## 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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### References:

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