Open Access
VOL. 74 | 2017 Deformation of morphisms onto Fano manifolds of Picard number 1 with linear varieties of minimal rational tangents
Jun-Muk Hwang

Editor(s) Keiji Oguiso, Caucher Birkar, Shihoko Ishii, Shigeharu Takayama

Adv. Stud. Pure Math., 2017: 237-264 (2017) DOI: 10.2969/aspm/07410237


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.


Published: 1 January 2017
First available in Project Euclid: 23 October 2018

zbMATH: 1388.14120
MathSciNet: MR3791216

Digital Object Identifier: 10.2969/aspm/07410237

Primary: 14J40

Keywords: minimal rational curves , Varieties of minimal rational tangents

Rights: Copyright © 2017 Mathematical Society of Japan


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