of a general element of an irreducible component, which we call the main component, of the moduli space of instanton sheaves. This component is defined inductively on the rank and the charge, a natural procedure based on elementary transformations along rational curves, starting with minimal instantons. The questions we address include the generic splitting over certain rational curves contained in $X$, as well as stable restriction to a K3 section $S$ of $X$. We obtain applications to Lagrangian subvarieties of moduli spaces of sheaves on $S$.

Finally, we study the acyclic extension associated with an instanton sheaf on Fano threefolds with curvilinear Kuznetsov component and give a monadic description in the case $H^3(X,\mathbb{Z})=0$.

Keywords

instanton bundle, Fano threefold, moduli space of instantons, restriction of stable sheaves, curvilinear Kuznetsov component, monads

Mathematical Subject Classification

Primary: 14F06, 14F08, 14J60

Secondary: 14D21

Milestones

Received: 23 April 2025

Revised: 26 May 2026

Accepted: 3 June 2026

Published: 8 September 2026