Tokyo Journal of Mathematics

Infinitesimal Deformations and Brauer Group of Some Generalized Calabi--Eckmann Manifolds

Indranil BISWAS, Mahan MJ, and Ajay Singh THAKUR

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Abstract

Let $X$ be a compact connected Riemann surface. Let $\xi_1: E_1\longrightarrow X$ and $\xi_2: E_2\,\longrightarrow X$ be holomorphic vector bundles of rank at least two. Given these together with a $\lambda \in {\mathbb C}$ with positive imaginary part, we construct a holomorphic fiber bundle $S^{\xi_1,\xi_2}_{\lambda}$ over $X$ whose fibers are the Calabi--Eckmann manifolds. We compute the Picard group of the total space of $S^{\xi_1,\xi_2}_{\lambda}$. We also compute the infinitesimal deformations of the total space of $S^{\xi_1,\xi_2}_{\lambda}$. The cohomological Brauer group of $S^{\xi_1,\xi_2}_{\lambda}$ is shown to be zero. In particular, the Brauer group of $S^{\xi_1,\xi_2}_{\lambda}$ vanishes.

Article information

Source
Tokyo J. Math., Volume 37, Number 1 (2014), 61-72.

Dates
First available in Project Euclid: 28 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1406552431

Digital Object Identifier
doi:10.3836/tjm/1406552431

Mathematical Reviews number (MathSciNet)
MR3264514

Zentralblatt MATH identifier
1330.14026

Subjects
Primary: 14F22: Brauer groups of schemes [See also 12G05, 16K50]
Secondary: 32Q55: Topological aspects of complex manifolds 32G05: Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15]

Citation

BISWAS, Indranil; MJ, Mahan; THAKUR, Ajay Singh. Infinitesimal Deformations and Brauer Group of Some Generalized Calabi--Eckmann Manifolds. Tokyo J. Math. 37 (2014), no. 1, 61--72. doi:10.3836/tjm/1406552431. https://projecteuclid.org/euclid.tjm/1406552431


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References

  • K. Akao, On deformations of the Calabi–Eckmann manifolds, Proc. Japan Acad. 51 (1975), 365–368.
  • E. Calabi and B. Eckmann, A class of compact, complex manifolds which are not algebraic, Ann. of Math. 58 (1953), 494–500.
  • A. Hatcher, Algebraic topology, Cambridge University Press, 2002.
  • F. Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
  • D. Huybrechts, Complex geometry. an introduction, Universitext, Springer-Verlag, Berlin, 2005.
  • S. Schröer, Topological methods for complex-analytic Brauer groups, Topology 44 (2005), 875–894.