Asymptotics of complete Kähler metrics of finite volume on quasiprojective manifolds

Let XX be a quasiprojective manifold given by the complement of a divisor D¯ with normal crossings in a smooth projective manifold X¯. Using a natural compactification of XX by a manifold with corners X˜, we describe the full asymptotic behavior at infinity of certain complete Kähler metrics of finite volume on XX. When these metrics evolve according to the Ricci flow, we prove that such asymptotic behaviors persist at later times by showing that the associated potential function is smooth up to the boundary on the compactification X˜. However, when the divisor D¯ is smooth with KX¯+[D¯]>0 so that the Ricci flow converges to a Kähler–Einstein metric, we show that this Kähler–Einstein metric has a rather different asymptotic behavior at infinity, since its associated potential function is polyhomogeneous with, in general, some logarithmic terms occurring in its expansion at the boundary.