Cylinders in Fano varieties (Q1983976)

A \textit{cylinder} in a complex projective variety \(X\) is a Zariski open subset \(U\) of \(X\) isomorphic to a product \(Z\times\mathbb{C}\) for some affine variety \(Z\). The article contains a survey of the current knowledge concerning the existence and non-existence of cylinders in mildly singular del Pezzo surfaces, in higher dimensional smooth Fano varieties and in total spaces of certain other Mori fiber spaces over positive dimensional bases. A particular emphasis is put on the question of existence of cylinders with respect to a given fixed ample polarization \(H\) on \(X\), the so-called \textit{\(H\)-polar cylinders}, which are cylinders \(U\) in \(X\) whose complements are the supports of effective \(\mathbb{Q}\)-divisors \(\mathbb{Q}\)-linearly equivalent to \(H\). A very natural case of special importance in the Fano context is that of anti-canonically polar cylinders. The article also reviews several inspiring connections to other classical problems such as rationality and \(K\)-stability of Fano varieties, existence and properties of compactification of the affine space \(\mathbb{C}^{n}\) into smooth and mildly singular Fano varieties and existence and properties of additive group actions on affine cones over polarized projective varieties.