Complex structures on product of circle bundles over complex manifolds

Parameswaran Sankaran[1]; Ajay Singh Thakur[2]

  • [1] The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
  • [2] Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post Bangalore 560059, India

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 4, page 1331-1366
  • ISSN: 0373-0956

Abstract

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Let L ¯ i → X i be a holomorphic line bundle over a compact complex manifold for i = 1 , 2 . Let S i denote the associated principal circle-bundle with respect to some hermitian inner product on L ¯ i . We construct complex structures on S = S 1 × S 2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that L ¯ i are equivariant ( ℂ * ) n i -bundles satisfying some additional conditions. The linear type complex structures are constructed assuming X i are (generalized) flag varieties and L ¯ i negative ample line bundles over X i . When H 1 ( X 1 ; ℝ ) = 0 and c 1 ( L ¯ 1 ) ∈ H 2 ( X 1 ; ℂ ) is non-zero, the compact manifold S does not admit any symplectic structure and hence it is non-Kähler with respect to any complex structure.We obtain a vanishing theorem for H q ( S ; 𝒪 S ) when X i are projective manifolds, L ¯ i ∨ are very ample and the cone over X i with respect to the projective imbedding defined by L ¯ i ∨ are Cohen-Macaulay. We obtain applications to the Picard group of S . When X i = G i / P i where P i are maximal parabolic subgroups and S is endowed with linear type complex structure with “vanishing unipotent part” we show that the field of meromorphic functions on S is purely transcendental over ℂ .

How to cite

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Sankaran, Parameswaran, and Thakur, Ajay Singh. "Complex structures on product of circle bundles over complex manifolds." Annales de l’institut Fourier 63.4 (2013): 1331-1366. <http://eudml.org/doc/275612>.

@article{Sankaran2013,
abstract = {Let $\bar\{L\}_i\rightarrow X_i$ be a holomorphic line bundle over a compact complex manifold for $i=1,2$. Let $S_i$ denote the associated principal circle-bundle with respect to some hermitian inner product on $\bar\{L\}_i$. We construct complex structures on $S=S_1\times S_2$ which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that $\bar\{L\}_i$ are equivariant $(\mathbb\{C\}^*)^\{n_i\}$-bundles satisfying some additional conditions. The linear type complex structures are constructed assuming $X_i$ are (generalized) flag varieties and $\bar\{L\}_i$ negative ample line bundles over $X_i$. When $H^1(X_1;\mathbb\{R\})=0$ and $c_1(\bar\{L\}_1)\in H^2(X_1;\mathbb\{C\})$ is non-zero, the compact manifold $S$ does not admit any symplectic structure and hence it is non-Kähler with respect to any complex structure.We obtain a vanishing theorem for $H^q(S;\mathcal\{O\}_S)$ when $X_i$ are projective manifolds, $\bar\{L\}_i^\vee $ are very ample and the cone over $X_i$ with respect to the projective imbedding defined by $\bar\{L\}_i^\vee $ are Cohen-Macaulay. We obtain applications to the Picard group of $S$. When $X_i=G_i/P_i$ where $P_i$ are maximal parabolic subgroups and $S$ is endowed with linear type complex structure with “vanishing unipotent part” we show that the field of meromorphic functions on $S$ is purely transcendental over $\mathbb\{C\}$.},
affiliation = {The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India; Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post Bangalore 560059, India},
author = {Sankaran, Parameswaran, Thakur, Ajay Singh},
journal = {Annales de l’institut Fourier},
keywords = {circle bundles; complex manifolds; homogeneous spaces; Picard groups; meromorphic function fields; elliptic surfaces; Hopf manifolds; Calabi-Eckmann manifolds; compact manifolds},
language = {eng},
number = {4},
pages = {1331-1366},
publisher = {Association des Annales de l’institut Fourier},
title = {Complex structures on product of circle bundles over complex manifolds},
url = {http://eudml.org/doc/275612},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Sankaran, Parameswaran
AU - Thakur, Ajay Singh
TI - Complex structures on product of circle bundles over complex manifolds
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 4
SP - 1331
EP - 1366
AB - Let $\bar{L}_i\rightarrow X_i$ be a holomorphic line bundle over a compact complex manifold for $i=1,2$. Let $S_i$ denote the associated principal circle-bundle with respect to some hermitian inner product on $\bar{L}_i$. We construct complex structures on $S=S_1\times S_2$ which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that $\bar{L}_i$ are equivariant $(\mathbb{C}^*)^{n_i}$-bundles satisfying some additional conditions. The linear type complex structures are constructed assuming $X_i$ are (generalized) flag varieties and $\bar{L}_i$ negative ample line bundles over $X_i$. When $H^1(X_1;\mathbb{R})=0$ and $c_1(\bar{L}_1)\in H^2(X_1;\mathbb{C})$ is non-zero, the compact manifold $S$ does not admit any symplectic structure and hence it is non-Kähler with respect to any complex structure.We obtain a vanishing theorem for $H^q(S;\mathcal{O}_S)$ when $X_i$ are projective manifolds, $\bar{L}_i^\vee $ are very ample and the cone over $X_i$ with respect to the projective imbedding defined by $\bar{L}_i^\vee $ are Cohen-Macaulay. We obtain applications to the Picard group of $S$. When $X_i=G_i/P_i$ where $P_i$ are maximal parabolic subgroups and $S$ is endowed with linear type complex structure with “vanishing unipotent part” we show that the field of meromorphic functions on $S$ is purely transcendental over $\mathbb{C}$.
LA - eng
KW - circle bundles; complex manifolds; homogeneous spaces; Picard groups; meromorphic function fields; elliptic surfaces; Hopf manifolds; Calabi-Eckmann manifolds; compact manifolds
UR - http://eudml.org/doc/275612
ER -

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