In this paper, on basis of three quadratic differential operators leaving the form degree of an arbitrary differential form unchanged, that is, the d'Alembertian operator and two combined ones from the Hodge coderivative and the exterior derivative, the usual Komar current for a Killing vector is formulated into another equivalent form. Then it is extended to more general currents in the absence of the linearity in the Killing vector field. Moreover, motivated by this equivalent of the usual Komar current, we put forward a conserved current corresponding to a generic vector with some constraint. Such a current can be generalized to the one with higher-order derivatives of the vector. The applications to some specific vector fields, such as the almost-Killing vectors, the conformal Killing vectors and the divergence-free vectors, are investigated. It is demonstrated that the above generalizations of the Killing vector can be uniformly described by a second-order derivative equation.

The purpose of this paper is to enhance the conventional Komar integral to asymptotically anti-de Sitter (AdS) black holes. In order to do so, we first obtain a potential that is the linear combination of the usual Komar potential with two third-order derivative terms generated by the action of the d'Alembertian operator and the exterior derivative upon a Killing vector. Then this higher-order corrected potential is extended to the Einstein gravity with a negative cosmological constant, yielding the potential that is the linear combination of the usual Komar one with it acted on by the d'Alembertian. It is demonstrated that the surface integral of the improved Komar potential can serve as a formula for conserved charges of asymptotically AdS spacetimes. Finally, to illustrate such a formula, we make use of it to compute the mass and the angular momentum of Schwarzschild-AdS black holes, regular AdS black holes, asymptotically AdS Kerr-Sen black holes, Kerr-NUT-AdS black holes, and Kerr-AdS black holes in arbitrary dimensions. The results are in agreement with the ones in the literature.