## Advanced Studies in Pure Mathematics

### Deformation of morphisms onto Fano manifolds of Picard number 1 with linear varieties of minimal rational tangents

Jun-Muk Hwang

#### Abstract

Let $X$ be a Fano manifold of Picard number 1, different from projective space. We study the question whether the space $\mathrm{Hom}^s(Y,X)$ of surjective morphisms from a projective manifold $Y$ to $X$ is homogeneous under the automorphism group $\mathrm{Aut}_o(X)$. An affirmative answer is given in [4] under the assumption that $X$ has a minimal dominating family $\mathcal{K}$ of rational curves whose variety of minimal rational tangents $\mathcal{C}_x$ at a general point $x \in X$ is non-linear or finite. In this paper, we study the case where $\mathcal{C}_x$ is linear of arbitrary dimension, which covers the cases unsettled in [4]. In this case, we will define a reduced divisor $B^{\mathcal{K}} \subset X$ and an irreducible subvariety $M^{\mathcal{K}} \subset \mathrm{Chow}(X)$ naturally associated to $\mathcal{K}$. We give a sufficient condition in terms of $\mathbf{B}^{\mathcal{K}}$ and $M^{\mathcal{K}}$ for the homogeneity of $\mathrm{Hom}^s(Y,X)$. This condition is satisfied if $\mathcal{C}_x$ is finite and our result generalizes [4]. A new ingredient, which is of independent interest, is a similar rigidity result for surjective morphisms to projective space in logarithmic setting.

#### Article information

Dates
Revised: 20 June 2014
First available in Project Euclid: 23 October 2018

https://projecteuclid.org/ euclid.aspm/1540319490

Digital Object Identifier
doi:10.2969/aspm/07410237

Mathematical Reviews number (MathSciNet)
MR3791216

Zentralblatt MATH identifier
1388.14120

Subjects
Primary: 14J40: $n$-folds ($n > 4$)

#### Citation

Hwang, Jun-Muk. Deformation of morphisms onto Fano manifolds of Picard number 1 with linear varieties of minimal rational tangents. Higher Dimensional Algebraic Geometry: In honour of Professor Yujiro Kawamata's sixtieth birthday, 237--264, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07410237. https://projecteuclid.org/euclid.aspm/1540319490