15 July 2015 G 2 -manifolds and associative submanifolds via semi-Fano 3 -folds
Alessio Corti, Mark Haskins, Johannes Nordström, Tommaso Pacini
Duke Math. J. 164(10): 1971-2092 (15 July 2015). DOI: 10.1215/00127094-3120743


We construct many new topological types of compact G 2 -manifolds, that is, 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 realized 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 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.


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Alessio Corti. Mark Haskins. Johannes Nordström. Tommaso Pacini. " G 2 -manifolds and associative submanifolds via semi-Fano 3 -folds." Duke Math. J. 164 (10) 1971 - 2092, 15 July 2015. https://doi.org/10.1215/00127094-3120743


Received: 25 June 2013; Revised: 13 August 2014; Published: 15 July 2015
First available in Project Euclid: 14 July 2015

zbMATH: 1343.53044
MathSciNet: MR3369307
Digital Object Identifier: 10.1215/00127094-3120743

Primary: 53C29
Secondary: 14J28 , 14J32 , 14J45 , 53C25 , 53C38

Keywords: associative submanifolds , calibrated submanifolds , compact $\mathrm{G}_{2}$-manifolds , Differential geometry , differential topology , Einstein and Ricci-flat manifolds , Fano and weak Fano varieties , lattice polarized K3 surfaces , noncompact Calabi–Yau manifolds , special and exceptional holonomy

Rights: Copyright © 2015 Duke University Press


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Vol.164 • No. 10 • 15 July 2015
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