TY - JOUR

T1 - Asymptotics of complete Kähler metrics of finite volume on quasiprojective manifolds

AU - Rochon, Frédéric

AU - Zhang, Zhou

PY - 2012

Y1 - 2012

N2 - Let X 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 X by a manifold with corners X~, we describe the full asymptotic behavior at infinity of certain complete Kähler metrics of finite volume on X. 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.

AB - Let X 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 X by a manifold with corners X~, we describe the full asymptotic behavior at infinity of certain complete Kähler metrics of finite volume on X. 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.

KW - Complex Monge-Ampere equation

KW - Kähler-Einstein metrics

KW - Kähler-Ricci flow

KW - Manifolds with corners

KW - Polyhomogeneity

KW - Quasiprojective manifolds

UR - http://www.scopus.com/inward/record.url?scp=84865997914&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2012.08.005

DO - 10.1016/j.aim.2012.08.005

M3 - Article

SN - 0001-8708

VL - 231

SP - 2892

EP - 2952

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 5

ER -