Stable vector bundles with \(\chi=0\) on a simple abelian surface (Q1818755)

The author produces a birational description of one component of \({\mathcal M}_X(2,D,0)\), the moduli space of stable rank \(2\) bundles with ample determinant \(D\) and Euler characteristic \(0\) over a simple abelian surface \(X\). Indeed if \(g=1+D^2/2\) and \(\phi_D:X\to \widehat X\) is the isogeny associated to \(D\), then he defines the divisor \(\widetilde D\) on \(\widehat X\) by the relation \(d\widetilde D=(\phi_D)_*D\). The author uses the Fourier transform from \(X\) to \(\widehat X\) to prove that \({\mathcal M}_X(2,D,0)\) has one component birational to the relative Jacobian \({\mathcal J}^{g-3}\) over \(\{ \widetilde D\}\), the space of curves algebraically equivalent to \(\widetilde D\).