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Large complex structure limits of K3 surfaces
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UNSPECIFIED. (2000) Large complex structure limits of K3 surfaces. JOURNAL OF DIFFERENTIAL GEOMETRY, 55 (3). pp. 475546. ISSN 0022040X
Full text not available from this repository.Abstract
Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we make a conjecture concerning the GromovHausdorff limits of CalabiYau nfolds (with Ricciflat Kahler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibrations associated with the large complex structure limit, namely an nsphere, and that the metric on this Sn is induced from a standard (singular) Riemannian metric on the base, the singularities of the metric corresponding to the limit discriminant locus of the fibrations. This conjecture is trivially true for elliptic curves; in this paper we prove it in the case of K3 surfaces. Using the standard description of mirror symmetry for K3 surfaces and the hyperkahler rotation trick, we reduce the problem to that of studying Kahler degenerations of elliptic K3 surfaces, with the Kahler class approaching the wall of the Kahler cone corresponding to the fibration and the volume normalized to be one. Here we are able to write down a remarkably accurate approximation to the Ricciflat metric: if the elliptic fibres are of area epsilon > 0, then the error is O(e(C/epsilon)) for some constant C > 0. This metric is obtained by gluing together a semiflat metric on the smooth part of the fibration with suitable OoguriVafa metrics near the singular fibres. For small epsilon, this is a sufficiently good approximation that the above conjecture is then an easy consequence.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics  
Journal or Publication Title:  JOURNAL OF DIFFERENTIAL GEOMETRY  
Publisher:  LEHIGH UNIV  
ISSN:  0022040X  
Official Date:  July 2000  
Dates: 


Volume:  55  
Number:  3  
Number of Pages:  72  
Page Range:  pp. 475546  
Publication Status:  Published  
URI:  http://wrap.warwick.ac.uk/id/eprint/11606 
Data sourced from Thomson Reuters' Web of Knowledge
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