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Volume minimization and conformally Kähler, Einstein-Maxwell geometry. (English) Zbl 1410.53071
In this technical article, the authors construct so-called conformally Kähler Einstein-Maxwell metrics on compact manifolds by finding the critical points of a certain volume functional.
More precisely, let $$(M,g)$$ be a compact Kähler manifold with its induced almost complex tensor field $$J$$. Then, a $$J$$-Hermitian manifold $$(M,\tilde{g})$$ is called a conformally Kähler Einstein-Maxwell (cKEM) manifold if (i) there exists a positive smooth function $$f$$ on $$M$$ such that $$g=f^2\tilde{g}$$; (ii) the Hamiltonian vector field $$K:=J\mathrm{grad}_gf$$ is Killing for both $$g$$ and $$\tilde{g}$$; (iii) the scalar curvature of $$\tilde{g}$$ is constant. (Regarding terminology, one can demonstrate that in four dimensions, out of a cKEM manifold one can obtain a solution of the Riemannian Einstein-Maxwell equations.) Let $$G\subset \mathrm{Aut}_r(M,g)$$ be a compact subgroup of the group of reduced automorphisms of $$(M,g)$$, with Lie algebra $${\mathfrak g}$$. Then, to every pair $$(K,a)\in {\mathfrak g}\times{\mathbb R}$$ one can construct a certain metric $$\tilde{g}_{K,a}$$ on $$M$$ and an associated functional on $${\mathfrak g}\times{\mathbb R}$$ given by $$(K,a)\mapsto \mathrm{Vol}(M, \tilde{g}_{K,a})$$. The main result of the paper is that cKEM metrics are critical points of this functional and a point $$(K,a)$$ is a critical point if and only if its so-called cKEM-Futaki invariant vanishes (see Theorem 1.1 and Section 1 for further explanations).
Section 2 contains the proof of the main result while Sections 3 and 4 are devoted to explicit examples of non-Kähler cKEM metrics and some computations with them.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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