GromovHausdorff collapse of CalabiYau manifolds
Kähler Geometry, Einstein Metrics, and Generalizations March 21, 2016  March 25, 2016
Location: SLMath: Eisenbud Auditorium
algebraic geometry and GAGA
complex differential geometry
mathematical physics
Kahler metric
mirror symmetry
GromovHausdorff metric
abelian varieties
Ricci flatness
51D25  Lattices of subspaces and geometric closure systems [See also 05B35]
51E14  Finite partial geometries (general), nets, partial spreads
14K10  Algebraic moduli of abelian varieties, classification [See also 11G15]
14K22  Complex multiplication and abelian varieties [See also 11G15]
14K20  Analytic theory of abelian varieties; abelian integrals and differentials
14J50  Automorphisms of surfaces and higherdimensional varieties
14L35  Classical groups (algebrogeometric aspects) [See also 20Gxx, 51N30]
14471
I will talk about joint work with Tosatti and Zhang on GromovHausdorff collapse of abelian fibred Calabiyau manifolds. One considers a CalabiYau manifold M with a holomorphic map M>N with general fibres being abelian varieties. Given a suitably chosen sequence of Ricciflat metrics so that the volume of the fibres goes to zero, we show that the Kahler form on M converges to the pullback of a form on N. Stronger results if the dimension of N is 1 imply that in fact M converges in the GromovHausdorff sense to N. This leads to an extension of an old collapsing result of myself and Wilson in the K3 case with no assumptions on the types of singular fibres
14471
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