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These are notes for the CIME school on Complex non-K\"ahler geometry fromJuly 9th to July 13th of 2018 in Cetraro, Italy. It is an overview of differentproperties of a class of non-K\"ahler compact complex manifolds called LVMBmanifolds, obtained as the Hausdorff space of leaves of systems of commutingcomplex linear equations in an open set in complex projective space ${{mathbbP}_{\mathbb C}}^{n-1}$

Intersection of quadrics in ${\mathbb{C}}^n$, moment-angle manifolds, complex manifolds and convex polytopes