Jordan Normal Forms of Endomorphisms of Vector Bundles over Curves and Applications to Moduli Space Automorphisms

Let X be a compact connected Riemann surface of genus g≥2 and let E be a holomorphic vector bundle of rank n over X. The compactness and connectedness of X imply that the characteristic polynomial of any holomorphic endomorphism φ∈H0(X,End(E)) has constant coefficients, a fact we call the Principle of Spectral Constancy. As a consequence, the eigenvalues of φ are globally constant over X, the primary decomposition of E with respect to φ consists of globally defined holomorphic subbundles, and the Jordan decomposition φ=φs+φn into semisimple and nilpotent parts is globally well defined as a decomposition of sections of End(E). This paper provides a systematic analysis of Jordan normal forms for endomorphisms of holomorphic vector bundles over X, relating the Jordan type of φ to the stability properties of E. In particular, it is proved that endomorphisms of stable bundles are necessarily scalar, that the Jordan decomposition of an endomorphism of a polystable bundle is determined componentwise by the classical Jordan normal forms of matrices in the associated endomorphism algebra, and that finite-order endomorphisms are always semisimple. These results are applied to the study of fixed points of automorphisms of the moduli space BX(SL(n,C)) of rank n and trivial determinant polystable vector bundles over X. Specifically, a new result establishes that the commutative subalgebra of H0(X,End(E)) generated by the endomorphism associated with a fixed-point condition is semisimple, so nilpotent endomorphisms of E are precisely those incompatible with the fixed-point structure.