A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves

Abstract

<p>Let $ X $ be a compact Riemann surface of genus $ g\geq 2 $ and $ M(G_2) $ be the moduli space of polystable principal $ G_2 $-bundles over $ X $. The Harder-Narasimhan types of the bundles induced a stratification of the moduli space $ M(G_2) $ called Shatz stratification. In this paper, a description of the Shatz strata of the unstable locus of $ M(G_2) $ corresponding to certain family of Harder-Narasimhan types (specifically, those of the form $ (\lambda, \mu, 0, -\mu, -\lambda) $ with $ \mu < \lambda\leq 0 $) was given. For this purpose, a family of vector bundles was constructed in which a 3-form and a 2-form were defined so that it was proved that they were strictly polystable principal $ G_2 $-bundles. From this, it was proved that, when the genus of $ X $ was $ g\geq 12 $, these Shatz strata were the disjoint union of a family of $ G_2 $-Hecke curves in $ M(G_2) $ that will be constructed along the paper. Therefore, the presented results provided an advance in the knowledge of the geometry of $ M(G_2) $ through the study of its Shatz strata and presented a methodological innovation, by using Hecke curves for this study.</p>