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Mathematics > Differential Geometry

arXiv:0710.4579 (math)
[Submitted on 25 Oct 2007 (v1), last revised 13 Apr 2009 (this version, v5)]

Title:Limits of Calabi-Yau metrics when the Kahler class degenerates

Authors:Valentino Tosatti
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Abstract: We study the behaviour of families of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry.
Comments: 22 pages; a new L^infty estimate in section 4; to appear in this http URL. 2009
Subjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG)
MSC classes: 32Q25; 14J32; 14J28; 32Q20
Cite as: arXiv:0710.4579 [math.DG]
  (or arXiv:0710.4579v5 [math.DG] for this version)
  https://doi.org/10.48550/arXiv.0710.4579
Journal reference: J. Eur. Math. Soc. (JEMS) 11 (2009), no.4, 755-776.
Related DOI: https://doi.org/10.4171/JEMS/165

Submission history

From: Valentino Tosatti [view email]
[v1] Thu, 25 Oct 2007 15:11:40 UTC (21 KB)
[v2] Thu, 6 Dec 2007 16:26:21 UTC (21 KB)
[v3] Sun, 15 Jun 2008 12:16:01 UTC (22 KB)
[v4] Thu, 31 Jul 2008 03:19:56 UTC (22 KB)
[v5] Mon, 13 Apr 2009 03:48:10 UTC (23 KB)
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