Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1

Jun-Muk Hwang[1]

  • [1] Korea Institute for Advanced Study 207-43 Cheongryangri-dong Seoul, 130-722 (Korea)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 3, page 815-823
  • ISSN: 0373-0956

Abstract

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Let X be a Fano manifold with b 2 = 1 different from the projective space such that any two surfaces in X have proportional fundamental classes in H 4 ( X , C ) . Let f : Y → X be a surjective holomorphic map from a projective variety Y . We show that all deformations of f with Y and X fixed, come from automorphisms of X . The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of X .

How to cite

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Hwang, Jun-Muk. "Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1." Annales de l’institut Fourier 57.3 (2007): 815-823. <http://eudml.org/doc/10243>.

@article{Hwang2007,
abstract = {Let $X$ be a Fano manifold with $b_2=1$ different from the projective space such that any two surfaces in $X$ have proportional fundamental classes in $H_4(X, \mathbf\{C\})$. Let $f:Y\rightarrow X$ be a surjective holomorphic map from a projective variety $Y$. We show that all deformations of $f$ with $Y$ and $X$ fixed, come from automorphisms of $X$. The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of $X$.},
affiliation = {Korea Institute for Advanced Study 207-43 Cheongryangri-dong Seoul, 130-722 (Korea)},
author = {Hwang, Jun-Muk},
journal = {Annales de l’institut Fourier},
keywords = {minimal rational curves; Fano manifold; deformation of holomorphic maps; deformations of holomorphic maps},
language = {eng},
number = {3},
pages = {815-823},
publisher = {Association des Annales de l’institut Fourier},
title = {Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1},
url = {http://eudml.org/doc/10243},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Hwang, Jun-Muk
TI - Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 3
SP - 815
EP - 823
AB - Let $X$ be a Fano manifold with $b_2=1$ different from the projective space such that any two surfaces in $X$ have proportional fundamental classes in $H_4(X, \mathbf{C})$. Let $f:Y\rightarrow X$ be a surjective holomorphic map from a projective variety $Y$. We show that all deformations of $f$ with $Y$ and $X$ fixed, come from automorphisms of $X$. The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of $X$.
LA - eng
KW - minimal rational curves; Fano manifold; deformation of holomorphic maps; deformations of holomorphic maps
UR - http://eudml.org/doc/10243
ER -

References

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  2. C. Araujo, Rational curves of minimal degree and characterization of projective spaces, Math. Annalen 335 (2006), 937-951 Zbl1109.14032MR2232023
  3. J.-M. Hwang, Geometry of minimal rational curves on Fano manifolds, ICTP Lect. Notes 6 (2001), 335-393, Abdus Salam Int. Cent. Theoret. Phys.,, Trieste Zbl1086.14506MR1919462
  4. J.-M. Hwang, On the degrees of Fano four-folds of Picard number 1, J. reine angew. Math. 556 (2003), 225-235 Zbl1016.14022MR1971147
  5. J.-M. Hwang, S. Kebekus, T. Peternell, Holomorphic maps onto varieties of non-negative Kodaira dimension, J. Alg. Geom. 15 (2006), 551-561 Zbl1112.14014MR2219848
  6. J.-M. Hwang, N. Mok, Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles, J. Alg. Geom. 12 (2003), 627-651 Zbl1038.14018MR1993759
  7. J.-M. Hwang, N. Mok, Birationality of the tangent map for minimal rational curves, Asian J. Math. 8 (2004), 51-64 Zbl1072.14015MR2128297
  8. C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Birkhäuser, Boston (1980) Zbl0438.32016MR561910

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