Extremal Almost-Kähler Metrics

Abstract

Abstract. We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kähler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermitian scalar curvature in Darboux coordinates which allows us to obtain examples of non-integrable extremal almost-Kähler metrics saturating LeBrun's estimates.

Stability under deformations of extremal almost-Kähler metrics in dimension $4.$

“…We can then apply Theorem 1.1 to produce compatible extremal metrics Kim and Sung [18] showed that, in any dimension, if one starts with a Kähler metric of constant scalar curvature with no holomorphic vector fields, one can construct infinite dimensional families of almost-Kähler metrics of constant hermitian scalar curvature which concide with the initial metric away from an open set. Similar existence result was presented in [22] when the initial Kähler metric is locally toric.…”

“…The assumption that T ⊂ Ham(M, ω) is a maximal torus is used only in the second part of Proposition 3.3. Indeed, the arguments in [22] show that z …”

Stability under deformations of Hermite-Einstein almost Kähler metrics

“…By Tian result [27], there exists a Kähler-Einstein metric except in the first Hirzebruch surface and its blown up at one point (actually these two surfaces are toric). In the latter two cases, the Futaki invariant of the anti-canonical class being non-zero implies that there is no toric HEAK metric [16].…”

“…The Hermitian scalar curvature s is defined as the contraction of ρ ∇ by ω and coincides with the (usual) Riemannian scalar curvature when the metric is Kähler. An almost-Kähler metric is called Hermite-Einstein [16] (HEAK for short) if the Hermitian Ricci form ρ ∇ satisfies ρ ∇ = s ∇ 2n ω (in particular s ∇ is a constant). Note that the terminology 'Hermite-Einstein' here does not imply the integrability of the almost-complex structure.…”

“…Furthermore, there is a natural action of the Hamiltonian group Ham(M, ω) on AK ω and it turns out that this action is Hamiltonian [5,8] with moment map identified with the Hermitian scalar curvature. A metric induced by a critical point of the square norm of the moment map J → M (s ∇ ) 2 ω n is called an extremal almost-Kähler metric [2,16,17]. Moreover, an almost-Kähler metric induced by J is extremal if and only if the symplectic gradient of its Hermitian scalar curvature is an infinitesimal isometry of J. Extremal almost-Kähler metrics are a generalization of Calabi extremal Kähler metrics [3].…”