Invariant syzygies for the Hermitian Dirac operator

Abstract

This paper is devoted to the algebraic analysis of the system of differential equations described by the Hermitian Dirac operators, which are two linear first order operators invariant with respect to the action of the unitary group. In the one variable case, we show that it is possible to give explicit formulae for all the maps of the resolution associated to the system. Moreover, we compute the minimal generators for the first syzygies also in the case of the Hermitian system in several vector variables. Finally, we study the removability of compact singularities. We also show a major difference with the orthogonal case: in the odd dimensional case it is possible to perform a reduction of the system which does not affect the behavior of the free resolution, while this is not always true for the case of even dimension.