Acyclic Toric Sheaves

Abstract Let $$\mathcal {E}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> be a torus-linearised reflexive sheaf over a smooth projective toric variety. Generalising a theorem of Perlman and Smith, we prove an explicit sufficient condition for $$\mathcal {E}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> to be acyclic via Weil decorations.