Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents

Jun-Muk Hwang

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 4, page 607-620
  • ISSN: 0012-9593

Abstract

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We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety Z , a family of minimal rational curves with Z -isotrivial varieties of minimal rational tangents is locally equivalent to the flat model. We show that this is the case when Z satisfies certain projective-geometric conditions, which hold for a non-singular hypersurface of degree ≥ 4 .

How to cite

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Hwang, Jun-Muk. "Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents." Annales scientifiques de l'École Normale Supérieure 43.4 (2010): 607-620. <http://eudml.org/doc/272168>.

@article{Hwang2010,
abstract = {We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$, a family of minimal rational curves with $Z$-isotrivial varieties of minimal rational tangents is locally equivalent to the flat model. We show that this is the case when $Z$ satisfies certain projective-geometric conditions, which hold for a non-singular hypersurface of degree $\ge 4$.},
author = {Hwang, Jun-Muk},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {equivalence problem; minimal rational curves},
language = {eng},
number = {4},
pages = {607-620},
publisher = {Société mathématique de France},
title = {Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents},
url = {http://eudml.org/doc/272168},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Hwang, Jun-Muk
TI - Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 4
SP - 607
EP - 620
AB - We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$, a family of minimal rational curves with $Z$-isotrivial varieties of minimal rational tangents is locally equivalent to the flat model. We show that this is the case when $Z$ satisfies certain projective-geometric conditions, which hold for a non-singular hypersurface of degree $\ge 4$.
LA - eng
KW - equivalence problem; minimal rational curves
UR - http://eudml.org/doc/272168
ER -

References

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  4. [4] J.-M. Hwang, Geometry of minimal rational curves on Fano manifolds, in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, 335–393. Zbl1086.14506MR1919462
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  7. [7] J.-M. Hwang & N. Mok, Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1, J. Math. Pures Appl.80 (2001), 563–575. Zbl1033.32013MR1842290
  8. [8] J.-M. Hwang & N. Mok, Birationality of the tangent map for minimal rational curves, Asian J. Math.8 (2004), 51–63. Zbl1072.14015MR2128297
  9. [9] J.-M. Hwang & N. Mok, Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation, Invent. Math.160 (2005), 591–645. Zbl1071.32022MR2178704
  10. [10] N. Mok, Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents, in Third International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 42, pt. 1 2, Amer. Math. Soc., 2008, 41–61. Zbl1182.14042MR2409622
  11. [11] S. Sternberg, Lectures on differential geometry, second éd., Chelsea Publishing Co., 1983. Zbl0518.53001MR891190

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