A refinement of Stein factorization and deformations of surjective morphism (Q1002440)

Let \(f: X \rightarrow Y\) be a surjective morphism between normal complex projective varieties. The Stein factorization separates \(f\) into a morphism \(X \rightarrow W\) with connected fibres and a finite morphism \(g: W \rightarrow Y\), where \(W\) is a normal variety. The main statement of this article shows that the morphism \(g\) itself can be factored into finite morphisms \(W \rightarrow Z\) and \(\beta: Z \rightarrow Y\) such that \(\beta\) is étale in codimension one and is maximal with this property. The construction of the variety \(Z\) is quite interesting: using positivity properties of the sheaf \(g_* \mathcal O_W\), the authors show that the first graded piece of the Harder-Narasimhan filtration of \(g_* \mathcal O_W\) is a \(\mathcal O_Y\)-algebra, the variety \(Z\) is the analytic spectrum of this algebra. The morphism \(\beta\) (the so-called maximal étale factorization) is invariant under deformation of \(f\): if \(f_t: X \rightarrow Y\) is a family of surjective morphisms such that \(f_0=f\), then \(f_t\) factors through \(\beta\) and \(\beta\) is the maximal étale factorization of \(f_t\). As an application of this refined Stein factorization, the authors continue the study of deformations of surjective morphisms initiated by \textit{J.-M. Hwang, S. Kebekus} and \textit{T. Peternell} [J. Algebr. Geom. 15, No. 3, 551--561 (2006; Zbl 1112.14014)]. There it was shown that if \(Y\) is not covered by rational curves, the deformations of the morphism \(f\) are unobstructed and the components of the Hom-scheme \(\Hom_f(X,Y)\) are smooth abelian varieties. At the other extreme if \(Y\) is rationally connected, such a description of the deformations of \(f\) no longer holds. The refined Stein factorization allows to interpolate between these two cases, i.e. to say something about targets \(Y\) that are uniruled but not rationally connected. It is well-known that such a variety \(Y\) admits a rational map \(q_Y : Y \dashrightarrow Q_Y\) such that the general fibre is rationally connected and the base \(Q_Y\) is not covered by rational curves. The authors show that an étale cover of the normalization of \(\Hom_f(X,Y)\) is a product of a torus and a normal variety \(H\) corresponding to the space parametrizing the deformations \(f_t\) of \(f\) over \(q_Y\), i.e. such that \(q_Y \circ f=q_Y \circ f_t\).