Complex structures on product of circle bundles over complex manifolds
Annales de l'Institut Fourier, Volume 63 (2013) no. 4, p. 1331-1366

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 $ℂ$.

Soient ${\overline{L}}_{i}\to {X}_{i}$ des fibrés en droites holomorphes sur des variétés complexes compactes, pour $i=1,2$. Soit ${S}_{i}$ le fibré en cercles associé par rapport à un produit scalaire hermitienne sur ${\overline{L}}_{i}$. On construit des structures complexes sur $S={S}_{1}×{S}_{2}$ dites de type scalaire, diagonal, ou linéaire. Bien que des structures de type scalaire existent toujours, on construit des structures plus générales de type diagonal mais non-scalaire dans le cas où les ${\overline{L}}_{i}$ sont des ${\left({ℂ}^{*}\right)}^{{n}_{i}}$ fibrés équivariants qui vérifient certaines hypothèses supplémentaires. Les structures complexes de type linéaire sont des variétés des drapeaux (généralisées) et les ${L}_{i}$ sont des fibrés en droites amples négatifs. Lorsque ${H}^{1}\left({X}_{1};ℝ\right)=0$ et ${c}_{1}\left({\overline{L}}_{1}\right)$ est non-nulle la variété compacte $S$ n’admet pas de structure symplectique et donc elle est non-Kählerienne par rapport à toute structure complexe.

On montre que ${H}^{q}\left(S;{𝒪}_{S}\right)$ s’annule quand les ${X}_{i}$ sont des variétés projectives, les ${\overline{L}}_{i}^{\vee }$ son très amples et le cône sur ${X}_{i}$ par rapport au plongement projectif défini par ${\overline{L}}_{i}^{\vee }$ sont Cohen-Macaulay. On applique ces résultats au groupe de Picard de $S$. Quand ${X}_{i}={G}_{i}/{P}_{i}$${P}_{i}$ sont les sousgroupes paraboliques maximaux et la variété $S$ est munie d’une structure complexe du type linéaire avec «  la partie unipotente nulle  » on montre que le corps des fonctions méromorphes sur $S$ est purement transcendental sur $ℂ$.

DOI : https://doi.org/10.5802/aif.2805
Classification:  32L05,  32J18,  32Q55
Keywords: circle bundles, complex manifolds, homogeneous spaces, Picard groups, meromorphic function fields
@article{AIF_2013__63_4_1331_0,
author = {Sankaran, Parameswaran and Thakur, Ajay Singh},
title = {Complex structures on product of circle bundles over complex manifolds},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {63},
number = {4},
year = {2013},
pages = {1331-1366},
doi = {10.5802/aif.2805},
mrnumber = {3137357},
zbl = {06359591},
language = {en},
url = {http://www.numdam.org/item/AIF_2013__63_4_1331_0}
}

Sankaran, Parameswaran; Thakur, Ajay Singh. Complex structures on product of circle bundles over complex manifolds. Annales de l'Institut Fourier, Volume 63 (2013) no. 4, pp. 1331-1366. doi : 10.5802/aif.2805. http://www.numdam.org/item/AIF_2013__63_4_1331_0/

[1] Arnold, V. I. Ordinary differential equations, Springer-Verlag, Berlin, Second printing of the 1992 edition. Universitext (2006) | MR 2242407 | Zbl 0432.34001

[2] Bǎnicǎ, C.; Stǎnǎşilǎ, O. Algebraic methods in the global theory of of complex spaces, John Wiley, London (1976) | MR 463470 | Zbl 0334.32001

[3] Borcea, C. Some remarks on deformations of Hopf manifolds, Rev. Roum. Math., Tome 26 (1981), pp. 1287-1294 | MR 646396 | Zbl 0543.32010

[4] Bosio, Frédéric Variétés complexes compactes: une généralisation de la construction de Meersseman et López de Medrano-Verjovsky, Ann. Inst. Fourier (Grenoble), Tome 51 (2001) no. 5, pp. 1259-1297 | Article | Numdam | MR 1860666 | Zbl 0994.32018

[5] Calabi, Eugenio; Eckmann, Beno A class of compact, complex manifolds which are not algebraic, Ann. of Math. (2), Tome 58 (1953), pp. 494-500 | Article | MR 57539 | Zbl 0051.40304

[6] Cassa, Antonio Formule di Künneth per la coomologia a valori in an fascio, Annali della Scuola Normale Superiore di Pisa, Tome 27 (1973), pp. 905-931 | Numdam | MR 374476 | Zbl 0335.55006

[7] Douady, A. (Séminaire H. Cartan, exp. 3 (1960/61))

[8] Haefliger, A. Deformations of transversely holomorphic flows on spheres and deformations of Hopf manifolds, Compositio Math., Tome 55 (1985), pp. 241-251 | Numdam | MR 795716 | Zbl 0582.32026

[9] Hopf, H. Zur Topologie der komplexen Mannigfaltigkeiten, New York, Courant Anniversary Volume (1948) | MR 23054 | Zbl 0033.02501

[10] Humphreys, J. E. Linear algebraic groups, Springer-Verlag, New York, Graduate Texts in Math (1975) | MR 396773 | Zbl 0471.20029

[11] Kodaira, K.; Spencer, D. C. On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math., Tome 71 (1960), pp. 43-76 | Article | MR 115189 | Zbl 0128.16902

[12] López De Medrano, Santiago; Verjovsky, Alberto A new family of complex, compact, non-symplectic manifolds, Bol. Soc. Brasil. Mat. (N.S.), Tome 28 (1997) no. 2, pp. 253-269 | Article | MR 1479504 | Zbl 0901.53021

[13] Lakshmibai, V; Seshadri, C. S. Singular locus of a Schubert variety, Bull. Amer. Math. Soc., Tome 11 (1984), pp. 363-366 | Article | MR 752799 | Zbl 0549.14016

[14] Loeb, Jean Jacques; Nicolau, Marcel Holomorphic flows and complex structures on products of odd-dimensional spheres, Math. Ann., Tome 306 (1996) no. 4, pp. 781-817 | Article | MR 1418353 | Zbl 0860.32001

[15] Meersseman, Laurent A new geometric construction of compact complex manifolds in any dimension, Math. Ann., Tome 317 (2000) no. 1, pp. 79-115 | Article | MR 1760670 | Zbl 0958.32013

[16] Meersseman, Laurent; Verjovsky, Alberto Holomorphic principal bundles over projective toric varieties, J. Reine Angew. Math., Tome 572 (2004), pp. 57-96 | MR 2076120 | Zbl 1070.14047

[17] Peternell, Th. Modifications, Several complex variables, VII, Springer, Berlin (Encyclopaedia Math. Sci.) Tome 74 (1994), pp. 285-317 | MR 1326617 | Zbl 0807.32028

[18] Pietsch, Albrecht Nuclear locally convex spaces, Springer-Verlag, New York – Heidelberg, Translated from the second German edition by William H. Ruckle. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, Tome 66 (1972) | MR 350360 | Zbl 0236.46001

[19] Ramanan, S.; Ramanathan, A. Projective normality of flag varieties and Schubert varieties, Invent. Math., Tome 79 (1985) no. 2, pp. 217-224 | Article | MR 778124 | Zbl 0553.14023

[20] Ramanathan, A. Schubert varieties are arithmetically Cohen-Macaulay, Invent. Math., Tome 80 (1985) no. 2, pp. 283-294 | Article | MR 788411 | Zbl 0541.14039

[21] Ramani, Vimala; Sankaran, Parameswaran Dolbeault cohomology of compact complex homogeneous manifolds, Proc. Indian Acad. Sci. Math. Sci., Tome 109 (1999) no. 1, pp. 11-21 | MR 1687024 | Zbl 0935.32017

[22] Sankaran, Parameswaran A coincidence theorem for holomorphic maps to $G/P$, Canad. Math. Bull., Tome 46 (2003) no. 2, pp. 291-298 | Article | MR 1981683 | Zbl 1038.55002

[23] Wang, Hsien-Chung Closed manifolds with homogeneous complex structure, Amer. J. Math., Tome 76 (1954), pp. 1-32 | Article | MR 66011 | Zbl 0055.16603