$L^2$-extension indices, sharper estimates and curvature positivity

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 Pr\'ekopa-type theorems and to study the positivity of a certain direct image sheaf. We also provide new characterizations of pluriharmonicity and curvature flatness.