Automorphism groups of spaces of minimal rational curves on Fano manifolds of Picard number $1$
Authors:
JunMuk Hwang and Ngaiming Mok
Journal:
J. Algebraic Geom. 13 (2004), 663673
DOI:
https://doi.org/10.1090/S1056391104003571
Published electronically:
February 18, 2004
MathSciNet review:
2072766
Fulltext PDF
Abstract  References  Additional Information
Abstract: Let $X$ be a Fano manifold of Picard number $1$ and $M$ an irreducible component of the space of minimal rational curves on $X$. It is a natural problem to understand the extent to which the geometry of $X$ is captured by the geometry of $M$. In this vein we raise the question as to whether the canonical map $\operatorname {Aut}_o(X) \to \operatorname {Aut}_o(M)$ is an isomorphism. After providing a number of examples showing that this may fail in general, we show that the map is indeed an isomorphism under the additional assumption that the subvariety of $M$ consisting of members passing through a general point $x \in X$ is irreducible and of dimension $\geq 2$.

[BS]BS Beltrametti, M. and Sommese, A.: The adjunction theory of complex projective varieties. Walter de Gruyter, 1995.
[DR]DR Desale, U.V. and Ramanan, S.: Classification of vector bundles of rank 2 on hyperelliptic curves. Invent. Math. 38 (1976) 161185.
[Ho]Ho Horikawa, E.: On deformations of holomorphic maps I. J. Math. Soc. Japan 25 (1973) 372396.
[HM1]HM1 Hwang, J.M. and Mok, N.: Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation. Invent. Math. 131 (1998) 393418.
[HM2]HM2 Hwang, J.M. and Mok, N.: CartanFubini type extension of holomorphic maps for Fano manifolds of Picard number 1. Journal Math. Pures Appl. 80 (2001) 563575.
[Is]Is Iskovskikh, V.A.: Anticanonical models of 3dimensional algebraic varieties. J. Soviet Math. 13 (1980) 745814.
[Ke]Ke Kebekus, S.: Families of singular rational curves. J. Alg. Geom. 11 (2002) 245256.
[Ko]Ko Kollár, J.: Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Band 32, SpringerVerlag, 1996.
[MU]MU Mukai, S. and Umemura, H.: Minimal rational threefolds. in Algebraic Geometry, Tokyo/Kyoto 1982. Lecture Notes in Math. 1016 (1983) 490518.
[NR]NR Narasimhan, M.S. and Ramanan, S.: Deformations of the moduli space of vector bundles over an algebraic curve. Ann. of Math. 101 (1975) 391417.
[SR]SR Semple, J. G. and Roth, L.: Introduction to algebraic geometry. Oxford University Press, 1985.
Additional Information
JunMuk Hwang
Affiliation:
Department of Mathematics, Korea Institute for Advanced Study, 20743 Cheongryangridong, Seoul 130012, Korea
MR Author ID:
362260
Email:
jmhwang@ns.kias.re.kr
Ngaiming Mok
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
MR Author ID:
191186
Email:
nmok@hkucc.hku.hk
Received by editor(s):
April 9, 2002
Published electronically:
February 18, 2004
Additional Notes:
The first author was supported by Grant No. 980701015L from the KOSEF. The second author was supported by a CERG of the Research Grants Council of Hong Kong