Differential Geometry and Topology

course: Calabi-Yau and G2 manifolds and their calibrated submanifolds

speaker: Alexei Kovalev (Cambridge University)

abstract: Week 2: 1. Asymptotically cylindrical Calabi-Yau metrics

Week 3: 2. From Fano 3-folds to compact G2-manifolds 3. Coassociative K3 fibrations of compact G2-manifolds 4. An example of special Lagrangian fibration

In the first talk, I shall describe a class of non-compact quasiprojective complex manifolds which topologically have a cylindrical end and admit a Ricci-flat Kähler metric with holonomy SU(n) asymptotic to a product metric on that cylindrical end. This is proved using a version of the Calabi conjecture for `asymptotically cylindrical' manifolds, building up on a previous work of Tian and Yau.

The Calabi-Yau manifolds with asymptotically cylindrical ends will be applied in the second talk where I shall explain a construction of Riemannian metrics with special holonomy G2 on compact 7-manifolds. This construction is based on a gluing theorem for appropriate elliptic partial differential equations on a carefully chosen generalized connected sum. Examples of G2-manifolds arising by this construction are obtained using the algebraic geometry of Fano threefolds and K3 surfaces and are topologically distinct from G2-manifolds previously constructed by Joyce.

The last two talks deal with coassociative submanifolds of G2-manifolds and special Lagrangian submanifolds of Calabi-Yau manifolds. These are examples of calibrated submanifolds (a distinguished class of minimal submanifolds introduced by Harvey and Lawson). I shall show that the `connected-sum' G2-manifolds are fibred by coassociative submanifolds with typical fibre diffeomorphic to a K3 surface.

The Schoen Calabi-Yau threefold can be (re)constructed analytically, by gluing a pair of asymptotically cylindrical Calabi-Yau threefolds. A smooth `approximately special Lagrangian' fibration map is then obtained on the Schoen Calabi-Yau and I shall explain how this leads to an example of special Lagrangian fibration required in the SYZ mirror symmetry conjecture.

timetable:
Fri 17 Sep, 9:45 - 11:15, Sala Conferenze Centro De Giorgi
Tue 21 Sep, 9:45 - 11:15, Sala Conferenze Centro De Giorgi
Wed 22 Sep, 11:15 - 12:15, Sala Conferenze Centro De Giorgi
Thu 23 Sep, 11:15 - 12:15, Sala Conferenze Centro De Giorgi
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