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 -