2009/05/31 by Valentino Tosatti
Mathematics · #Adiabatic process #Algebraic Geometry and Number Theory #Base (topology) #Curvature #Fibration #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Hermitian manifold #Holomorphic function #Homotopy #Kähler manifold #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Metric (unit) #Physics #Pure mathematics #Ricci curvature #Ricci flow #Space (punctuation) #Volume form #Zero (linguistics) #math.DG #msc:14J32 #msc:32Q20 #msc:32Q25 #msc:53C25
paper · pdf · doi:10.4310/jdg/1274707320
published as J. Differential Geom. 84 (2010), no.2, 427-453 · 26 pages; final version to appear in J. Differential Geom
arxiv created 2009/10/23 · crossref issued 2010/02/01 · crossref published 2010/02/01 · crossref published-print 2010/02/01 · openalex publication_date 2010/02/01 · crossref created 2017/03/16 · arxiv updated 2018/04/19 · crossref deposited 2021/05/09 · openalex created_date 2025/10/10 · crossref indexed 2026/07/31 · openalex updated_date 2026/08/05
We study adiabatic limits of Ricci-flat Kähler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampère equation, we show that the Ricci-flat metrics collapse (away from the singular fibers) to a metric on the base of the fibration. This metric has Ricci curvature equal to a Weil- Petersson metric that measures the variation of complex structure of the Calabi-Yau fibers. This generalizes results of Gross-Wilson for K3 surfaces to higher dimensions.