The cohomology groups \(H^ 1({\mathbb{P}}^ 3-{\mathbb{P}}^ 1,{\mathcal O}(m))\) (Q2639348)

In my paper in J. Geom. Phys. 3, 191-210 (1986; Zbl 0603.53011), some isomorphism among the cohomology groups \(H^ 1({\mathcal U},{\mathcal O}(-n-2))\) and the spaces \({\mathcal I}^ n_ s(C)\) of holomorphic functions on the cone \[ C=\{z\in {\mathbb{C}}^ 4:\;z_{00}\cdot z_{11}-z_{01}\cdot z_{10}=0\} \] with vanishing order at least n on a plane S of C are introduced with a sketch of proof. The present article develops the proof using a procedure inspired by a method invented by \textit{J. Frenkel} [Bull. Soc. Math. Fr. 85, 135-220 (1957; Zbl 0082.377)]; the isomorphisms so obtained give new representations of the spaces of holomorphic solutions for the Dirac equations [cf. the author (loc. cit.)].