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SUMMARY:Hodge theory of Calabi-Yau fibrations - Matt Kerr (Durham)
DTSTART:20081112T141500Z
DTEND:20081112T151500Z
UID:TALK13468@talks.cam.ac.uk
CONTACT:Burt Totaro
DESCRIPTION:Abstract:  This talk is about 1-parameter families of elliptic
  curves\, K3 surfaces\, and Calabi-Yau 3-folds -- objects which arise\, fo
 r example\, in the theory of modular forms and in mirror symmetry -- with 
 particular\nattention to the role played by singular fibers.  Instead of l
 ooking at the geometry of the family directly\, one often studies the asso
 ciated "variation of Hodge [i.e.\, complex analytic] structure"\, and the 
 degrees\nof related vector bundles on the parameter space are a tool for s
 tudying global behavior.\n\nIn his classic study of minimal elliptic fibra
 tions\, Kodaira described all possible singular fibers and their relation 
 to the A-D-E classification\n(from Lie/singularity theory).  We will first
  recall this and how one can relate fiber types to the Euler characteristi
 c of the total space and the\ndegree of the Hodge bundles.\n\nWhat is inte
 resting is how these relations generalize (or fail to generalize) to highe
 r dimensions (K3\, CY 3-fold)\, and the related nonexistence (or existence
 ) of non-isotrivial families with no singular\nfibers.  We will describe s
 ome results along these (global) lines\; if time permits\, we will sketch 
 how they fit with our earlier classification of\n(local) degenerations of 
 CY threefold VHS's related to mirror symmetry\, and lead to a Torelli theo
 rem for the mirror quintic family.
LOCATION:MR13\, CMS
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