Decomposition of compact complex varieties and the cancellation problem (Q793869)

Let \({\mathcal T}_ n\) consist of all compact complex varieties of the form \(T\times F/graph \chi\), where \(T={\mathbb{C}}^ n/\Gamma\) is an n-dimensional torus, F a compact complex variety, and \(\chi\) a homomorphism of some finite subgroup of T into Aut(F). The main results of this paper are the following: Cancellation theorem. Let X,Y,Z be compact complex varieties such that \(X\times Z\cong Y\times Z.\) If \(\{\) X,Y,\(Z\} \not\subset {\mathcal T}_ n\) for all \(n>0\), then \(X\cong Y\). - In particular, every \(Z\not\in \cup_{n>0}{\mathcal T}_ n\) cancels. Decomposition theorem. Let X be a compact complex variety which is not contained in any \({\mathcal T}_ n\), \(n>0\). Then X admits a unique decomposition \(X\cong X_ 1\times...\times X_ k\) into indecomposable positive-dimensional factors.