WorkshopsCounting associatives in connected-sum G_2-manifolds
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Associative 3-manifolds are a kind of calibrated submanifolds of manifolds with holonomy G_2. The deformations of an associative may be obstructed, but if they are not then the associative is rigid, i.e. the moduli space is discrete. Kovalev constructed examples of compact G_2-manifolds as "twisted connected sums" of asymptotically cylindrical Calabi-Yau 3-folds, in turn constructed from Fano 3-folds. I will describe work with Corti, Haskins and Pacini to construct compact G_2-manifolds by the same method but starting from weak Fano 3-folds, which are more plentiful. Moreover, rigid complex curves in the weak Fano give rise to associative submanifolds, and these are the first examples of associatives in compact G_2-manifolds that are known to be rigid. In at least some special cases, all associatives in a homology class arise this way, allowing them to be counted.
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Johannes Nordström
2011-07-04
15:15:00 - 16:15:00
R101 , Astronomy and Mathematics Building
Associative 3-manifolds are a kind of calibrated submanifolds of manifolds with holonomy G_2. The deformations of an associative may be obstructed, but if they are not then the associative is rigid, i.e. the moduli space is discrete. Kovalev constructed examples of compact G_2-manifolds as "twisted connected sums" of asymptotically cylindrical Calabi-Yau 3-folds, in turn constructed from Fano 3-folds. I will describe work with Corti, Haskins and Pacini to construct compact G_2-manifolds by the same method but starting from weak Fano 3-folds, which are more plentiful. Moreover, rigid complex curves in the weak Fano give rise to associative submanifolds, and these are the first examples of associatives in compact G_2-manifolds that are known to be rigid. In at least some special cases, all associatives in a homology class arise this way, allowing them to be counted.