论文标题
$ g_2 $ -manifolds从K3表面的构建，$ \ mathbb {z}^2_2 $ - action
A construction of $G_2$-manifolds from K3 surfaces with a $\mathbb{Z}^2_2$-action
论文作者
论文摘要
K3 Surface $ S $和Flat 3维圆环$ t^3 $的产物是具有hulonomy $ su（2）$的多种多样。由于$ su（2）$是$ g_2 $的子组，因此$ s \ times t^3 $带有无扭矩的$ g_2 $结构。我们假设$ s $允许使用某些属性的$ \ mathbb {z}^2_2 $的操作。将此操作扩展到$ s \ times t^3 $有几种可能性。 Joyce和Karigiannis的最新结果使我们能够解决$（s \ times t^3）/\ Mathbb {z}^2_2 $的奇异性，从而使我们获得平稳的$ G_2 $ -Manifolds。我们将标的$（S \ times t^3）/\ Mathbb {z}^2_2 $在某些限制下进行分类，并计算相应的$ G_2 $ -MANIFOLDS的BETTI数字。此外，我们研究了一个非亚洲群体的一类商。我们的几个示例具有$（b^2，b^3）$的新值。
A product of a K3 surface $S$ and a flat 3-dimensional torus $T^3$ is a manifold with holonomy $SU(2)$. Since $SU(2)$ is a subgroup of $G_2$, $S\times T^3$ carries a torsion-free $G_2$-structure. We assume that $S$ admits an action of $\mathbb{Z}^2_2$ with certain properties. There are several possibilities to extend this action to $S\times T^3$. A recent result of Joyce and Karigiannis allows us to resolve the singularities of $(S\times T^3)/\mathbb{Z}^2_2$ such that we obtain smooth $G_2$-manifolds. We classify the quotients $(S\times T^3)/\mathbb{Z}^2_2$ under certain restrictions and compute the Betti numbers of the corresponding $G_2$-manifolds. Moreover, we study a class of quotients by a non-abelian group. Several of our examples have new values of $(b^2,b^3)$.
