Lahdili Abdellah

Aarhus University | AU · Department of Mathematics

We show that a compact weighted extremal Kahler manifold (as defined by the third named author) has coercive weighted Mabuchi energy with respect to a maximal complex torus in the reduced group of complex automorphisms. This provides a vast extension and a unification of a number of results concerning Kahler metrics satisfying special curvature con...

We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal K\"ahler metrics on a compact K\"ahler manifold introduced in our previous work. This extends a result by Berman--Berndtsson and Chen--Paun--Zeng in the extremal K\"ahler case. Furthermore, we show that a weighted extre...

We introduce a notion of a Kähler metric with constant weighted scalar curvature on a compact Kähler manifold X, depending on a fixed real torus T in the reduced group of automorphisms of X, and two smooth (weight) functions v > 0 and w, defined on the momentum image (with respect to a given Kähler class α on X)) of X in the dual Lie algebra of T....

We introduce a notion of a K\"ahler metric with constant weighted scalar curvature on a compact K\"ahler manifold $X$, depending on a fixed real torus $\mathbb{T}$ in the reduced group of automorphisms of $X$, and two smooth (weight) functions $\mathrm{v}>0$ and $\mathrm{w}$, defined on the momentum image (with respect to a given K\"ahler class $\a...

We prove that if a closed polarized complex manifold admits a conformally K\"ahler, Einstein--Maxwell metric, or more generally, a K\"ahler metric of constant $(\xi, a, p)$-scalar curvature in the sense of [ACGL, lahdili], then it minimizes the $(\xi,a,p)$-Mabuchi functional introduced in [lahdili]. Our method of proof extends the approach introduc...

We obtain a structure theorem for the group of holomorphic automorphisms of a conformally K\"ahler, Einstein--Maxwell metric, extending the classical results of Matsushima~\cite{M}, Licherowicz~\cite{L} and Calabi~\cite{calabi} in the K\"ahler--Einstein, cscK, and extremal K\"ahler cases. Combined with previous results of LeBrun~\cite{LeB1}, Aposto...