Basic results on irregular varieties via Fourier-Mukai methods (Q2918466)

Let \(A\) be an abelian variety, \(\hat A\) its dual abelian variety and \(\mathcal P\) the normalized Poincaré line bundle on \(A\times \hat A\) so that \(\mathcal P\) parametrizes deformations of the trivial line bundle \(\mathcal O _A\). The Fourier-Mukai transform is an isomorphism between the derived categories of \(A\) and \(\hat A\) given by \(\mathbf{Rq}_*(p^*(\cdot )\otimes \mathcal P ):\mathbf D (A)\to \mathbf D (\hat A)\) where \(p:A\times \hat A \to A\) and \(q:A\times \hat A \to \hat A\) are the projections. In recent years the Fourier-Mukai functor has proven to be an invaluable tool in studying the geometry of irregular varieties, that is varieties with a non-trivial morphism to an abelian variety. In the paper under review the author revisits some of the basic results of the theory of irregular varieties and gives simplified proofs of several important theorems on the geometry of irregular varieties due to Chen-Hacon and to Ein-LazarsfeldNEWLINENEWLINEFor the entire collection see [Zbl 1242.14002].