## Tokyo Journal of Mathematics

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

#### 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

https://projecteuclid.org/euclid.tjm/1406552431

Digital Object Identifier
doi:10.3836/tjm/1406552431

Mathematical Reviews number (MathSciNet)
MR3264514

Zentralblatt MATH identifier
1330.14026

#### 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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