Sheaves of slice regular functions

Abstract

AbstractSlice regular functions have been introduced in 20 as solutions of a special partial differential operator with variable coefficients. As such they do not naturally form a sheaf. In this paper we use a modified definition of slice regularity, see 21, to introduce the sheaf of slice regular functions with values in in the algebra of quaternions and, more in general, in a Clifford algebra and we study its cohomological properties. We show that the first cohomology group with coefficients in the sheaf of slice regular functions vanishes for any open set in the space of quaternions (resp. the space of paravectors in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+1}$\end{document}). However, we prove that not all the open sets are domains of slice regularity but only those special sets which are axially symmetric, i.e., invariant with respect to rotations that fix the real axis.