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

TL;DR

This paper provides an overview of LVMB manifolds, a class of non-K"ahler compact complex manifolds, exploring their properties and connections to convex polytopes and moment-angle manifolds.

Contribution

It offers a comprehensive survey of LVMB manifolds, highlighting their geometric structures and relationships with convex polytopes and moment-angle manifolds.

Findings

LVMB manifolds are non-K"ahler compact complex manifolds.

They can be described as spaces of leaves of linear systems in projective space.

Connections between complex geometry and convex polytopes are elucidated.

Abstract

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