We introduce the notion of Hitchin variety over C. Let L be a holomorphic line bundle X. investigate space all global sections sheaf differential operators Dk(L) and symmetric powers first order Sk(D1(L)) X show that for projective Hithcin both spaces are one dimensional. As an application, we C(L) connections on does not admit any non-constant regular function.

These are extended notes of my talk at the IMPANGA seminar in Warsaw on October 11, 2019. The goal was to give a non-technical and arithmetically motivated introduction definition limiting mixed Hodge structure by Schmid Deligne. We state several assertions terms natural classical theory ordinary differential operators prove them using elementary arguments. References geometric context only mad...