Generalized positive scalar curvature on \(\text{spin}^c\) manifolds (Q6642872)

The authors consider a compact \(\text{spin}^c\) manifold \((M,g)\) whose auxiliary line bundle has a connection \(A\). They introduce the notion of generalized scalar curvature which is expressed in terms of the scalar curvature of the manifold and the operator norm of Clifford multiplication \(c(\Omega)\), where \(\Omega\) is the curvature of the connection \(A\). They show that the positivity of the generalized scalar curvature is equivalent to the positivity of a twisted scalar curvature that they have introduced in a previous paper [\textit{B. Botvinnik} and \textit{J. Rosenberg}, J. Reine Angew. Math. 803, 103--136 (2023; Zbl 1530.53065)]. Then, they use the trichotomy theorem of Kazdan-Warner concerning the existence of metrics with positive scalar curvature to get a corresponding result concerning the generalized scalar curvature on \(\text{spin}^c\) manifolds.