Abstrakt
The Hartog's type phenomena in several complex variables are best understood in terms of the Dolbeault sequence. A lot of attention was paid in the last decades to its analogue in the function theory of several Clifford variables, i.e. the Dirac operator in several variables.
A so-called BGG resolution of this operator is then an analogue to the Dolbeault sequence. The complete description is known in dimension 4.
Much less is known in higher dimensions. The case of three variables was described completely by F.
Colombo, I. Sabadini, F.
Sommen, D. C.
Struppa. The full description of the complex for all dimensions is not known at present.
In the case of the stable rank (i.e., when the number of variables is less or equal to the half of the even dimension), certain progress has been done. In the paper, we construct the resolution for the case of k variables in the stable range, we show the case of k = 4 in details, and we show the exactness of this sequence.
The tools used in the construction are the Penrose transform, Cech cohomology and Leray theorem.