Spectral Structures on Vector Bundles

Abstract

We develop an intrinsic theory of spectral structures on holomorphic vector bundles over compact Riemann surfaces, generalising eigenspace decompositions without eigenvalues. Their geometric realisations are presentations of vector bundles as pushforwards of line bundles along finite branched covers. We construct their analytic moduli stack locally of finite presentation and identify it with the relative Picard stack of the universal family of finite covers. We prove that the forgetful morphism to the stack of vector bundles is representable by analytic spaces and is smooth and surjective on a distinguished smooth open substack. Consequently, every holomorphic family of vector bundles locally admits spectral presentations along smooth connected simply branched covers, and these presentations can be deformed alongside the underlying bundles. The relative Picard description yields explicit local dimension formulas and recovers the stack of regular Higgs bundles as base change along the Hitchin base. We also characterise the additional data supplied to a spectral structure by a regular Higgs field as a global twisted generator, and give examples of smooth connected spectral structures which cannot arise from any Higgs field.