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Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles. (English) Zbl 1038.14018

Two main results are proved here, both concerning Fano \(n\)-manifolds \(X\) that contain a rational curve \(C\) with trivial normal bundle. By this one means that if \(\mu: \widetilde C \to C\) is the normalisation, then \[ 0\to \Theta_{\widetilde C}\to \mu^* \Theta_X\to {\mathcal O}_C^{n-1}\to 0 \] is exact. The first result is that, if \(f: X'\to X\) is a generically finite holomorphic map from any projective manifold \(X'\), then \(f\) is rigid up to automorphisms of \(X\); that is, if we try to deform \(X\) or \(f\) keeping \(X'\) fixed we get nothing new except by composing \(f\) with an automorphism of \(X\). In particular the set of such maps \(f\) is countable (again, modulo automorphisms of \(X\)). The second result is that if \(X'\) is also Fano with \(\rho(X')=1\), and \(f: X'\to X\) is finite, then \(\deg f\) is bounded by a constant depending on \(X'\) alone.
From this it follows that, given a Fano manifold \(X'\) with \(\rho(X')=1\), there are only finitely many \(X\) as above which are images of \(X'\). This, and more, was proved in the case \(n=3\) by E. Amerik [Doc. Math., J. DMV 2, 195–211 (1997; Zbl 0922.14007)] and others, but using the classification of Fano 3-folds.
The method here is, very roughly, to deform \(C\) in \(X\) in an \((n-1)\)-dimensional family, thus obtaining a finite map \(\phi: F \to X\), and to study the branching behaviour of \(\phi\). This contrasts strongly with the authors’ earlier paper [J.-M. Hwang and N. Mok, J. Math. Pures Appl., IX. Sér. 80, 563–575 (2001; Zbl 1033.32013)]: Some of the results proved there are quite similar, but the methods are completely different and rely precisely on the absence of a curve with trivial normal bundle.

MSC:

14J45 Fano varieties
14J40 \(n\)-folds (\(n>4\))
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