Let $M_{\mathrm{\Sigma }}$ be an $n$-dimensional Thom–Mather stratified space of depth $1$. We denote by $\beta M$ the singular locus and by $L$ the associated link. In this paper, we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge $\alpha$-class ${\alpha }_w(M_{\mathrm{\Sigma }})\in {\mathrm{\text{KO}}}_n$. In order to establish a sufficient condition, we need to assume additional structure: we assume that the link of $M_{\mathrm{\Sigma }}$ is a homogeneous space of positive scalar curvature, $L=G/K$, where the semisimple compact Lie group $G$ acts transitively on $L$ by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when $M_{\mathrm{\Sigma }}$ and $\beta M$ are spin, we reinterpret our obstruction in terms of two $\alpha$-classes associated to the resolution of $M_{\mathrm{\Sigma }}$, $M$, and to the singular locus $\beta M$. Finally, when $M_{\mathrm{\Sigma }}$, $\beta M$, $L$ and $G$ are simply connected and $dimM$ is big enough, and when some other conditions on $L$ (satisfied in a large number of cases) hold, we establish the main result of this paper, showing that the vanishing of these two $\alpha$-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.

让${米}_{\mathrm{\Sigma }}$豆$n$-dimensional Thom-Mather 深度分层空间$\mathrm{1个}$. 我们用$\beta 米$奇点和通过$\mathrm{大号}$关联的链接。在这篇论文中，我们研究了这样一个空间何时可以被赋予正标量曲率的楔度量的问题。我们将这个问题与最近关于分层空间指数理论的工作联系起来，首先根据楔形来阻碍这种度量的存在$\alpha$-班级${\alpha }_w({米}_{\mathrm{\Sigma }})\in {\mathrm{\text{高}}}_n$. 为了建立充分条件，我们需要假设额外的结构：我们假设${米}_{\mathrm{\Sigma }}$是正标量曲率的齐次空间，$\mathrm{大号}=G/钾$, 其中半单紧李群$G$传递作用于$\mathrm{大号}$通过等距。这种流形的例子包括紧致的半单李群和紧致类型的黎曼对称空间。在这些假设下，当${米}_{\mathrm{\Sigma }}$和$\beta 米$是自旋，我们从两个方面重新解释我们的阻碍$\alpha$-与分辨率相关的类${米}_{\mathrm{\Sigma }}$,$米$, 以及奇异轨迹$\beta 米$. 最后，当${米}_{\mathrm{\Sigma }}$,$\beta 米$,$\mathrm{大号}$和$G$简单地连接并且$暗淡米$足够大，并且当一些其他条件$\mathrm{大号}$（在大量情况下都满意）成立，我们建立了本文的主要结果，表明这两个消失$\alpha$-classes 也足以存在一个适应性良好的正标量曲率楔形度量。