Splitting of supervector bundles on projective superspaces

Abstract We provide a splitting criterion for supervector bundles over the projective superspaces $$\mathbb {P}^{n|m}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>|</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:math> . More precisely, we prove that a rank p | q supervector bundle on $$\mathbb {P}^{n|m}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>|</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:math> with vanishing intermediate cohomology is isomorphic to the direct sum of even and odd line bundles, provided that $$n \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . For $$n=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> we provide an example of a supervector bundle that cannot be written as a sum of line bundles.