Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles

By Jun-Muk Hwang, N. Mok. 2003

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Let X be a Fano manifold of Picard number 1 admitting a rational curve with trivial normal bundle and f : X′ → X be a generically finite surjective holomorphic map from a projective manifold X′ onto X. When the domain manifold X′ is fixed and the target manifold X is a priori allowed to deform we prove that the holomorphic map f : X′ → X is locally rigid up to biholomorphisms of target manifolds. This result complements, with a completely different method of proof, an earlier local rigidity theorem of ours (see J. Math. Pures Appl. 80 (2001), 563– 575) for the analogous situation where the target manifold X is a Fano manifold of Picard number 1 on which there is no rational curve with trivial normal bundle. In another direction, given a Fano manifold X′ of Picard number 1, we prove a finiteness result for generically finite surjective holomorphic maps of X′ onto Fano manifolds (necessarily of Picard number 1) admitting rational curves with trivial normal bundles. As a consequence, any 3-dimensional Fano manifold of Picard number 1 can only dominate a finite number of isomorphism classes of projective

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0dcb91181f857b26153fa0fc96658bf9ea5fe6b5

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Polarized endomorphisms of complex normal varieties (2009, background): “Especially, the following cases of varieties are well studied: projective surfaces ([44], [17]), homogeneous manifolds ([49], [11]), some special Fano threefolds ([28]), projective bundles ([2]), and projective threefolds with non-negative Kodaira dimension ([16], [18]).”