Splitting of supervector bundles on projective superspaces

We provide a splitting criterion for supervector bundles over the projective superspaces ${\mathbb{P}}^{n\mathrm{\mid }m}$. More precisely, we prove that a rank $p\mathrm{\mid }q$ supervector bundle on ${\mathbb{P}}^{n\mathrm{\mid }m}$ with vanishing intermediate cohomology is isomorphic to the direct sum of even and odd line bundles, provided that $n\ge 2$. For $n=1$ we provide an example of a supervector bundle that cannot be written as a sum of line bundles.