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**Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1.**
*(English)*
Zbl 1126.32011

Let \(f:Y \rightarrow X\) be a surjective holomorphic map from a compact complex manifold onto a projective manifold. An obvious way to deform the map \(f\) is to take a family of automorphisms \(g_t\) of \(X\) such that \(g_0\) is the identity; the morphisms \(g_t \circ f\) then define a family of deformations of \(f\). J.-M. Hwang, S. Kebekus and T. Peternell [J. Algebr. Geom. 15, No. 3, 551–561 (2006; Zbl 1112.14014)] have shown that if \(X\) is not uniruled, then (up to an étale cover) all the deformations of \(f\) arise in this way. It is natural to ask what happens in a completely opposite situation, for example if \(X\) is a Fano manifold.

If \(X\) is the projective space \(\mathbb P^n\), it is not true in general that all the deformations of \(f\) come from its automorphisms; if \(Y\) is a projective manifold embedded in some projective space \(\mathbb P^N\) for \(N\) large, then there are many ways to project \(Y\) onto \(\mathbb P^n \subset \mathbb P^N\). Nevertheless there is a good reason to believe that the projective space is rather exceptional in this regard. In fact it is conjectured that the projective space is the only Fano manifold with second Betti number one such that the variety of minimal rational tangents of \(X\) at a general point is a union of linear subspaces (recall that given a family of minimal rational curves on \(X\), the variety of minimal rational tangents is the subset of \(\mathbb P(T_{X,x})\) obtained as the closure of the tangent directions of the minimal rational curves passing through the point \(x\) and smooth at this point). Furthermore, J.-M. Hwang and N. Mok [Asian J. Math. 8, No. 1, 51–64 (2004; Zbl 1072.14015)] have shown that if the variety of minimal rational tangents of \(X\) at a general point is not a union of linear subspaces, then the deformations of \(f\) come from the automorphisms of \(X\). This result motivates the (hopefully easier) conjecture that if \(X\) is a Fano manifold with second Betti number one that is not the projective space, then all the deformations of morphisms onto \(X\) come from its automorphisms.

The paper under review gives partial answers to the two conjectures mentioned. Firstly, if \(X\) is a Fano manifold with second Betti number one such that the variety of minimal tangents is a union of linear subspaces of dimension at least \(\frac{\dim X-1}{2}\), then \(X\) is the projective space. Secondly, if \(X\) is a Fano manifold with second Betti number one such that the cohomology classes of any two surfaces in \(X\) are proportional (e.g., if the fourth Betti number is one), then all the deformations of morphisms onto \(X\) come from its automorphisms. These two statements are obtained by analysing the geometry of the multi-valued foliation defined by the minimal tangent varieties in the case where they are a union of linear subspaces.

If \(X\) is the projective space \(\mathbb P^n\), it is not true in general that all the deformations of \(f\) come from its automorphisms; if \(Y\) is a projective manifold embedded in some projective space \(\mathbb P^N\) for \(N\) large, then there are many ways to project \(Y\) onto \(\mathbb P^n \subset \mathbb P^N\). Nevertheless there is a good reason to believe that the projective space is rather exceptional in this regard. In fact it is conjectured that the projective space is the only Fano manifold with second Betti number one such that the variety of minimal rational tangents of \(X\) at a general point is a union of linear subspaces (recall that given a family of minimal rational curves on \(X\), the variety of minimal rational tangents is the subset of \(\mathbb P(T_{X,x})\) obtained as the closure of the tangent directions of the minimal rational curves passing through the point \(x\) and smooth at this point). Furthermore, J.-M. Hwang and N. Mok [Asian J. Math. 8, No. 1, 51–64 (2004; Zbl 1072.14015)] have shown that if the variety of minimal rational tangents of \(X\) at a general point is not a union of linear subspaces, then the deformations of \(f\) come from the automorphisms of \(X\). This result motivates the (hopefully easier) conjecture that if \(X\) is a Fano manifold with second Betti number one that is not the projective space, then all the deformations of morphisms onto \(X\) come from its automorphisms.

The paper under review gives partial answers to the two conjectures mentioned. Firstly, if \(X\) is a Fano manifold with second Betti number one such that the variety of minimal tangents is a union of linear subspaces of dimension at least \(\frac{\dim X-1}{2}\), then \(X\) is the projective space. Secondly, if \(X\) is a Fano manifold with second Betti number one such that the cohomology classes of any two surfaces in \(X\) are proportional (e.g., if the fourth Betti number is one), then all the deformations of morphisms onto \(X\) come from its automorphisms. These two statements are obtained by analysing the geometry of the multi-valued foliation defined by the minimal tangent varieties in the case where they are a union of linear subspaces.

Reviewer: Andreas Höring (Paris)

### MSC:

32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |

14J45 | Fano varieties |

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\textit{J.-M. Hwang}, Ann. Inst. Fourier 57, No. 3, 815--823 (2007; Zbl 1126.32011)

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