## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Higher Dimensional Algebraic Geometry: In honour of Professor Yujiro Kawamata's sixtieth birthday, K. Oguiso, C. Birkar, S. Ishii and S. Takayama, eds. (Tokyo: Mathematical Society of Japan, 2017), 237 - 264

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

#### 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**

Received: 29 April 2013

Revised: 20 June 2014

First available in Project Euclid:
23 October 2018

**Permanent link to this document**

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 ($n > 4$)

**Keywords**

varieties of minimal rational tangents minimal rational curves

#### 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