Some Examples of Isomorphisms Induced by Fourier-mukai Functors

In order to investigate sheaves on abelian varieties, Mukai [Mu1] introduced a very powerful tool called Fourier-Mukai functor. As an application, Mukai ([Mu1], [Mu4]) computed some moduli spaces of stable sheaves on abelian varieties. Recently, Dekker [D] found some examples of isomorphisms of moduli spaces of sheaves induced by Fourier-Mukai functors. As an application, he proved that moduli spaces of sheaves on abelian surfaces are deformation equivalent to Hilbert schemes of points (see Theorem 1.4). Recently Fourier-Mukai functor was generalized to more general situations (e.g. [Br1], [Br2], [Mu6], [Mu7]). Next task is to construct many examples of birational maps of moduli spaces of sheaves. In this note, we restrict ourselves to an abelian or a K3 surface of Picard number 1 and give some examples of birational maps of moduli spaces of sheaves induced by Fourier-Mukai functor (Theorem 2.3). More precisely, we shall find some examples of sheaves which satisfy WITi. In general, Fourier-Mukai functor does not induce isomorphisms of moduli spaces of sheaves. Motivated by recent work of Markman [Mr], we also consider the composition of Fourier-Mukai functor and “taking-dual” functor. Then we can get isomorphisms in these cases. As an application, we get another proof of Dekker’s result (Theorem 1.4). Our condition and method are similar to [Y4]. In section 1, we shall treat original Fourier-Mukai functor. In section 2, we shall explain how to generalize it to more general situations.