Gromov-Hausdorff collapse of Calabi-Yau 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
Gromov-Hausdorff 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 higher-dimensional varieties
14L35 - Classical groups (algebro-geometric aspects) [See also 20Gxx, 51N30]
14471
I will talk about joint work with Tosatti and Zhang on Gromov-Hausdorff collapse of abelian fibred Calabi-yau manifolds. One considers a Calabi-Yau manifold M with a holomorphic map M->N with general fibres being abelian varieties. Given a suitably chosen sequence of Ricci-flat metrics so that the volume of the fibres goes to zero, we show that the Kahler form on M converges to the pull-back of a form on N. Stronger results if the dimension of N is 1 imply that in fact M converges in the Gromov-Hausdorff 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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