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Stability and existence of critical Kaehler metrics on ruled manifolds. (English) Zbl 1180.53039

Recall the following fundamental result of Lichnerowicz: the Lie algebra \(h(M)\) of holomorphic vector fields on a compact Kähler manifold \((M,w)\) with constant scalar curvature decomposes as \(h(M)=h_0(M)\oplus c(M)\), where \(h_0(M)= \{z\in h(M):z\lrcorner w \text{ is }\overline\partial\text{-exact}\}\) and \(c(M)=\{z\in h(M):z \lrcorner w\in H^{0,1}(M,\mathbb{C})\}\). The author discusses

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J37 Perturbations of PDEs on manifolds; asymptotics
32W50 Other partial differential equations of complex analysis in several variables
53C55 Global differential geometry of Hermitian and Kählerian manifolds
19B14 Stability for linear groups
53D20 Momentum maps; symplectic reduction
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