# Complex structures on product of circle bundles over complex manifolds

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

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## Abstract

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Let ${\overline{L}}_{i}\to {X}_{i}$ 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 ${\overline{L}}_{i}$. 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 ${\overline{L}}_{i}$ are equivariant ${\left({ℂ}^{*}\right)}^{{n}_{i}}$-bundles satisfying some additional conditions. The linear type complex structures are constructed assuming ${X}_{i}$ are (generalized) flag varieties and ${\overline{L}}_{i}$ negative ample line bundles over ${X}_{i}$. When ${H}^{1}\left({X}_{1};ℝ\right)=0$ and ${c}_{1}\left({\overline{L}}_{1}\right)\in {H}^{2}\left({X}_{1};ℂ\right)$ 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 ${H}^{q}\left(S;{𝒪}_{S}\right)$ when ${X}_{i}$ are projective manifolds, ${\overline{L}}_{i}^{\vee }$ are very ample and the cone over ${X}_{i}$ with respect to the projective imbedding defined by ${\overline{L}}_{i}^{\vee }$ 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 $ℂ$.

## 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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