# 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

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topSankaran, 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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