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arXiv · 1808.07144

A compact $G_2$-calibrated manifold with first Betti number $b_1=1$

Marisa Fernández·Anna Fino·Alexei Kovalev·Vicente Muñoz
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Abstract

We construct a compact formal 7-manifold with a closed $G_2$-structure and with first Betti number $b_1=1$, which does not admit any torsion-free $G_2$-structure, that is, it does not admit any $G_2$-structure such that the holonomy group of the associated metric is a subgroup of $G_2$. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed $G_2$-structure and, for each of those 3-tori, we show a 3-dimensional family of non-trivial associative deformations. We also construct a fibration of our 7-manifold over $S^2\times S^1$ with generic fiber a (non-calibrated) coassociative 4-torus and some singular fibers.

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Marisa Fernández, Anna Fino, Alexei Kovalev, Vicente Muñoz. 2018-08-21. A compact $G_2$-calibrated manifold with first Betti number $b_1=1$. https://doi.org/10.1016/j.aim.2021.107623

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