We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $G_2$-manifolds. In the AC case, we show that if the rate of convergence $\nu$ to the cone at infinity is generic in a precise sense and lies in the interval $(-4, 0)$, then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates $\nu < -4$ in the AC case, and for generic positive rates of convergence to the cones at the singular points in the CS case, the deformation theory is in general obstructed. We describe the obstruction spaces explicitly in terms of the spectrum of the Laplacian on the link of the cones on the ends, and compute the virtual dimension of the moduli space. We also present many applications of these results, including: the uniqueness of the Bryant--Salamon AC $G_2$-manifolds via local rigidity and the cohomogeneity one property of AC $G_2$-manifolds asymptotic to homogeneous cones; the smoothness of the CS moduli space if the singularities are modeled on particular $G_2$-cones; and the proof of existence of a "good gauge" needed for desingularization of CS $G_2$-manifolds. Finally, we discuss some open problems.

中文翻译：

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$\mathrm{G}_2$ conifolds 的变形理论

我们考虑渐近圆锥（AC）和圆锥奇异（CS）$G_2$-流形的变形理论。在 AC 的情况下，我们表明，如果收敛速度 $\nu$ 到无穷远的锥体在精确意义上是通用的并且位于 $(-4, 0)$ 区间内，那么模空间是平滑的，我们根据拓扑和分析数据计算其维度。对于 AC 情况下的通用速率 $\nu < -4$，以及 CS 情况下在奇异点处收敛到锥体的通用正速率，变形理论通常受到阻碍。我们根据两端锥体链接上的拉普拉斯算子的频谱明确描述障碍空间，并计算模空间的虚拟维数。我们还介绍了这些结果的许多应用，包括：Bryant--Salamon AC$G_2$-流形的唯一性通过局部刚性和同质性 AC$G_2$-流形渐近到齐次锥的性质；如果奇点在特定的 $G_2$-锥体上建模，则 CS 模空间的平滑度；以及对 CS $G_2$-流形进行去奇异化所需的“良好规范”的存在证明。最后，我们讨论一些未解决的问题。