Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians

Given a complex surface X with singularities of class T and no nontrivial holomorphic vector field, endowed with a Kähler class Ω0, we consider smoothings (Mt,Ωt), where Ωt is a Kähler class on Mt degenerating to Ω0. Under an hypothesis of nondegeneracy of the smoothing at each singular point, we pr...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2015-11, Vol.285, p.980-1024
Main Authors: Biquard, Olivier, Rollin, Yann
Format: Article
Language:English
Subjects:
Differential Geometry
Mathematics
Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians
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Summary:Given a complex surface X with singularities of class T and no nontrivial holomorphic vector field, endowed with a Kähler class Ω0, we consider smoothings (Mt,Ωt), where Ωt is a Kähler class on Mt degenerating to Ω0. Under an hypothesis of nondegeneracy of the smoothing at each singular point, we prove that if X admits a constant scalar curvature Kähler metric in Ω0, then Mt admits a constant scalar curvature Kähler metric in Ωt for small t. In addition, we construct small Lagrangian stationary spheres which represent Lagrangian vanishing cycles when t is small.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2015.08.013