Complex structures on product of circle bundles over complex manifolds

[Structures complexes sur les produit de fibrés en cercles au-dessus des variétés complexes]

Sankaran, Parameswaran ; Thakur, Ajay Singh

Annales de l'Institut Fourier, Tome 63 (2013), p. 1331-1366 / Harvested from Numdam

Résumé

Soient $\bar{L}_i\rightarrow X_i$ 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 $\bar{L}_i$. On construit des structures complexes sur $S=S_1\times S_2$ 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 $\bar{L}_i$ sont des $(\mathbb{C}^*)^{n_i}$ 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 $H^1(X_1;\mathbb{R})=0$ et $c_1(\bar{L}_1)$ 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 $H^q(S;\mathcal{O}_S)$ s’annule quand les $X_i$ sont des variétés projectives, les $\bar{L}_i^\vee $ son très amples et le cône sur $X_i$ par rapport au plongement projectif défini par $\bar{L}_i^\vee $ sont Cohen-Macaulay. On applique ces résultats au groupe de Picard de $S$. Quand $X_i=G_i/P_i$ où $P_i$ 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 $\mathbb{C}$.

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 $\mathbb{C}$.

Détail

Publié le : 2013-01-01

MR 3137357 | Zbl 06359591

DOI : https://doi.org/10.5802/aif.2805
Classification:  32L05,  32J18,  32Q55
Mots clés: fibré en cercles, variétés complexes, espaces homogénes, groupes de Picard, corpses des fonctions meromorphes

Bibliographie

[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