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This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier–Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier–Mukai transforms. As a consequence, an affirmative answer is given to the conjecture proposed by Mukai in the 1980s. This paper also includes an explicit description of the birational correspondence between the moduli spaces of stable sheaves and the Hilbert schemes.

Semi-homogeneous sheaves, Fourier–Mukai transforms and moduli of stable sheaves on abelian surfaces