A weight two phenomenon for the moduli of rank one local systems on open varieties

ar X iv :0 71 0. 28 00 v1 [ ma th. AG ] 15 O ct 20 07 A weight two phenomenon for the moduli of rank one local systems on open varieties Carlos Simpson Abstract. The twistor space of representations on an open variety maps to a weight two space of local monodromy transformations around a divisor com- ponent at infinty. The space of ?-invariant sections of this slope-two bundle over the twistor line is a real 3 dimensional space whose parameters correspond to the complex residue of the Higgs field, and the real parabolic weight of a harmonic bundle. 1. Introduction Let X be a smooth projective variety and D ? X a reduced effective divisor with simple normal crossings. We would like to define a Deligne glueing for the Hitchin twistor space of the moduli of local systems over X ? D. Making the construction presents new difficulties which are not present in the case of compact base, so we only treat the case of local systems of rank 1. Every local system comes from a vector bundle on X with connection logarithmic along D, however one can make local meromorphic gauge transformations near components of D, and this changes the structure of the bundle as well as the eigenvalues of the residue of the connection. The change in eigenvalues is by subtracting an integer.

• tate twistor


• local monodromy


• also defined over


• tangent space


• space constructed


• deligne glueing


• direct gener- alization


• no reasonable algebraic


• every local


• transformation affects