Locally conformally Kähler metrics on Hopf surfaces

Paul Gauduchon; Liviu Ornea

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 4, page 1107-1127
  • ISSN: 0373-0956

Abstract

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A primary Hopf surface is a compact complex surface with universal cover ℂ 2 - { ( 0 , 0 ) } and cyclic fundamental group generated by the transformation ( u , v ) ↦ ( α u + λ v m , β v ) , m ∈ ℤ , and α , β , λ ∈ ℂ such that ∣ α ∣ ≥ ∣ β ∣ > 1 and ( α - β m ) λ = 0 . Being diffeomorphic with S 3 × S 1 Hopf surfaces cannot admit any Kähler metric. However, it was known that for λ = 0 and ∣ α ∣ = ∣ β ∣ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for all primary Hopf surfaces of class 1 ( λ = 0 ). We also show that these metrics are obtained via a Riemannian suspension over S 1 , by deforming the canonical Sasakian structure of S 3 by a Hermitian quadratic form of ℂ 2 . We finally infer the existence of a locally conformally Kähler metric for all primary Hopf surfaces by a deformation argument due to C. LeBrun.

How to cite

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Gauduchon, Paul, and Ornea, Liviu. "Locally conformally Kähler metrics on Hopf surfaces." Annales de l'institut Fourier 48.4 (1998): 1107-1127. <http://eudml.org/doc/75311>.

@article{Gauduchon1998,
abstract = {A primary Hopf surface is a compact complex surface with universal cover $\{\Bbb C\}^2-\lbrace (0,0)\rbrace $ and cyclic fundamental group generated by the transformation $(u,v)\mapsto (\alpha u + \lambda v^m, \beta v)$, $m\in \{\Bbb Z\}$, and $\alpha ,~ \beta ,~ \lambda ~\in \{\Bbb C\}$ such that $\mid \alpha \mid \ge \mid \beta \mid &gt;1$ and $(\alpha -\beta ^m)\lambda =0$. Being diffeomorphic with $S^3\times S^1$ Hopf surfaces cannot admit any Kähler metric. However, it was known that for $\lambda =0$ and $\mid \alpha \mid =\mid \beta \mid $ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for all primary Hopf surfaces of class $1$ ($\lambda = 0$). We also show that these metrics are obtained via a Riemannian suspension over $S^1$, by deforming the canonical Sasakian structure of $S^3$ by a Hermitian quadratic form of $\{\Bbb C\}^2$. We finally infer the existence of a locally conformally Kähler metric for all primary Hopf surfaces by a deformation argument due to C. LeBrun.},
author = {Gauduchon, Paul, Ornea, Liviu},
journal = {Annales de l'institut Fourier},
keywords = {Hopf surface; locally conformal Kähler metric; Sasakian structure; contact form; Killing vector field; deformation},
language = {eng},
number = {4},
pages = {1107-1127},
publisher = {Association des Annales de l'Institut Fourier},
title = {Locally conformally Kähler metrics on Hopf surfaces},
url = {http://eudml.org/doc/75311},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Gauduchon, Paul
AU - Ornea, Liviu
TI - Locally conformally Kähler metrics on Hopf surfaces
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 4
SP - 1107
EP - 1127
AB - A primary Hopf surface is a compact complex surface with universal cover ${\Bbb C}^2-\lbrace (0,0)\rbrace $ and cyclic fundamental group generated by the transformation $(u,v)\mapsto (\alpha u + \lambda v^m, \beta v)$, $m\in {\Bbb Z}$, and $\alpha ,~ \beta ,~ \lambda ~\in {\Bbb C}$ such that $\mid \alpha \mid \ge \mid \beta \mid &gt;1$ and $(\alpha -\beta ^m)\lambda =0$. Being diffeomorphic with $S^3\times S^1$ Hopf surfaces cannot admit any Kähler metric. However, it was known that for $\lambda =0$ and $\mid \alpha \mid =\mid \beta \mid $ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for all primary Hopf surfaces of class $1$ ($\lambda = 0$). We also show that these metrics are obtained via a Riemannian suspension over $S^1$, by deforming the canonical Sasakian structure of $S^3$ by a Hermitian quadratic form of ${\Bbb C}^2$. We finally infer the existence of a locally conformally Kähler metric for all primary Hopf surfaces by a deformation argument due to C. LeBrun.
LA - eng
KW - Hopf surface; locally conformal Kähler metric; Sasakian structure; contact form; Killing vector field; deformation
UR - http://eudml.org/doc/75311
ER -

References

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