Infinitesimal deformations and Brauer group of some generalized Calabi-Eckmann manifolds (Q743668)

The first examples of simply connected compact complex manifolds that are not Kähler were introduced in [\textit{E. Calabi} and \textit{B. Eckmann}, Ann. Math. (2) 58, 494--500 (1953; Zbl 0051.40304)]. The underlying topological space of these Calabi--Eckmann manifolds \(M_{m,n}^\lambda\) are products of odd-dimensional spheres \(S^{2m-1}\times S^{2n-1}\) for integers \(m,n\geq 2\). The complex structure arises by viewing this as a quotient of \(\mathbb{C}^m\smallsetminus\left\{0\right\}\times\mathbb{C}^n\smallsetminus\left\{0\right\}\) by an action of the additive group \(\mathbb{C}\), which depends on a complex parameter \(\lambda\) from the upper half plane. The quotient comes with a natural projection \(M_{m,n}^\lambda\rightarrow\mathbb{P}^m\times \mathbb{P}^n\), whose fibers are elliptic curve with constant \(j\)-invariant. Here the authors study families of Calabi--Eckmann manifolds \(S\rightarrow X\), naturally attached to vector bundles \(E_1\), \(E_2\) of respective rank \(m\) and \(n\) over a smooth projective curve \(X\). Using spectral sequences, they compute the Picard group of \(S\). For simple vector bundle over curves of genus \(g\geq 2\), they furthermore compute the first cohomology of the tangent sheaf. Finally, they show that the Brauer group vanishes.