TY - JOUR
T1 - Collapsing of abelian fibered calabi-yau manifolds
AU - Gross, Mark
AU - Tosatti, Valentino
AU - Zhang, Yuguang
PY - 2013/2
Y1 - 2013/2
N2 - We study the collapsing behavior of Ricci-flat Kähler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkähler manifolds in the large complex structure limit. In this setting, we prove an analogue of a result of Gross and Wilson for K3 surfaces, which is motivated by the Strominger-Yau-Zaslow picture of mirror symmetry.
AB - We study the collapsing behavior of Ricci-flat Kähler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkähler manifolds in the large complex structure limit. In this setting, we prove an analogue of a result of Gross and Wilson for K3 surfaces, which is motivated by the Strominger-Yau-Zaslow picture of mirror symmetry.
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U2 - 10.1215/00127094-2019703
DO - 10.1215/00127094-2019703
M3 - Article
AN - SCOPUS:84877045038
SN - 0012-7094
VL - 162
SP - 517
EP - 551
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 3
ER -