Non-Finitely Generated Canonical Rings: A Singular Variety and a Non- Algebraic Manifold

A fundamental result by Birkar, Cascini, Hacon, and McKernan (2006) states that the canonical ring of a smooth projective algebraic variety is finitely generated. This property, however, may fail if one weakens the hypothesis of the theorem. The talk will showcase two examples: the first example will explore a normal projective algebraic variety (which is necessarily singular) and in the second example a non-algebraic complex manifold will be presented.