Johannes Nordström

G2-manifolds and associative submanifolds via semi-Fano 3-folds

We construct many new topological types of compact G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau 3-folds built from semi-Fano 3-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 7-manifolds completely; we find that many 2-connected 7-manifolds can be realised as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G_2-metrics. Many of the G_2-manifolds we construct contain compact rigid associative 3-folds, which play an important role in the the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G_2-metrics. By varying the semi-Fanos used to build different G_2-metrics on the same 7-manifold we can change the number of rigid associative 3-folds we produce.

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

We prove the existence of asymptotically cylindrical (ACyl) Calabi–Yau 3–folds starting with (almost) any deformation family of smooth weak Fano 3–folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previously only a few hundred ACyl Calabi–Yau 3–folds were known. We pay particular attention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. Semi-Fano 3–folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3–folds, but are far more numerous than genuine Fano 3–folds. Also, unlike Fanos they often contain ℙ1s with normal bundle O(−1) ⊕O(−1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi–Yau 3–folds. We introduce some general methods to compute the basic topological invariants of ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples. All the features of the ACyl Calabi–Yau 3–folds studied here find application in [arXiv:1207.4470] where we construct many new compact G2–manifolds using Kovalev’s twisted connected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds are particularly well-adapted for this purpose.

Asymptotically cylindrical 7-manifolds of holonomy G2 with applications to compact irreducible G2-manifolds

We construct examples of exponentially asymptotically cylindrical (EAC) Riemannian 7-manifolds with holonomy group equal to G2. To our knowledge, these are the first such examples. We also obtain EAC coassociative calibrated submanifolds. Finally, we apply our results to show that one of the compact G2-manifolds constructed by Joyce by desingularisation of a flat orbifold T7/Γ can be deformed to give one of the compact G2-manifolds obtainable as a generalized connected sum of two EAC SU(3)-manifolds via the method of Kovalev (J Reine Angew Math 565:125–160, 2003).