The moduli space of torsion-free \(G_{2}\) structures
Let \(\mathfrak{M}\) be the moduli space of torsion free \(G_2\) structures on a compact oriented manifold \(M\). The natural cohomology map \(\pi^3:\mathfrak{M}\to H^3(M,{\mathbb R})\) is known to be a local diffeomorphism [\textit{D. D. Joyce}, Compact manifolds with special holonomy. Oxford Mathematical Monographs (2000; Zbl 1027.53052); \textit{R. L. Bryant}, Ann. Math. (2) 126, 525--576 (1987; Zbl 0637.53042)]. Let \(\mathfrak{M}_1\subset\mathfrak{M}\) be the subset of \(G_2\) structures with volume \((M)=1\). The author shows that every nonzero element of \(H^4(M,{\mathbb R})=H^3(M,{\mathbb R})^*\) is a Morse function on \(\mathfrak{M}_1\), when composed with \(\pi^3\), and computes its Hessian. The result implies a special case of Torelli's theorem: if \(H^1(M,{\mathbb R})=0\) and \(H^3(M,{\mathbb R})=2\), the cohomology map \(\pi^3:\mathfrak{M}\to H^3(M,{\mathbb R})\) is one to one on each connected component of \(\mathfrak{M}\). The author formulates a compactness conjecture on the set of \(G_2\) structures of volume\((M)=1\) with bounded \(L^2\) norm of curvature. If this conjecture were true, it would imply that every connected component of \(\mathfrak{M}\) is contractible, and that every compact \(G_2\) manifold supports a \(G_2\) structure whose fundamental \(4\)-form represents the negative of the (nonzero) first Pontryagin class of \({M}\). He also observes that when \(H^1(M,{\mathbb R})=0\), and the volume of the torus \(H^3(M,{\mathbb R})/H^3(M,{\mathbb Z})\) is constant along \(\mathfrak{M}_1\), the locus \(\pi^3(\mathfrak{M}_1)\subset H^3(M,{\mathbb R})\) is a hyperbolic affine sphere.
scientific article; zbMATH DE number 2175213
scientific article