In this paper, we introduce a new concept of L 2 -extension indices. This index is a function that gives the minimum constant with respect to the L 2 -estimate of an Ohsawa–Takegoshi-type extension at each point. By using this notion, we propose a new way to study the positivity of curvature. We prove that there is an equivalence between how sharp the L 2 -extension is and how positive the curvature is. New examples of sharper L 2 -extensions are also systematically given. As applications, we use the L 2 -extension index to study Prékopa-type theorems and to study the positivity of a certain direct image sheaf. We also provide new characterizations of pluriharmonicity and curvature flatness.